Parallel Algorithm Solves Coupled Differential Equations

Science.gov (United States)

Hayashi, A.

1987-01-01

Numerical methods adapted to concurrent processing. Algorithm solves set of coupled partial differential equations by numerical integration. Adapted to run on hypercube computer, algorithm separates problem into smaller problems solved concurrently. Increase in computing speed with concurrent processing over that achievable with conventional sequential processing appreciable, especially for large problems.

SIMULTANEOUS DIFFERENTIAL EQUATION COMPUTER

Science.gov (United States)

Collier, D.M.; Meeks, L.A.; Palmer, J.P.

1960-05-10

A description is given for an electronic simulator for a system of simultaneous differential equations, including nonlinear equations. As a specific example, a homogeneous nuclear reactor system including a reactor fluid, heat exchanger, and a steam boiler may be simulated, with the nonlinearity resulting from a consideration of temperature effects taken into account. The simulator includes three operational amplifiers, a multiplier, appropriate potential sources, and interconnecting R-C networks.

A Photon Free Method to Solve Radiation Transport Equations

International Nuclear Information System (INIS)

Chang, B

2006-01-01

The multi-group discrete-ordinate equations of radiation transfer is solved for the first time by Newton's method. It is a photon free method because the photon variables are eliminated from the radiation equations to yield a N group XN direction smaller but equivalent system of equations. The smaller set of equations can be solved more efficiently than the original set of equations. Newton's method is more stable than the Semi-implicit Linear method currently used by conventional radiation codes

Solving Absolute Value Equations Algebraically and Geometrically

Science.gov (United States)

Shiyuan, Wei

2005-01-01

The way in which students can improve their comprehension by understanding the geometrical meaning of algebraic equations or solving algebraic equation geometrically is described. Students can experiment with the conditions of the absolute value equation presented, for an interesting way to form an overall understanding of the concept.

Variational iteration method for solving coupled-KdV equations

International Nuclear Information System (INIS)

Assas, Laila M.B.

2008-01-01

In this paper, the He's variational iteration method is applied to solve the non-linear coupled-KdV equations. This method is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. This technique provides a sequence of functions which converge to the exact solution of the coupled-KdV equations. This procedure is a powerful tool for solving coupled-KdV equations

Exp-function method for solving Fisher's equation

Energy Technology Data Exchange (ETDEWEB)

Zhou, X-W [Department of Mathematics, Kunming Teacher' s College, Kunming, Yunnan 650031 (China)], E-mail: km_xwzhou@163.com

2008-02-15

There are many methods to solve Fisher's equation, but each method can only lead to a special solution. In this paper, a new method, namely the exp-function method, is employed to solve the Fisher's equation. The obtained result includes all solutions in open literature as special cases, and the generalized solution with some free parameters might imply some fascinating meanings hidden in the Fisher's equation.

Solving Variable Coefficient Fourth-Order Parabolic Equation by ...

African Journals Online (AJOL)

Solving Variable Coefficient Fourth-Order Parabolic Equation by Modified initial guess Variational ... variable coefficient fourth order parabolic partial differential equations. The new method shows rapid convergence to the exact solution.

Sinc-collocation method for solving the Blasius equation

International Nuclear Information System (INIS)

Parand, K.; Dehghan, Mehdi; Pirkhedri, A.

2009-01-01

Sinc-collocation method is applied for solving Blasius equation which comes from boundary layer equations. It is well known that sinc procedure converges to the solution at an exponential rate. Comparison with Howarth and Asaithambi's numerical solutions reveals that the proposed method is of high accuracy and reduces the solution of Blasius' equation to the solution of a system of algebraic equations.

Control functions in nonseparable simultaneous equations models

OpenAIRE

Blundell, R.; Matzkin, R. L.

2014-01-01

The control function approach (Heckman and Robb (1985)) in a system of linear simultaneous equations provides a convenient procedure to estimate one of the functions in the system using reduced form residuals from the other functions as additional regressors. The conditions on the structural system under which this procedure can be used in nonlinear and nonparametric simultaneous equations has thus far been unknown. In this paper, we define a new property of functions called control function ...

Spectrophotometric estimation of ethamsylate and mefenamic acid from a binary mixture by dual wavelength and simultaneous equation methods

OpenAIRE

Goyal Anju; Singhvi I

2008-01-01

Two simple, accurate, economical and reproducible spectrophotometric methods for simultaneous estimation of two-component drug mixture of ethamsylate and mefenamic acid in combined tablet dosage form have been developed. The first developed method involves formation and solving of simultaneous equation using 287.6 nm and 313.2 nm as two wavelengths. Second developed method is based on two wavelength calculation. Two wavelengths selected for estimation of ethamsylate were 274.4 nm and 301.2 nm...

Matrix-type multiple reciprocity boundary element method for solving three-dimensional two-group neutron diffusion equations

International Nuclear Information System (INIS)

Itagaki, Masafumi; Sahashi, Naoki.

1997-01-01

The multiple reciprocity boundary element method has been applied to three-dimensional two-group neutron diffusion problems. A matrix-type boundary integral equation has been derived to solve the first and the second group neutron diffusion equations simultaneously. The matrix-type fundamental solutions used here satisfy the equation which has a point source term and is adjoint to the neutron diffusion equations. A multiple reciprocity method has been employed to transform the matrix-type domain integral related to the fission source into an equivalent boundary one. The higher order fundamental solutions required for this formulation are composed of a series of two types of analytic functions. The eigenvalue itself is also calculated using only boundary integrals. Three-dimensional test calculations indicate that the present method provides stable and accurate solutions for criticality problems. (author)

Solving the Schroedinger equation using Smolyak interpolants

International Nuclear Information System (INIS)

Avila, Gustavo; Carrington, Tucker Jr.

2013-01-01

In this paper, we present a new collocation method for solving the Schroedinger equation. Collocation has the advantage that it obviates integrals. All previous collocation methods have, however, the crucial disadvantage that they require solving a generalized eigenvalue problem. By combining Lagrange-like functions with a Smolyak interpolant, we device a collocation method that does not require solving a generalized eigenvalue problem. We exploit the structure of the grid to develop an efficient algorithm for evaluating the matrix-vector products required to compute energy levels and wavefunctions. Energies systematically converge as the number of points and basis functions are increased

A New Fractional Projective Riccati Equation Method for Solving Fractional Partial Differential Equations

International Nuclear Information System (INIS)

Feng Qing-Hua

2014-01-01

In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann—Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method, we apply this method to solve the space-time fractional Whitham—Broer—Kaup (WBK) equations and the nonlinear fractional Sharma—Tasso—Olever (STO) equation, and as a result, some new exact solutions for them are obtained. (general)

Experimental quantum computing to solve systems of linear equations.

Science.gov (United States)

Cai, X-D; Weedbrook, C; Su, Z-E; Chen, M-C; Gu, Mile; Zhu, M-J; Li, Li; Liu, Nai-Le; Lu, Chao-Yang; Pan, Jian-Wei

2013-06-07

Solving linear systems of equations is ubiquitous in all areas of science and engineering. With rapidly growing data sets, such a task can be intractable for classical computers, as the best known classical algorithms require a time proportional to the number of variables N. A recently proposed quantum algorithm shows that quantum computers could solve linear systems in a time scale of order log(N), giving an exponential speedup over classical computers. Here we realize the simplest instance of this algorithm, solving 2×2 linear equations for various input vectors on a quantum computer. We use four quantum bits and four controlled logic gates to implement every subroutine required, demonstrating the working principle of this algorithm.

Method of mechanical quadratures for solving singular integral equations of various types

Science.gov (United States)

Sahakyan, A. V.; Amirjanyan, H. A.

2018-04-01

The method of mechanical quadratures is proposed as a common approach intended for solving the integral equations defined on finite intervals and containing Cauchy-type singular integrals. This method can be used to solve singular integral equations of the first and second kind, equations with generalized kernel, weakly singular equations, and integro-differential equations. The quadrature rules for several different integrals represented through the same coefficients are presented. This allows one to reduce the integral equations containing integrals of different types to a system of linear algebraic equations.

Solving the Sea-Level Equation in an Explicit Time Differencing Scheme

Science.gov (United States)

Klemann, V.; Hagedoorn, J. M.; Thomas, M.

2016-12-01

In preparation of coupling the solid-earth to an ice-sheet compartment in an earth-system model, the dependency of initial topography on the ice-sheet history and viscosity structure has to be analysed. In this study, we discuss this dependency and how it influences the reconstruction of former sea level during a glacial cycle. The modelling is based on the VILMA code in which the field equations are solved in the time domain applying an explicit time-differencing scheme. The sea-level equation is solved simultaneously in the same explicit scheme as the viscoleastic field equations (Hagedoorn et al., 2007). With the assumption of only small changes, we neglect the iterative solution at each time step as suggested by e.g. Kendall et al. (2005). Nevertheless, the prediction of the initial paleo topography in case of moving coastlines remains to be iterated by repeated integration of the whole load history. The sensitivity study sketched at the beginning is accordingly motivated by the question if the iteration of the paleo topography can be replaced by a predefined one. This study is part of the German paleoclimate modelling initiative PalMod. Lit:Hagedoorn JM, Wolf D, Martinec Z, 2007. An estimate of global mean sea-level rise inferred from tide-gauge measurements using glacial-isostatic models consistent with the relative sea-level record. Pure appl. Geophys. 164: 791-818, doi:10.1007/s00024-007-0186-7Kendall RA, Mitrovica JX, Milne GA, 2005. On post-glacial sea level - II. Numerical formulation and comparative reesults on spherically symmetric models. Geophys. J. Int., 161: 679-706, doi:10.1111/j.365-246.X.2005.02553.x

Nodal spectrum method for solving neutron diffusion equation

International Nuclear Information System (INIS)

Sanchez, D.; Garcia, C. R.; Barros, R. C. de; Milian, D.E.

1999-01-01

Presented here is a new numerical nodal method for solving static multidimensional neutron diffusion equation in rectangular geometry. Our method is based on a spectral analysis of the nodal diffusion equations. These equations are obtained by integrating the diffusion equation in X, Y directions and then considering flat approximations for the current. These flat approximations are the only approximations that are considered in this method, as a result the numerical solutions are completely free from truncation errors. We show numerical results to illustrate the methods accuracy for coarse mesh calculations

Solving the Schroedinger equation using the finite difference time domain method

International Nuclear Information System (INIS)

Sudiarta, I Wayan; Geldart, D J Wallace

2007-01-01

In this paper, we solve the Schroedinger equation using the finite difference time domain (FDTD) method to determine energies and eigenfunctions. In order to apply the FDTD method, the Schroedinger equation is first transformed into a diffusion equation by the imaginary time transformation. The resulting time-domain diffusion equation is then solved numerically by the FDTD method. The theory and an algorithm are provided for the procedure. Numerical results are given for illustrative examples in one, two and three dimensions. It is shown that the FDTD method accurately determines eigenfunctions and energies of these systems

Lagrange-Noether method for solving second-order differential equations

Institute of Scientific and Technical Information of China (English)

Wu Hui-Bin; Wu Run-Heng

2009-01-01

The purpose of this paper is to provide a new method called the Lagrange-Noether method for solving second-order differential equations. The method is,firstly,to write the second-order differential equations completely or partially in the form of Lagrange equations,and secondly,to obtain the integrals of the equations by using the Noether theory of the Lagrange system. An example is given to illustrate the application of the result.

Multiparameter extrapolation and deflation methods for solving equation systems

Directory of Open Access Journals (Sweden)

A. J. Hughes Hallett

1984-01-01

Full Text Available Most models in economics and the applied sciences are solved by first order iterative techniques, usually those based on the Gauss-Seidel algorithm. This paper examines the convergence of multiparameter extrapolations (accelerations of first order iterations, as an improved approximation to the Newton method for solving arbitrary nonlinear equation systems. It generalises my earlier results on single parameter extrapolations. Richardson's generalised method and the deflation method for detecting successive solutions in nonlinear equation systems are also presented as multiparameter extrapolations of first order iterations. New convergence results are obtained for those methods.

[Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (1)].

Science.gov (United States)

Murase, Kenya

2014-01-01

Utilization of differential equations and methods for solving them in medical physics are presented. First, the basic concept and the kinds of differential equations were overviewed. Second, separable differential equations and well-known first-order and second-order differential equations were introduced, and the methods for solving them were described together with several examples. In the next issue, the symbolic and series expansion methods for solving differential equations will be mainly introduced.

Application of Monte Carlo method to solving boundary value problem of differential equations

International Nuclear Information System (INIS)

Zuo Yinghong; Wang Jianguo

2012-01-01

This paper introduces the foundation of the Monte Carlo method and the way how to generate the random numbers. Based on the basic thought of the Monte Carlo method and finite differential method, the stochastic model for solving the boundary value problem of differential equations is built. To investigate the application of the Monte Carlo method to solving the boundary value problem of differential equations, the model is used to solve Laplace's equations with the first boundary condition and the unsteady heat transfer equation with initial values and boundary conditions. The results show that the boundary value problem of differential equations can be effectively solved with the Monte Carlo method, and the differential equations with initial condition can also be calculated by using a stochastic probability model which is based on the time-domain finite differential equations. Both the simulation results and theoretical analyses show that the errors of numerical results are lowered as the number of simulation particles is increased. (authors)

A novel algebraic procedure for solving non-linear evolution equations of higher order

International Nuclear Information System (INIS)

Huber, Alfred

2007-01-01

We report here a systematic approach that can easily be used for solving non-linear partial differential equations (nPDE), especially of higher order. We restrict the analysis to the so called evolution equations describing any wave propagation. The proposed new algebraic approach leads us to traveling wave solutions and moreover, new class of solution can be obtained. The crucial step of our method is the basic assumption that the solutions satisfy an ordinary differential equation (ODE) of first order that can be easily integrated. The validity and reliability of the method is tested by its application to some non-linear evolution equations. The important aspect of this paper however is the fact that we are able to calculate distinctive class of solutions which cannot be found in the current literature. In other words, using this new algebraic method the solution manifold is augmented to new class of solution functions. Simultaneously we would like to stress the necessity of such sophisticated methods since a general theory of nPDE does not exist. Otherwise, for practical use the algebraic construction of new class of solutions is of fundamental interest

Solving differential–algebraic equation systems by means of index reduction methodology

DEFF Research Database (Denmark)

Sørensen, Kim; Houbak, Niels; Condra, Thomas

2006-01-01

of a number of differential equations and algebraic equations — a so called DAE system. Two of the DAE systems are of index 1 and they can be solved by means of standard DAE-solvers. For the actual application, the equation systems are integrated by means of MATLAB’s solver: ode23t, that solves moderately...... stiff ODEs and index 1 DAEs by means of the trapezoidal rule. The last sub-model that models the boilers steam drum consist of two differential and three algebraic equations. The index of this model is greater than 1, which means that ode23t cannot integrate this equation system. In this paper......, it is shown how the equation system, by means of an index reduction methodology, can be reduced to a system of ordinary differential equations — ODEs....

Shooting method for third order simultaneous ordinary differential equations with application to magnetohydrodynamic boundary layer

International Nuclear Information System (INIS)

Srivastava, A.C.; Hazarika, G.C.

1990-01-01

An algorithm based on the shooting method has been developed for the solution of a two-point boundary value problem consisting of a system of third order simultaneous ordinary differential equations. The Falkner-Skan equations for electrically conducting viscous fluid with applied magnetic field has been solved by using this algorithm for various values of the wedge angle and magnetic parameters. The shooting method seems to be well convergent for a system as the results are in good agreement with those obtained by other methods. It is observed that both viscous boundary layer and magnetic boundary layer decrease while velocity as well as magnetic field increase with the increase of the wedge angle. (author). 6 tabs., 7 refs

Solving the Helmholtz equation in conformal mapped ARROWstructures using homotopy perturbation method

DEFF Research Database (Denmark)

Reck, Kasper; Thomsen, Erik Vilain; Hansen, Ole

2011-01-01

. The solution of the mapped Helmholtz equation is found by solving an infinite series of Poisson equations using two dimensional Fourier series. The solution is entirely based on analytical expressions and is not mesh dependent. The analytical results are compared to a numerical (finite element method) solution......The scalar wave equation, or Helmholtz equation, describes within a certain approximation the electromagnetic field distribution in a given system. In this paper we show how to solve the Helmholtz equation in complex geometries using conformal mapping and the homotopy perturbation method...

Modified Chebyshev Collocation Method for Solving Differential Equations

Directory of Open Access Journals (Sweden)

M Ziaul Arif

2015-05-01

Full Text Available This paper presents derivation of alternative numerical scheme for solving differential equations, which is modified Chebyshev (Vieta-Lucas Polynomial collocation differentiation matrices. The Scheme of modified Chebyshev (Vieta-Lucas Polynomial collocation method is applied to both Ordinary Differential Equations (ODEs and Partial Differential Equations (PDEs cases. Finally, the performance of the proposed method is compared with finite difference method and the exact solution of the example. It is shown that modified Chebyshev collocation method more effective and accurate than FDM for some example given.

Solving (2 + 1)-dimensional sine-Poisson equation by a modified variable separated ordinary differential equation method

International Nuclear Information System (INIS)

Ka-Lin, Su; Yuan-Xi, Xie

2010-01-01

By introducing a more general auxiliary ordinary differential equation (ODE), a modified variable separated ordinary differential equation method is presented for solving the (2 + 1)-dimensional sine-Poisson equation. As a result, many explicit and exact solutions of the (2 + 1)-dimensional sine-Poisson equation are derived in a simple manner by this technique. (general)

An exact method for solving logical loops in reliability analysis

International Nuclear Information System (INIS)

Matsuoka, Takeshi

2009-01-01

This paper presents an exact method for solving logical loops in reliability analysis. The systems that include logical loops are usually described by simultaneous Boolean equations. First, present a basic rule of solving simultaneous Boolean equations. Next, show the analysis procedures for three-component system with external supports. Third, more detailed discussions are given for the establishment of logical loop relation. Finally, take up two typical structures which include more than one logical loop. Their analysis results and corresponding GO-FLOW charts are given. The proposed analytical method is applicable to loop structures that can be described by simultaneous Boolean equations, and it is very useful in evaluating the reliability of complex engineering systems.

Fibonacci-regularization method for solving Cauchy integral equations of the first kind

Directory of Open Access Journals (Sweden)

Mohammad Ali Fariborzi Araghi

2017-09-01

Full Text Available In this paper, a novel scheme is proposed to solve the first kind Cauchy integral equation over a finite interval. For this purpose, the regularization method is considered. Then, the collocation method with Fibonacci base function is applied to solve the obtained second kind singular integral equation. Also, the error estimate of the proposed scheme is discussed. Finally, some sample Cauchy integral equations stem from the theory of airfoils in fluid mechanics are presented and solved to illustrate the importance and applicability of the given algorithm. The tables in the examples show the efficiency of the method.

Solving differential-algebraic equation systems by means of index reduction methodology

DEFF Research Database (Denmark)

Sørensen, Kim; Houbak, Niels; Condra, Thomas Joseph

2006-01-01

of a number of differential equations and algebraic equations - a so called DAE system. Two of the DAE systems are of index 1 and they can be solved by means of standard DAE-solvers. For the actual application, the equation systems are integrated by means of MATLAB’s solver: ode23t, that solves moderately...... stiff ODE’s and index 1 DAE’s by means of the trapezoidal rule. The last sub-model that models the boilers steam drum consist of two differential and three algebraic equations. The index of this model is greater than 1, which means that ode23t cannot integrate this equation system. In this paper......, it is shown how the equation system, by means of an index reduction methodology, can be reduced to a system of Ordinary- Differential-Equations - ODE’s....

Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening

Directory of Open Access Journals (Sweden)

Thomas Gomez

2018-04-01

Full Text Available Atomic structure of N-electron atoms is often determined by solving the Hartree-Fock equations, which are a set of integro-differential equations. The integral part of the Hartree-Fock equations treats electron exchange, but the Hartree-Fock equations are not often treated as an integro-differential equation. The exchange term is often approximated as an inhomogeneous or an effective potential so that the Hartree-Fock equations become a set of ordinary differential equations (which can be solved using the usual shooting methods. Because the Hartree-Fock equations are an iterative-refinement method, the inhomogeneous term relies on the previous guess of the wavefunction. In addition, there are numerical complications associated with solving inhomogeneous differential equations. This work uses matrix methods to solve the Hartree-Fock equations as an integro-differential equation. It is well known that a derivative operator can be expressed as a matrix made of finite-difference coefficients; energy eigenvalues and eigenvectors can be obtained by using linear-algebra packages. The integral (exchange part of the Hartree-Fock equation can be approximated as a sum and written as a matrix. The Hartree-Fock equations can be solved as a matrix that is the sum of the differential and integral matrices. We compare calculations using this method against experiment and standard atomic structure calculations. This matrix method can also be used to solve for free-electron wavefunctions, thus improving how the atoms and free electrons interact. This technique is important for spectral line broadening in two ways: it improves the atomic structure calculations, and it improves the motion of the plasma electrons that collide with the atom.

Solving the Richardson equations close to the critical points

Energy Technology Data Exchange (ETDEWEB)

DomInguez, F [Departamento de Matematicas, Universidad de Alcala, 28871 Alcala de Henares (Spain); Esebbag, C [Departamento de Matematicas, Universidad de Alcala, 28871 Alcala de Henares (Spain); Dukelsky, J [Instituto de Estructura de la Materia, CSIC, Serrano 123, 28006 Madrid (Spain)

2006-09-15

We study the Richardson equations close to the critical values of the pairing strength g{sub c}, where the occurrence of divergences precludes numerical solutions. We derive a set of equations for determining the critical g values and the non-collapsing pair energies. Studying the behaviour of the solutions close to the critical points, we develop a procedure to solve numerically the Richardson equations for arbitrary coupling strength.

Ten-Year-Old Students Solving Linear Equations

Science.gov (United States)

Brizuela, Barbara; Schliemann, Analucia

2004-01-01

In this article, the authors seek to re-conceptualize the perspective regarding students' difficulties with algebra. While acknowledging that students "do" have difficulties when learning algebra, they also argue that the generally espoused criteria for algebra as the ability to work with the syntactical rules for solving equations is…

Students' errors in solving linear equation word problems: Case ...

African Journals Online (AJOL)

The study examined errors students make in solving linear equation word problems with a view to expose the nature of these errors and to make suggestions for classroom teaching. A diagnostic test comprising 10 linear equation word problems, was administered to a sample (n=130) of senior high school first year Home ...

Adams Predictor-Corrector Systems for Solving Fuzzy Differential Equations

Directory of Open Access Journals (Sweden)

Dequan Shang

2013-01-01

Full Text Available A predictor-corrector algorithm and an improved predictor-corrector (IPC algorithm based on Adams method are proposed to solve first-order differential equations with fuzzy initial condition. These algorithms are generated by updating the Adams predictor-corrector method and their convergence is also analyzed. Finally, the proposed methods are illustrated by solving an example.

The Use of Transformations in Solving Equations

Science.gov (United States)

Libeskind, Shlomo

2010-01-01

Many workshops and meetings with the US high school mathematics teachers revealed a lack of familiarity with the use of transformations in solving equations and problems related to the roots of polynomials. This note describes two transformational approaches to the derivation of the quadratic formula as well as transformational approaches to…

Final report [on solving the multigroup diffusion equations

International Nuclear Information System (INIS)

Birkhoff, G.

1975-01-01

Progress achieved in the development of variational methods for solving the multigroup neutron diffusion equations is described. An appraisal is made of the extent to which improved variational methods could advantageously replace difference methods currently used

On method of solving third-order ordinary differential equations directly using Bernstein polynomials

Science.gov (United States)

Khataybeh, S. N.; Hashim, I.

2018-04-01

In this paper, we propose for the first time a method based on Bernstein polynomials for solving directly a class of third-order ordinary differential equations (ODEs). This method gives a numerical solution by converting the equation into a system of algebraic equations which is solved directly. Some numerical examples are given to show the applicability of the method.

Semiconductor device simulation by a new method of solving poisson, Laplace and Schrodinger equations

International Nuclear Information System (INIS)

Sharifi, M. J.; Adibi, A.

2000-01-01

In this paper, we have extended and completed our previous work, that was introducing a new method for finite differentiation. We show the applicability of the method for solving a wide variety of equations such as poisson, Laplace and Schrodinger. These equations are fundamental to the most semiconductor device simulators. In a section, we solve the Shordinger equation by this method in several cases including the problem of finding electron concentration profile in the channel of a HEMT. In another section, we solve the Poisson equation by this method, choosing the problem of SBD as an example. Finally we solve the Laplace equation in two dimensions and as an example, we focus on the VED. In this paper, we have shown that, the method can get stable and precise results in solving all of these problems. Also the programs which have been written based on this method become considerably faster, more clear, and more abstract

A method for solving neutron transport equation

International Nuclear Information System (INIS)

Dimitrijevic, Z.

1993-01-01

The procedure for solving the transport equation by directly integrating for case one-dimensional uniform multigroup medium is shown. The solution is expressed in terms of linear combination of function H n (x,μ), and the coefficient is determined from given conditions. The solution is applied for homogeneous slab of critical thickness. (author)

Integral transform method for solving time fractional systems and fractional heat equation

Directory of Open Access Journals (Sweden)

Arman Aghili

2014-01-01

Full Text Available In the present paper, time fractional partial differential equation is considered, where the fractional derivative is defined in the Caputo sense. Laplace transform method has been applied to obtain an exact solution. The authors solved certain homogeneous and nonhomogeneous time fractional heat equations using integral transform. Transform method is a powerful tool for solving fractional singular Integro - differential equations and PDEs. The result reveals that the transform method is very convenient and effective.

Solving the Coupled System Improves Computational Efficiency of the Bidomain Equations

KAUST Repository

Southern, J.A.

2009-10-01

The bidomain equations are frequently used to model the propagation of cardiac action potentials across cardiac tissue. At the whole organ level, the size of the computational mesh required makes their solution a significant computational challenge. As the accuracy of the numerical solution cannot be compromised, efficiency of the solution technique is important to ensure that the results of the simulation can be obtained in a reasonable time while still encapsulating the complexities of the system. In an attempt to increase efficiency of the solver, the bidomain equations are often decoupled into one parabolic equation that is computationally very cheap to solve and an elliptic equation that is much more expensive to solve. In this study, the performance of this uncoupled solution method is compared with an alternative strategy in which the bidomain equations are solved as a coupled system. This seems counterintuitive as the alternative method requires the solution of a much larger linear system at each time step. However, in tests on two 3-D rabbit ventricle benchmarks, it is shown that the coupled method is up to 80% faster than the conventional uncoupled method-and that parallel performance is better for the larger coupled problem.

Solving the Coupled System Improves Computational Efficiency of the Bidomain Equations

KAUST Repository

Southern, J.A.; Plank, G.; Vigmond, E.J.; Whiteley, J.P.

2009-01-01

The bidomain equations are frequently used to model the propagation of cardiac action potentials across cardiac tissue. At the whole organ level, the size of the computational mesh required makes their solution a significant computational challenge. As the accuracy of the numerical solution cannot be compromised, efficiency of the solution technique is important to ensure that the results of the simulation can be obtained in a reasonable time while still encapsulating the complexities of the system. In an attempt to increase efficiency of the solver, the bidomain equations are often decoupled into one parabolic equation that is computationally very cheap to solve and an elliptic equation that is much more expensive to solve. In this study, the performance of this uncoupled solution method is compared with an alternative strategy in which the bidomain equations are solved as a coupled system. This seems counterintuitive as the alternative method requires the solution of a much larger linear system at each time step. However, in tests on two 3-D rabbit ventricle benchmarks, it is shown that the coupled method is up to 80% faster than the conventional uncoupled method-and that parallel performance is better for the larger coupled problem.

Approximate analytical methods for solving ordinary differential equations

CERN Document Server

Radhika, TSL; Rani, T Raja

2015-01-01

Approximate Analytical Methods for Solving Ordinary Differential Equations (ODEs) is the first book to present all of the available approximate methods for solving ODEs, eliminating the need to wade through multiple books and articles. It covers both well-established techniques and recently developed procedures, including the classical series solution method, diverse perturbation methods, pioneering asymptotic methods, and the latest homotopy methods.The book is suitable not only for mathematicians and engineers but also for biologists, physicists, and economists. It gives a complete descripti

Nonlinearly Activated Neural Network for Solving Time-Varying Complex Sylvester Equation.

Science.gov (United States)

Li, Shuai; Li, Yangming

2013-10-28

The Sylvester equation is often encountered in mathematics and control theory. For the general time-invariant Sylvester equation problem, which is defined in the domain of complex numbers, the Bartels-Stewart algorithm and its extensions are effective and widely used with an O(n³) time complexity. When applied to solving the time-varying Sylvester equation, the computation burden increases intensively with the decrease of sampling period and cannot satisfy continuous realtime calculation requirements. For the special case of the general Sylvester equation problem defined in the domain of real numbers, gradient-based recurrent neural networks are able to solve the time-varying Sylvester equation in real time, but there always exists an estimation error while a recently proposed recurrent neural network by Zhang et al [this type of neural network is called Zhang neural network (ZNN)] converges to the solution ideally. The advancements in complex-valued neural networks cast light to extend the existing real-valued ZNN for solving the time-varying real-valued Sylvester equation to its counterpart in the domain of complex numbers. In this paper, a complex-valued ZNN for solving the complex-valued Sylvester equation problem is investigated and the global convergence of the neural network is proven with the proposed nonlinear complex-valued activation functions. Moreover, a special type of activation function with a core function, called sign-bi-power function, is proven to enable the ZNN to converge in finite time, which further enhances its advantage in online processing. In this case, the upper bound of the convergence time is also derived analytically. Simulations are performed to evaluate and compare the performance of the neural network with different parameters and activation functions. Both theoretical analysis and numerical simulations validate the effectiveness of the proposed method.

Normal scheme for solving the transport equation independently of spatial discretization

International Nuclear Information System (INIS)

Zamonsky, O.M.

1993-01-01

To solve the discrete ordinates neutron transport equation, a general order nodal scheme is used, where nodes are allowed to have different orders of approximation and the whole system reaches a final order distribution. Independence in the election of system discretization and order of approximation is obtained without loss of accuracy. The final equations and the iterative method to reach a converged order solution were implemented in a two-dimensional computer code to solve monoenergetic, isotropic scattering, external source problems. Two benchmark problems were solved using different automatic selection order methods. Results show accurate solutions without spatial discretization, regardless of the initial selection of distribution order. (author)

Sufficient Descent Conjugate Gradient Methods for Solving Convex Constrained Nonlinear Monotone Equations

Directory of Open Access Journals (Sweden)

San-Yang Liu

2014-01-01

Full Text Available Two unified frameworks of some sufficient descent conjugate gradient methods are considered. Combined with the hyperplane projection method of Solodov and Svaiter, they are extended to solve convex constrained nonlinear monotone equations. Their global convergence is proven under some mild conditions. Numerical results illustrate that these methods are efficient and can be applied to solve large-scale nonsmooth equations.

Solving the interval type-2 fuzzy polynomial equation using the ranking method

Science.gov (United States)

Rahman, Nurhakimah Ab.; Abdullah, Lazim

2014-07-01

Polynomial equations with trapezoidal and triangular fuzzy numbers have attracted some interest among researchers in mathematics, engineering and social sciences. There are some methods that have been developed in order to solve these equations. In this study we are interested in introducing the interval type-2 fuzzy polynomial equation and solving it using the ranking method of fuzzy numbers. The ranking method concept was firstly proposed to find real roots of fuzzy polynomial equation. Therefore, the ranking method is applied to find real roots of the interval type-2 fuzzy polynomial equation. We transform the interval type-2 fuzzy polynomial equation to a system of crisp interval type-2 fuzzy polynomial equation. This transformation is performed using the ranking method of fuzzy numbers based on three parameters, namely value, ambiguity and fuzziness. Finally, we illustrate our approach by numerical example.

A predictor-corrector scheme for solving the Volterra integral equation

KAUST Repository

Al Jarro, Ahmed

2011-08-01

The occurrence of late time instabilities is a common problem of almost all time marching methods developed for solving time domain integral equations. Implicit marching algorithms are now considered stable with various efforts that have been developed for removing low and high frequency instabilities. On the other hand, literature on stabilizing explicit schemes, which might be considered more efficient since they do not require a matrix inversion at each time step, is practically non-existent. In this work, a stable but still explicit predictor-corrector scheme is proposed for solving the Volterra integral equation and its efficacy is verified numerically. © 2011 IEEE.

Solving Fuzzy Fractional Differential Equations Using Zadeh's Extension Principle

Science.gov (United States)

Ahmad, M. Z.; Hasan, M. K.; Abbasbandy, S.

2013-01-01

We study a fuzzy fractional differential equation (FFDE) and present its solution using Zadeh's extension principle. The proposed study extends the case of fuzzy differential equations of integer order. We also propose a numerical method to approximate the solution of FFDEs. To solve nonlinear problems, the proposed numerical method is then incorporated into an unconstrained optimisation technique. Several numerical examples are provided. PMID:24082853

A method of solving simple harmonic oscillator Schroedinger equation

Science.gov (United States)

Maury, Juan Carlos F.

1995-01-01

A usual step in solving totally Schrodinger equation is to try first the case when dimensionless position independent variable w is large. In this case the Harmonic Oscillator equation takes the form (d(exp 2)/dw(exp 2) - w(exp 2))F = 0, and following W.K.B. method, it gives the intermediate corresponding solution F = exp(-w(exp 2)/2), which actually satisfies exactly another equation, (d(exp 2)/dw(exp 2) + 1 - w(exp 2))F = 0. We apply a different method, useful in anharmonic oscillator equations, similar to that of Rampal and Datta, and although it is slightly more complicated however it is also more general and systematic.

Using Computer Symbolic Algebra to Solve Differential Equations.

Science.gov (United States)

Mathews, John H.

1989-01-01

This article illustrates that mathematical theory can be incorporated into the process to solve differential equations by a computer algebra system, muMATH. After an introduction to functions of muMATH, several short programs for enhancing the capabilities of the system are discussed. Listed are six references. (YP)

Representation of Students in Solving Simultaneous Linear Equation Problems Based on Multiple Intelligence

Science.gov (United States)

Yanti, Y. R.; Amin, S. M.; Sulaiman, R.

2018-01-01

This study described representation of students who have musical, logical-mathematic and naturalist intelligence in solving a problem. Subjects were selected on the basis of multiple intelligence tests (TPM) consists of 108 statements, with 102 statements adopted from Chislet and Chapman and 6 statements equal to eksistensial intelligences. Data were analyzed based on problem-solving tests (TPM) and interviewing. See the validity of the data then problem-solving tests (TPM) and interviewing is given twice with an analyzed using the representation indikator and the problem solving step. The results showed that: the stage of presenting information known, stage of devising a plan, and stage of carrying out the plan those three subjects were using same form of representation. While he stage of presenting information asked and stage of looking back, subject of logical-mathematic was using different forms of representation with subjects of musical and naturalist intelligence. From this research is expected to provide input to the teacher in determining the learning strategy that will be used by considering the representation of students with the basis of multiple intelligences.

Solving Eigenvalue response matrix equations with Jacobian-Free Newton-Krylov methods

International Nuclear Information System (INIS)

Roberts, Jeremy A.; Forget, Benoit

2011-01-01

The response matrix method for reactor eigenvalue problems is motivated as a technique for solving coarse mesh transport equations, and the classical approach of power iteration (PI) for solution is described. The method is then reformulated as a nonlinear system of equations, and the associated Jacobian is derived. A Jacobian-Free Newton-Krylov (JFNK) method is employed to solve the system, using an approximate Jacobian coupled with incomplete factorization as a preconditioner. The unpreconditioned JFNK slightly outperforms PI, and preconditioned JFNK outperforms both PI and Steffensen-accelerated PI significantly. (author)

Taylor's series method for solving the nonlinear point kinetics equations

International Nuclear Information System (INIS)

Nahla, Abdallah A.

2011-01-01

Highlights: → Taylor's series method for nonlinear point kinetics equations is applied. → The general order of derivatives are derived for this system. → Stability of Taylor's series method is studied. → Taylor's series method is A-stable for negative reactivity. → Taylor's series method is an accurate computational technique. - Abstract: Taylor's series method for solving the point reactor kinetics equations with multi-group of delayed neutrons in the presence of Newtonian temperature feedback reactivity is applied and programmed by FORTRAN. This system is the couples of the stiff nonlinear ordinary differential equations. This numerical method is based on the different order derivatives of the neutron density, the precursor concentrations of i-group of delayed neutrons and the reactivity. The r th order of derivatives are derived. The stability of Taylor's series method is discussed. Three sets of applications: step, ramp and temperature feedback reactivities are computed. Taylor's series method is an accurate computational technique and stable for negative step, negative ramp and temperature feedback reactivities. This method is useful than the traditional methods for solving the nonlinear point kinetics equations.

Multivariate Padé Approximation for Solving Nonlinear Partial Differential Equations of Fractional Order

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Veyis Turut

2013-01-01

Full Text Available Two tecHniques were implemented, the Adomian decomposition method (ADM and multivariate Padé approximation (MPA, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo sense. First, the fractional differential equation has been solved and converted to power series by Adomian decomposition method (ADM, then power series solution of fractional differential equation was put into multivariate Padé series. Finally, numerical results were compared and presented in tables and figures.

Solving the discrete KdV equation with homotopy analysis method

International Nuclear Information System (INIS)

Zou, L.; Zong, Z.; Wang, Z.; He, L.

2007-01-01

In this Letter, we apply the homotopy analysis method to differential-difference equations. We take the discrete KdV equation as an example, and successfully obtain double periodic wave solutions and solitary wave solutions. It illustrates the validity and the great potential of the homotopy analysis method in solving discrete KdV equation. Comparisons are made between the results of the proposed method and exact solutions. The results reveal that the proposed method is very effective and convenient

Spectrophotometric estimation of ethamsylate and mefenamic Acid from a binary mixture by dual wavelength and simultaneous equation methods.

Science.gov (United States)

Goyal, Anju; Singhvi, I

2008-01-01

Two simple, accurate, economical and reproducible spectrophotometric methods for simultaneous estimation of two-component drug mixture of ethamsylate and mefenamic acid in combined tablet dosage form have been developed. The first developed method involves formation and solving of simultaneous equation using 287.6 nm and 313.2 nm as two wavelengths. Second developed method is based on two wavelength calculation. Two wavelengths selected for estimation of ethamsylate were 274.4 nm and 301.2 nm while that for mefenamic acid were 304.8 nm and 320.4 nm. Both the developed methods obey Beer's law in the concentration ranges employed for the respective methods. The results of analysis were validated statistically and by recovery studies.

A Model for Solving the Maxwell Quasi Stationary Equations in a 3-Phase Electric Reduction Furnace

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S. Ekrann

1982-10-01

Full Text Available A computer code has been developed for the approximate computation of electric and magnetic fields within an electric reduction furnace. The paper describes the numerical methods used to solve Maxwell's quasi-stationary equations, which are the governing equations for this problem. The equations are discretized by a staggered grid finite difference technique. The resulting algebraic equations are solved by iterating between computations of electric and magnetic quantities. This 'outer' iteration converges only when the skin depth is larger or of about the same magnitude as the linear dimensions of the computational domain. In solving for electric quantities with magnetic quantities being regarded as known, and vice versa, the central computational task is the solution of a Poisson equation for a scalar potential. These equations are solved by line successive overrelaxation combined with a rebalancing technique.

Matrix form of Legendre polynomials for solving linear integro-differential equations of high order

Science.gov (United States)

Kammuji, M.; Eshkuvatov, Z. K.; Yunus, Arif A. M.

2017-04-01

This paper presents an effective approximate solution of high order of Fredholm-Volterra integro-differential equations (FVIDEs) with boundary condition. Legendre truncated series is used as a basis functions to estimate the unknown function. Matrix operation of Legendre polynomials is used to transform FVIDEs with boundary conditions into matrix equation of Fredholm-Volterra type. Gauss Legendre quadrature formula and collocation method are applied to transfer the matrix equation into system of linear algebraic equations. The latter equation is solved by Gauss elimination method. The accuracy and validity of this method are discussed by solving two numerical examples and comparisons with wavelet and methods.

Simultaneous and semi-alternating projection algorithms for solving split equality problems.

Science.gov (United States)

Dong, Qiao-Li; Jiang, Dan

2018-01-01

In this article, we first introduce two simultaneous projection algorithms for solving the split equality problem by using a new choice of the stepsize, and then propose two semi-alternating projection algorithms. The weak convergence of the proposed algorithms is analyzed under standard conditions. As applications, we extend the results to solve the split feasibility problem. Finally, a numerical example is presented to illustrate the efficiency and advantage of the proposed algorithms.

Splitting Method for Solving the Coarse-Mesh Discretized Low-Order Quasi-Diffusion Equations

International Nuclear Information System (INIS)

Hiruta, Hikaru; Anistratov, Dmitriy Y.; Adams, Marvin L.

2005-01-01

In this paper, the development is presented of a splitting method that can efficiently solve coarse-mesh discretized low-order quasi-diffusion (LOQD) equations. The LOQD problem can reproduce exactly the transport scalar flux and current. To solve the LOQD equations efficiently, a splitting method is proposed. The presented method splits the LOQD problem into two parts: (a) the D problem that captures a significant part of the transport solution in the central parts of assemblies and can be reduced to a diffusion-type equation and (b) the Q problem that accounts for the complicated behavior of the transport solution near assembly boundaries. Independent coarse-mesh discretizations are applied: the D problem equations are approximated by means of a finite element method, whereas the Q problem equations are discretized using a finite volume method. Numerical results demonstrate the efficiency of the methodology presented. This methodology can be used to modify existing diffusion codes for full-core calculations (which already solve a version of the D problem) to account for transport effects

A highly accurate method to solve Fisher's equation

Indian Academy of Sciences (India)

The solution of the Helmholtz equation was approximated by a sixth-order compact finite difference. (CFD6) method in [29]. In [30], a CFD6 scheme has been presented to ... efficiency of the proposed method are reported in §3. Finally .... our discussion, one can apply the proposed method to solve the more general problem.

Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation

Science.gov (United States)

Agarwal, P.; El-Sayed, A. A.

2018-06-01

In this paper, a new numerical technique for solving the fractional order diffusion equation is introduced. This technique basically depends on the Non-Standard finite difference method (NSFD) and Chebyshev collocation method, where the fractional derivatives are described in terms of the Caputo sense. The Chebyshev collocation method with the (NSFD) method is used to convert the problem into a system of algebraic equations. These equations solved numerically using Newton's iteration method. The applicability, reliability, and efficiency of the presented technique are demonstrated through some given numerical examples.

New method for solving three-dimensional Schroedinger equation

International Nuclear Information System (INIS)

Melezhik, V.S.

1990-01-01

The method derived recently for solving a multidimensional scattering problem is applied to a three-dimensional Schroedinger equation. As compared with direct three-dimensional calculations of finite elements and finite differences, this approach gives sufficiently accurate upper and lower approximations to the helium-atom binding energy, which demonstrates its efficiency. 15 refs.; 1 fig.; 2 tabs

A neuro approach to solve fuzzy Riccati differential equations

Energy Technology Data Exchange (ETDEWEB)

Shahrir, Mohammad Shazri, E-mail: mshazri@gmail.com [InstitutSainsMatematik, Universiti Malaya 50603 Kuala Lumpur, Wilayah Persekutuan Kuala Lumpur (Malaysia); Telekom Malaysia, R& D TM Innovation Centre, LingkaranTeknokrat Timur, 63000 Cyberjaya, Selangor (Malaysia); Kumaresan, N., E-mail: drnk2008@gmail.com; Kamali, M. Z. M.; Ratnavelu, Kurunathan [InstitutSainsMatematik, Universiti Malaya 50603 Kuala Lumpur, Wilayah Persekutuan Kuala Lumpur (Malaysia)

2015-10-22

There are many applications of optimal control theory especially in the area of control systems in engineering. In this paper, fuzzy quadratic Riccati differential equation is estimated using neural networks (NN). Previous works have shown reliable results using Runge-Kutta 4th order (RK4). The solution can be achieved by solving the 1st Order Non-linear Differential Equation (ODE) that is found commonly in Riccati differential equation. Research has shown improved results relatively to the RK4 method. It can be said that NN approach shows promising results with the advantage of continuous estimation and improved accuracy that can be produced over RK4.

A Meshfree Quasi-Interpolation Method for Solving Burgers’ Equation

Directory of Open Access Journals (Sweden)

Mingzhu Li

2014-01-01

Full Text Available The main aim of this work is to consider a meshfree algorithm for solving Burgers’ equation with the quartic B-spline quasi-interpolation. Quasi-interpolation is very useful in the study of approximation theory and its applications, since it can yield solutions directly without the need to solve any linear system of equations and overcome the ill-conditioning problem resulting from using the B-spline as a global interpolant. The numerical scheme is presented, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable. Compared to other numerical methods, the main advantages of our scheme are higher accuracy and lower computational complexity. Meanwhile, the algorithm is very simple and easy to implement and the numerical experiments show that it is feasible and valid.

Reproducing Kernel Method for Solving Nonlinear Differential-Difference Equations

Directory of Open Access Journals (Sweden)

Reza Mokhtari

2012-01-01

Full Text Available On the basis of reproducing kernel Hilbert spaces theory, an iterative algorithm for solving some nonlinear differential-difference equations (NDDEs is presented. The analytical solution is shown in a series form in a reproducing kernel space, and the approximate solution , is constructed by truncating the series to terms. The convergence of , to the analytical solution is also proved. Results obtained by the proposed method imply that it can be considered as a simple and accurate method for solving such differential-difference problems.

A New Approach and Solution Technique to Solve Time Fractional Nonlinear Reaction-Diffusion Equations

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Inci Cilingir Sungu

2015-01-01

Full Text Available A new application of the hybrid generalized differential transform and finite difference method is proposed by solving time fractional nonlinear reaction-diffusion equations. This method is a combination of the multi-time-stepping temporal generalized differential transform and the spatial finite difference methods. The procedure first converts the time-evolutionary equations into Poisson equations which are then solved using the central difference method. The temporal differential transform method as used in the paper takes care of stability and the finite difference method on the resulting equation results in a system of diagonally dominant linear algebraic equations. The Gauss-Seidel iterative procedure then used to solve the linear system thus has assured convergence. To have optimized convergence rate, numerical experiments were done by using a combination of factors involving multi-time-stepping, spatial step size, and degree of the polynomial fit in time. It is shown that the hybrid technique is reliable, accurate, and easy to apply.

Isotope decay equations solved by means of a recursive method

International Nuclear Information System (INIS)

Grant, Carlos

2009-01-01

The isotope decay equations have been solved using forward finite differences taking small time steps, among other methods. This is the case of the cell code WIMS, where it is assumed that concentrations of all fissionable isotopes remain constant during the integration interval among other simplifications. Even when the problem could be solved running through a logical tree, all algorithms used for resolution of these equations used an iterative programming formulation. That happened because nearly all computer languages used up to a recent past by the scientific programmers did not support recursion, such as the case of the old versions of FORTRAN or BASIC. Nowadays also an integral form of the depletion equations is used in Monte Carlo simulation. In this paper we propose another programming solution using a recursive algorithm, running through all descendants of each isotope and adding their contributions to all isotopes in each generation. The only assumption made for this solution is that fluxes remain constant during the whole time step. Recursive process is interrupted when a stable isotope was attained or the calculated contributions are smaller than a given precision. These algorithms can be solved by means an exact analytic method that can have some problems when circular loops appear for isotopes with alpha decay, and a more general polynomial method. Both methods are shown. (author)

Solving ordinary differential equations by electrical analogy: a multidisciplinary teaching tool

Science.gov (United States)

Sanchez Perez, J. F.; Conesa, M.; Alhama, I.

2016-11-01

Ordinary differential equations are the mathematical formulation for a great variety of problems in science and engineering, and frequently, two different problems are equivalent from a mathematical point of view when they are formulated by the same equations. Students acquire the knowledge of how to solve these equations (at least some types of them) using protocols and strict algorithms of mathematical calculation without thinking about the meaning of the equation. The aim of this work is that students learn to design network models or circuits in this way; with simple knowledge of them, students can establish the association of electric circuits and differential equations and their equivalences, from a formal point of view, that allows them to associate knowledge of two disciplines and promote the use of this interdisciplinary approach to address complex problems. Therefore, they learn to use a multidisciplinary tool that allows them to solve these kinds of equations, even students of first course of engineering, whatever the order, grade or type of non-linearity. This methodology has been implemented in numerous final degree projects in engineering and science, e.g., chemical engineering, building engineering, industrial engineering, mechanical engineering, architecture, etc. Applications are presented to illustrate the subject of this manuscript.

Direct method of solving finite difference nonlinear equations for multicomponent diffusion in a gas centrifuge

International Nuclear Information System (INIS)

Potemki, Valeri G.; Borisevich, Valentine D.; Yupatov, Sergei V.

1996-01-01

This paper describes the the next evolution step in development of the direct method for solving systems of Nonlinear Algebraic Equations (SNAE). These equations arise from the finite difference approximation of original nonlinear partial differential equations (PDE). This method has been extended on the SNAE with three variables. The solving SNAE bases on Reiterating General Singular Value Decomposition of rectangular matrix pencils (RGSVD-algorithm). In contrast to the computer algebra algorithm in integer arithmetic based on the reduction to the Groebner's basis that algorithm is working in floating point arithmetic and realizes the reduction to the Kronecker's form. The possibilities of the method are illustrated on the example of solving the one-dimensional diffusion equation for 3-component model isotope mixture in a ga centrifuge. The implicit scheme for the finite difference equations without simplifying the nonlinear properties of the original equations is realized. The technique offered provides convergence to the solution for the single run. The Toolbox SNAE is developed in the framework of the high performance numeric computation and visualization software MATLAB. It includes more than 30 modules in MATLAB language for solving SNAE with two and three variables. (author)

Solving Nonlinear Partial Differential Equations with Maple and Mathematica

CERN Document Server

Shingareva, Inna K

2011-01-01

The emphasis of the book is given in how to construct different types of solutions (exact, approximate analytical, numerical, graphical) of numerous nonlinear PDEs correctly, easily, and quickly. The reader can learn a wide variety of techniques and solve numerous nonlinear PDEs included and many other differential equations, simplifying and transforming the equations and solutions, arbitrary functions and parameters, presented in the book). Numerous comparisons and relationships between various types of solutions, different methods and approaches are provided, the results obtained in Maple an

Solving nonlinear evolution equation system using two different methods

Science.gov (United States)

Kaplan, Melike; Bekir, Ahmet; Ozer, Mehmet N.

2015-12-01

This paper deals with constructing more general exact solutions of the coupled Higgs equation by using the (G0/G, 1/G)-expansion and (1/G0)-expansion methods. The obtained solutions are expressed by three types of functions: hyperbolic, trigonometric and rational functions with free parameters. It has been shown that the suggested methods are productive and will be used to solve nonlinear partial differential equations in applied mathematics and engineering. Throughout the paper, all the calculations are made with the aid of the Maple software.

Effective methods of solving of model equations of certain class of thermal systems

International Nuclear Information System (INIS)

Lach, J.

1985-01-01

A number of topics connected with solving of model equations of certain class of thermal systems by the method of successive approximations is touched. A system of partial differential equations of the first degree, appearing most frequently in practical applications of heat and mass transfer theory is reduced to an equivalent system of Volterra integral equations of the second kind. Among a few sample applications the thermal processes appearing in the fuel channel of nuclear reactor are solved. The theoretical analysis is illustrated by the results of numerical calculations given in tables and diagrams. 111 refs., 17 figs., 16 tabs. (author)

Localization of the eigenvalues of linear integral equations with applications to linear ordinary differential equations.

Science.gov (United States)

Sloss, J. M.; Kranzler, S. K.

1972-01-01

The equivalence of a considered integral equation form with an infinite system of linear equations is proved, and the localization of the eigenvalues of the infinite system is expressed. Error estimates are derived, and the problems of finding upper bounds and lower bounds for the eigenvalues are solved simultaneously.

Approximate Method for Solving the Linear Fuzzy Delay Differential Equations

Directory of Open Access Journals (Sweden)

S. Narayanamoorthy

2015-01-01

Full Text Available We propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using Adomian decomposition method. The detailed algorithm of the approach is provided. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show this proper features of this proposed method, numerical example is illustrated.

Nonlinear evolution equations and solving algebraic systems: the importance of computer algebra

International Nuclear Information System (INIS)

Gerdt, V.P.; Kostov, N.A.

1989-01-01

In the present paper we study the application of computer algebra to solve the nonlinear polynomial systems which arise in investigation of nonlinear evolution equations. We consider several systems which are obtained in classification of integrable nonlinear evolution equations with uniform rank. Other polynomial systems are related with the finding of algebraic curves for finite-gap elliptic potentials of Lame type and generalizations. All systems under consideration are solved using the method based on construction of the Groebner basis for corresponding polynomial ideals. The computations have been carried out using computer algebra systems. 20 refs

New approach to solve fully fuzzy system of linear equations using ...

Indian Academy of Sciences (India)

This paper proposes two new methods to solve fully fuzzy system of linear equations. The fuzzy system has been converted to a crisp system of linear equations by using single and double parametric form of fuzzy numbers to obtain the non-negative solution. Double parametric form of fuzzy numbers is defined and applied ...

Recovering the source and initial value simultaneously in a parabolic equation

International Nuclear Information System (INIS)

Zheng, Guang-Hui; Wei, Ting

2014-01-01

In this paper, we consider an inverse problem to simultaneously reconstruct the source term and initial data associated with a parabolic equation based on the additional temperature data at a terminal time t = T and the temperature data on an accessible part of a boundary. The conditional stability and uniqueness of the inverse problem are established. We apply a variational regularization method to recover the source and initial value. The existence, uniqueness and stability of the minimizer of the corresponding variational problem are obtained. Taking the minimizer as a regularized solution for the inverse problem, under an a priori and an a posteriori parameter choice rule, the convergence rates of the regularized solution under a source condition are also given. Furthermore, the source condition is characterized by an optimal control approach. Finally, we use a conjugate gradient method and a stopping criterion given by Morozov's discrepancy principle to solve the variational problem. Numerical experiments are provided to demonstrate the feasibility of the method. (papers)

Solving the linear inviscid shallow water equations in one dimension, with variable depth, using a recursion formula

Science.gov (United States)

Hernandez-Walls, R.; Martín-Atienza, B.; Salinas-Matus, M.; Castillo, J.

2017-11-01

When solving the linear inviscid shallow water equations with variable depth in one dimension using finite differences, a tridiagonal system of equations must be solved. Here we present an approach, which is more efficient than the commonly used numerical method, to solve this tridiagonal system of equations using a recursion formula. We illustrate this approach with an example in which we solve for a rectangular channel to find the resonance modes. Our numerical solution agrees very well with the analytical solution. This new method is easy to use and understand by undergraduate students, so it can be implemented in undergraduate courses such as Numerical Methods, Lineal Algebra or Differential Equations.

Solving the linear inviscid shallow water equations in one dimension, with variable depth, using a recursion formula

International Nuclear Information System (INIS)

Hernandez-Walls, R; Martín-Atienza, B; Salinas-Matus, M; Castillo, J

2017-01-01

When solving the linear inviscid shallow water equations with variable depth in one dimension using finite differences, a tridiagonal system of equations must be solved. Here we present an approach, which is more efficient than the commonly used numerical method, to solve this tridiagonal system of equations using a recursion formula. We illustrate this approach with an example in which we solve for a rectangular channel to find the resonance modes. Our numerical solution agrees very well with the analytical solution. This new method is easy to use and understand by undergraduate students, so it can be implemented in undergraduate courses such as Numerical Methods, Lineal Algebra or Differential Equations. (paper)

Solving Partial Differential Equations Using a New Differential Evolution Algorithm

Directory of Open Access Journals (Sweden)

Natee Panagant

2014-01-01

Full Text Available This paper proposes an alternative meshless approach to solve partial differential equations (PDEs. With a global approximate function being defined, a partial differential equation problem is converted into an optimisation problem with equality constraints from PDE boundary conditions. An evolutionary algorithm (EA is employed to search for the optimum solution. For this approach, the most difficult task is the low convergence rate of EA which consequently results in poor PDE solution approximation. However, its attractiveness remains due to the nature of a soft computing technique in EA. The algorithm can be used to tackle almost any kind of optimisation problem with simple evolutionary operation, which means it is mathematically simpler to use. A new efficient differential evolution (DE is presented and used to solve a number of the partial differential equations. The results obtained are illustrated and compared with exact solutions. It is shown that the proposed method has a potential to be a future meshless tool provided that the search performance of EA is greatly enhanced.

Perceptual support promotes strategy generation: Evidence from equation solving.

Science.gov (United States)

Alibali, Martha W; Crooks, Noelle M; McNeil, Nicole M

2017-08-30

Over time, children shift from using less optimal strategies for solving mathematics problems to using better ones. But why do children generate new strategies? We argue that they do so when they begin to encode problems more accurately; therefore, we hypothesized that perceptual support for correct encoding would foster strategy generation. Fourth-grade students solved mathematical equivalence problems (e.g., 3 + 4 + 5 = 3 + __) in a pre-test. They were then randomly assigned to one of three perceptual support conditions or to a Control condition. Participants in all conditions completed three mathematical equivalence problems with feedback about correctness. Participants in the experimental conditions received perceptual support (i.e., highlighting in red ink) for accurately encoding the equal sign, the right side of the equation, or the numbers that could be added to obtain the correct solution. Following this intervention, participants completed a problem-solving post-test. Among participants who solved the problems incorrectly at pre-test, those who received perceptual support for correctly encoding the equal sign were more likely to generate new, correct strategies for solving the problems than were those who received feedback only. Thus, perceptual support for accurate encoding of a key problem feature promoted generation of new, correct strategies. Statement of Contribution What is already known on this subject? With age and experience, children shift to using more effective strategies for solving math problems. Problem encoding also improves with age and experience. What the present study adds? Support for encoding the equal sign led children to generate correct strategies for solving equations. Improvements in problem encoding are one source of new strategies. © 2017 The British Psychological Society.

Development of a set of benchmark problems to verify numerical methods for solving burnup equations

International Nuclear Information System (INIS)

Lago, Daniel; Rahnema, Farzad

2017-01-01

Highlights: • Description transmutation chain benchmark problems. • Problems for validating numerical methods for solving burnup equations. • Analytical solutions for the burnup equations. • Numerical solutions for the burnup equations. - Abstract: A comprehensive set of transmutation chain benchmark problems for numerically validating methods for solving burnup equations was created. These benchmark problems were designed to challenge both traditional and modern numerical methods used to solve the complex set of ordinary differential equations used for tracking the change in nuclide concentrations over time due to nuclear phenomena. Given the development of most burnup solvers is done for the purpose of coupling with an established transport solution method, these problems provide a useful resource in testing and validating the burnup equation solver before coupling for use in a lattice or core depletion code. All the relevant parameters for each benchmark problem are described. Results are also provided in the form of reference solutions generated by the Mathematica tool, as well as additional numerical results from MATLAB.

Simultaneous Heat and Mass Transfer Model for Convective Drying of Building Material

Science.gov (United States)

Upadhyay, Ashwani; Chandramohan, V. P.

2018-04-01

A mathematical model of simultaneous heat and moisture transfer is developed for convective drying of building material. A rectangular brick is considered for sample object. Finite-difference method with semi-implicit scheme is used for solving the transient governing heat and mass transfer equation. Convective boundary condition is used, as the product is exposed in hot air. The heat and mass transfer equations are coupled through diffusion coefficient which is assumed as the function of temperature of the product. Set of algebraic equations are generated through space and time discretization. The discretized algebraic equations are solved by Gauss-Siedel method via iteration. Grid and time independent studies are performed for finding the optimum number of nodal points and time steps respectively. A MATLAB computer code is developed to solve the heat and mass transfer equations simultaneously. Transient heat and mass transfer simulations are performed to find the temperature and moisture distribution inside the brick.

Solution of the stellar structure equations in Eulerian coordinates

International Nuclear Information System (INIS)

Deupree, R.G.

1976-01-01

The equations of hydrostatic and thermal equilibrium, assuming only radiative energy transport and spherical symmetry, are solved in Eulerian coordinates by a suitable modification of the Henyey method. An Eulerian approach may possibly be more suitably extended to more spatial dimensions than the usual Lagrangian procedure. The principle advantage of this method is that the equations of hydrostatic and thermal equilibrium and Poisson's equation may be solved simultaneously

A predictor-corrector scheme for solving the Volterra integral equation

KAUST Repository

Al Jarro, Ahmed; Bagci, Hakan

2011-01-01

The occurrence of late time instabilities is a common problem of almost all time marching methods developed for solving time domain integral equations. Implicit marching algorithms are now considered stable with various efforts that have been

Solving the KPI wave equation with a moving adaptive FEM grid

Directory of Open Access Journals (Sweden)

Granville Sewell

2013-04-01

Full Text Available The Kadomtsev-Petviashvili I (KPI equation is the difficult nonlinear wave equation $U_{xt} + 6U_x^2 + 6UU_{xx} + U_{xxxx} = 3U_{yy}.$ We solve this equation using PDE2D (www.pde2d.com with initial conditions consisting of two lump solitons, which collide and reseparate. Since the solution has steep, moving, peaks, an adaptive finite element grid is used with a grading which moves with the peaks.

An Accurate Approximate-Analytical Technique for Solving Time-Fractional Partial Differential Equations

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M. Bishehniasar

2017-01-01

Full Text Available The demand of many scientific areas for the usage of fractional partial differential equations (FPDEs to explain their real-world systems has been broadly identified. The solutions may portray dynamical behaviors of various particles such as chemicals and cells. The desire of obtaining approximate solutions to treat these equations aims to overcome the mathematical complexity of modeling the relevant phenomena in nature. This research proposes a promising approximate-analytical scheme that is an accurate technique for solving a variety of noninteger partial differential equations (PDEs. The proposed strategy is based on approximating the derivative of fractional-order and reducing the problem to the corresponding partial differential equation (PDE. Afterwards, the approximating PDE is solved by using a separation-variables technique. The method can be simply applied to nonhomogeneous problems and is proficient to diminish the span of computational cost as well as achieving an approximate-analytical solution that is in excellent concurrence with the exact solution of the original problem. In addition and to demonstrate the efficiency of the method, it compares with two finite difference methods including a nonstandard finite difference (NSFD method and standard finite difference (SFD technique, which are popular in the literature for solving engineering problems.

On choosing a nonlinear initial iterate for solving the 2-D 3-T heat conduction equations

International Nuclear Information System (INIS)

An Hengbin; Mo Zeyao; Xu Xiaowen; Liu Xu

2009-01-01

The 2-D 3-T heat conduction equations can be used to approximately describe the energy broadcast in materials and the energy swapping between electron and photon or ion. To solve the equations, a fully implicit finite volume scheme is often used as the discretization method. Because the energy diffusion and swapping coefficients have a strongly nonlinear dependence on the temperature, and some physical parameters are discontinuous across the interfaces between the materials, it is a challenge to solve the discretized nonlinear algebraic equations. Particularly, as time advances, the temperature varies so greatly in the front of energy that it is difficult to choose an effective initial iterate when the nonlinear algebraic equations are solved by an iterative method. In this paper, a method of choosing a nonlinear initial iterate is proposed for iterative solving this kind of nonlinear algebraic equations. Numerical results show the proposed initial iterate can improve the computational efficiency, and also the convergence behavior of the nonlinear iteration.

A method for solving the KDV equation and some numerical experiments

International Nuclear Information System (INIS)

Chang Jinjiang.

1993-01-01

In this paper, by means of difference method for discretization of space partial derivatives of KDV equation, an initial value problem in ordinary differential equations of large dimensions is produced. By using this ordinary differential equations the existence and the uniqueness of the solution of the KDV equation and the conservation of scheme are proved. This ordinary differential equation can be solved by using implicit Runge-Kutta methods, so a new method for finding the numerical solution of the KDV equation is presented. Numerical experiments not only describe in detail the procedure of two solitons collision, soliton reflex and soliton produce, but also show that this method is very effective. (author). 7 refs, 3 figs

Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (4).

Science.gov (United States)

Murase, Kenya

2016-01-01

Partial differential equations are often used in the field of medical physics. In this (final) issue, the methods for solving the partial differential equations were introduced, which include separation of variables, integral transform (Fourier and Fourier-sine transforms), Green's function, and series expansion methods. Some examples were also introduced, in which the integral transform and Green's function methods were applied to solving Pennes' bioheat transfer equation and the Fourier series expansion method was applied to Navier-Stokes equation for analyzing the wall shear stress in blood vessels.Finally, the author hopes that this series will be helpful for people who engage in medical physics.

An efficient numerical technique for solving navier-stokes equations for rotating flows

International Nuclear Information System (INIS)

Haroon, T.; Shah, T.M.

2000-01-01

This paper simulates an industrial problem by solving compressible Navier-Stokes equations. The time-consuming tri-angularization process of a large-banded matrix, performed by memory economical Frontal Technique. This scheme successfully reduces the time for I/O operations even for as large as (40, 000 x 40, 000) matrix. Previously, this industrial problem can solved by using modified Newton's method with Gaussian elimination technique for the large matrix. In the present paper, the proposed Frontal Technique is successfully used, together with Newton's method, to solve compressible Navier-Stokes equations for rotating cylinders. By using the Frontal Technique, the method gives the solution within reasonably acceptance computational time. Results are compared with the earlier works done, and found computationally very efficient. Some features of the solution are reported here for the rotating machines. (author)

A control volume based finite difference method for solving the equilibrium equations in terms of displacements

DEFF Research Database (Denmark)

Hattel, Jesper; Hansen, Preben

1995-01-01

This paper presents a novel control volume based FD method for solving the equilibrium equations in terms of displacements, i.e. the generalized Navier equations. The method is based on the widely used cv-FDM solution of heat conduction and fluid flow problems involving a staggered grid formulati....... The resulting linear algebraic equations are solved by line-Gauss-Seidel....

Simultaneous State and Parameter Estimation Using Maximum Relative Entropy with Nonhomogenous Differential Equation Constraints

Directory of Open Access Journals (Sweden)

Adom Giffin

2014-09-01

Full Text Available In this paper, we continue our efforts to show how maximum relative entropy (MrE can be used as a universal updating algorithm. Here, our purpose is to tackle a joint state and parameter estimation problem where our system is nonlinear and in a non-equilibrium state, i.e., perturbed by varying external forces. Traditional parameter estimation can be performed by using filters, such as the extended Kalman filter (EKF. However, as shown with a toy example of a system with first order non-homogeneous ordinary differential equations, assumptions made by the EKF algorithm (such as the Markov assumption may not be valid. The problem can be solved with exponential smoothing, e.g., exponentially weighted moving average (EWMA. Although this has been shown to produce acceptable filtering results in real exponential systems, it still cannot simultaneously estimate both the state and its parameters and has its own assumptions that are not always valid, for example when jump discontinuities exist. We show that by applying MrE as a filter, we can not only develop the closed form solutions, but we can also infer the parameters of the differential equation simultaneously with the means. This is useful in real, physical systems, where we want to not only filter the noise from our measurements, but we also want to simultaneously infer the parameters of the dynamics of a nonlinear and non-equilibrium system. Although there were many assumptions made throughout the paper to illustrate that EKF and exponential smoothing are special cases ofMrE, we are not “constrained”, by these assumptions. In other words, MrE is completely general and can be used in broader ways.

On extension of solutions of a simultaneous system of iterative functional equations

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Janusz Matkowski

2009-01-01

Full Text Available Some sufficient conditions which allow to extend every local solution of a simultaneous system of equations in a single variable of the form \\[ \\varphi(x = h (x, \\varphi[f_1(x],\\ldots,\\varphi[f_m(x],\\] \\[\\varphi(x = H (x, \\varphi[F_1(x],\\ldots,\\varphi[F_m(x],\\] to a global one are presented. Extensions of solutions of functional equations, both in single and in several variables, play important role (cf. for instance [M. Kuczma, Functional equations in a single variable, Monografie Mat. 46, Polish Scientific Publishers, Warsaw, 1968, M. Kuczma, B. Choczewski, R. Ger, Iterative functional equations, Encyclopedia of Mathematics and Its Applications v. 32, Cambridge, 1990, J. Matkowski, Iteration groups, commuting functions and simultaneous systems of linear functional equations, Opuscula Math. 28 (2008 4, 531-541].

Solving Fokker-Planck Equations on Cantor Sets Using Local Fractional Decomposition Method

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Shao-Hong Yan

2014-01-01

Full Text Available The local fractional decomposition method is applied to approximate the solutions for Fokker-Planck equations on Cantor sets with local fractional derivative. The obtained results give the present method that is very effective and simple for solving the differential equations on Cantor set.

MPFA algorithm for solving stokes-brinkman equations on quadrilateral grids

KAUST Repository

Iliev, Oleg; Kirsch, Ralf; Lakdawala, Zahra; Printsypar, Galina

2014-01-01

This work is concerned with the development of a robust and accurate numerical method for solving the Stokes-Brinkman system of equations, which describes a free fluid flow coupled with a flow in porous media. Quadrilateral boundary fitted grid

Intuitive physics knowledge, physics problem solving and the role of mathematical equations

Directory of Open Access Journals (Sweden)

Laura Buteler

2012-09-01

Full Text Available The present work explores the role that mathematical equations play in modifying students’ physical intuition (diSessa, 1993. The work is carried out assuming that students achieve a great deal of the refinement in their physical intuitions during problem solving (Sherin, 2006. The study is guided by the question of how the use of mathematical equations contributes to this refinement. The authors aim at expanding on Sherin´s (2006 hypothesis, suggesting a more bounding relation between physical intuitions and mathematics. In this scenario, intuitions play a more compelling role in “deciding” which equations are acceptable and which are not. Our hypothesis is constructed on the basis of three cases: the first published by Sherin (2006 and two more from registries of our own. The three cases are compared and analyzed in relation to the role of mathematical equations in refining – or not – the intuitive knowledge students bring to play during problem solving.

Students' errors in solving linear equation word problems: Case ...

African Journals Online (AJOL)

kofi.mereku

Development in most areas of life is based on effective knowledge of science and ... Problem solving, as used in mathematics education literature, refers ... word problems, on the other hand, are those linear equation tasks or ... taught LEWPs in the junior high school, many of them reach the senior high school without a.

Various Newton-type iterative methods for solving nonlinear equations

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Manoj Kumar

2013-10-01

Full Text Available The aim of the present paper is to introduce and investigate new ninth and seventh order convergent Newton-type iterative methods for solving nonlinear equations. The ninth order convergent Newton-type iterative method is made derivative free to obtain seventh-order convergent Newton-type iterative method. These new with and without derivative methods have efficiency indices 1.5518 and 1.6266, respectively. The error equations are used to establish the order of convergence of these proposed iterative methods. Finally, various numerical comparisons are implemented by MATLAB to demonstrate the performance of the developed methods.

An efficient numerical method for solving the Boltzmann equation in multidimensions

Science.gov (United States)

Dimarco, Giacomo; Loubère, Raphaël; Narski, Jacek; Rey, Thomas

2018-01-01

In this paper we deal with the extension of the Fast Kinetic Scheme (FKS) (Dimarco and Loubère, 2013 [26]) originally constructed for solving the BGK equation, to the more challenging case of the Boltzmann equation. The scheme combines a robust and fast method for treating the transport part based on an innovative Lagrangian technique supplemented with conservative fast spectral schemes to treat the collisional operator by means of an operator splitting approach. This approach along with several implementation features related to the parallelization of the algorithm permits to construct an efficient simulation tool which is numerically tested against exact and reference solutions on classical problems arising in rarefied gas dynamic. We present results up to the 3 D × 3 D case for unsteady flows for the Variable Hard Sphere model which may serve as benchmark for future comparisons between different numerical methods for solving the multidimensional Boltzmann equation. For this reason, we also provide for each problem studied details on the computational cost and memory consumption as well as comparisons with the BGK model or the limit model of compressible Euler equations.

A Python Program for Solving Schro¨dinger's Equation in Undergraduate Physical Chemistry

Science.gov (United States)

Srnec, Matthew N.; Upadhyay, Shiv; Madura, Jeffry D.

2017-01-01

In undergraduate physical chemistry, Schrödinger's equation is solved for a variety of cases. In doing so, the energies and wave functions of the system can be interpreted to provide connections with the physical system being studied. Solving this equation by hand for a one-dimensional system is a manageable task, but it becomes time-consuming…

MPFA algorithm for solving stokes-brinkman equations on quadrilateral grids

KAUST Repository

Iliev, Oleg

2014-01-01

This work is concerned with the development of a robust and accurate numerical method for solving the Stokes-Brinkman system of equations, which describes a free fluid flow coupled with a flow in porous media. Quadrilateral boundary fitted grid with a sophisticated finite volume method, namely MPFA O-method, is used to discretize the system of equations. Numerical results for two examples are presented, namely, channel flow and flow in a ring with a rolled porous medium. © Springer International Publishing Switzerland 2014.

A matrix formalism to solve interface condition equations in a reactor system

Energy Technology Data Exchange (ETDEWEB)

Matausek, M V [Boris Kidric Institute of Nuclear Sciences Vinca, Beograd (Yugoslavia)

1970-05-15

When a nuclear reactor or a reactor lattice cell is treated by an approximate procedure to solve the neutron transport equation, as the last computational step often appears a problem of solving systems of algebraic equations stating the interface and boundary conditions for the neutron flux moments. These systems have usually the coefficient matrices of the block-bi diagonal type, containing thus a large number of zero elements. In the present report it is shown how such a system can be solved efficiently accounting for all the zero elements both in the coefficient matrix and in the free term vector. The procedure is presented here for the case of multigroup P{sub 3} calculation of neutron flux distribution in a cylindrical reactor lattice cell. Compared with the standard gaussian elimination method, this procedure is more advantageous both in respect to the number of operations needed to solve a given problem and in respect to the computer memory storage requirements. A similar formalism can also be applied for other approximate methods, for instance for multigroup diffusion treatment of a multi zone reactor. (author)

Conditions for the existence of control functions in nonseparable simultaneous equations models

OpenAIRE

Blundell, Richard; Matzkin, Rosa L.

2010-01-01

The control function approach (Heckman and Robb (1985)) in a system of linear simultaneous equations provides a convenient procedure to estimate one of the functions in the system using reduced form residuals from the other functions as additional regressors. The conditions on the structural system under which this procedure can be used in nonlinear and nonparametric simultaneous equations has thus far been unknown. In this note, we define a new property of functions called control function s...

Space-time spectral collocation algorithm for solving time-fractional Tricomi-type equations

Directory of Open Access Journals (Sweden)

Abdelkawy M.A.

2016-01-01

Full Text Available We introduce a new numerical algorithm for solving one-dimensional time-fractional Tricomi-type equations (T-FTTEs. We used the shifted Jacobi polynomials as basis functions and the derivatives of fractional is evaluated by the Caputo definition. The shifted Jacobi Gauss-Lobatt algorithm is used for the spatial discretization, while the shifted Jacobi Gauss-Radau algorithmis applied for temporal approximation. Substituting these approximations in the problem leads to a system of algebraic equations that greatly simplifies the problem. The proposed algorithm is successfully extended to solve the two-dimensional T-FTTEs. Extensive numerical tests illustrate the capability and high accuracy of the proposed methodologies.

A New Numerical Technique for Solving Systems Of Nonlinear Fractional Partial Differential Equations

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Mountassir Hamdi Cherif

2017-11-01

Full Text Available In this paper, we apply an efficient method called the Aboodh decomposition method to solve systems of nonlinear fractional partial differential equations. This method is a combined form of Aboodh transform with Adomian decomposition method. The theoretical analysis of this investigated for systems of nonlinear fractional partial differential equations is calculated in the explicit form of a power series with easily computable terms. Some examples are given to shows that this method is very efficient and accurate. This method can be applied to solve others nonlinear systems problems.

Numerical solution of Boltzmann's equation

International Nuclear Information System (INIS)

Sod, G.A.

1976-04-01

The numerical solution of Boltzmann's equation is considered for a gas model consisting of rigid spheres by means of Hilbert's expansion. If only the first two terms of the expansion are retained, Boltzmann's equation reduces to the Boltzmann-Hilbert integral equation. Successive terms in the Hilbert expansion are obtained by solving the same integral equation with a different source term. The Boltzmann-Hilbert integral equation is solved by a new very fast numerical method. The success of the method rests upon the simultaneous use of four judiciously chosen expansions; Hilbert's expansion for the distribution function, another expansion of the distribution function in terms of Hermite polynomials, the expansion of the kernel in terms of the eigenvalues and eigenfunctions of the Hilbert operator, and an expansion involved in solving a system of linear equations through a singular value decomposition. The numerical method is applied to the study of the shock structure in one space dimension. Numerical results are presented for Mach numbers of 1.1 and 1.6. 94 refs, 7 tables, 1 fig

Finite element method with quadratic quadrilateral unit for solving two dimensional incompressible N-S equation

International Nuclear Information System (INIS)

Tao Ganqiang; Yu Qing; Xiao Xiao

2011-01-01

Viscous and incompressible fluid flow is important for numerous engineering mechanics problems. Because of high non linear and incompressibility for Navier-Stokes equation, it is very difficult to solve Navier-Stokes equation by numerical method. According to its characters of Navier-Stokes equation, quartic derivation controlling equation of the two dimensional incompressible Navier-Stokes equation is set up firstly. The method solves the problem for dealing with vorticity boundary and automatically meets incompressibility condition. Then Finite Element equation for Navier-Stokes equation is proposed by using quadratic quadrilateral unit with 8 nodes in which the unit function is quadratic and non linear.-Based on it, the Finite Element program of quadratic quadrilateral unit with 8 nodes is developed. Lastly, numerical experiment proves the accuracy and dependability of the method and also shows the method has good application prospect in computational fluid mechanics. (authors)

Application of Trotter approximation for solving time dependent neutron transport equation

International Nuclear Information System (INIS)

Stancic, V.

1987-01-01

A method is proposed to solve multigroup time dependent neutron transport equation with arbitrary scattering anisotropy. The recurrence relation thus obtained is simple, numerically stable and especially suitable for treatment of complicated geometries. (author)

An Unconditionally Stable Method for Solving the Acoustic Wave Equation

Directory of Open Access Journals (Sweden)

Zhi-Kai Fu

2015-01-01

Full Text Available An unconditionally stable method for solving the time-domain acoustic wave equation using Associated Hermit orthogonal functions is proposed. The second-order time derivatives in acoustic wave equation are expanded by these orthogonal basis functions. By applying Galerkin temporal testing procedure, the time variable can be eliminated from the calculations. The restriction of Courant-Friedrichs-Levy (CFL condition in selecting time step for analyzing thin layer can be avoided. Numerical results show the accuracy and the efficiency of the proposed method.

Factors Affecting Differential Equation Problem Solving Ability of Students at Pre-University Level: A Conceptual Model

Science.gov (United States)

Aisha, Bibi; Zamri, Sharifa NorulAkmar Syed; Abdallah, Nabeel; Abedalaziz, Mohammad; Ahmad, Mushtaq; Satti, Umbreen

2017-01-01

In this study, different factors affecting students' differential equations (DEs) solving abilities were explored at pre university level. To explore main factors affecting students' differential equations problem solving ability, articles for a 19-year period, from 1996 to 2015, were critically reviewed and analyzed. It was revealed that…

An algorithm for solving an arbitrary triangular fully fuzzy Sylvester matrix equations

Science.gov (United States)

Daud, Wan Suhana Wan; Ahmad, Nazihah; Malkawi, Ghassan

2017-11-01

Sylvester matrix equations played a prominent role in various areas including control theory. Considering to any un-certainty problems that can be occurred at any time, the Sylvester matrix equation has to be adapted to the fuzzy environment. Therefore, in this study, an algorithm for solving an arbitrary triangular fully fuzzy Sylvester matrix equation is constructed. The construction of the algorithm is based on the max-min arithmetic multiplication operation. Besides that, an associated arbitrary matrix equation is modified in obtaining the final solution. Finally, some numerical examples are presented to illustrate the proposed algorithm.

Application of Central Upwind Scheme for Solving Special Relativistic Hydrodynamic Equations

Science.gov (United States)

Yousaf, Muhammad; Ghaffar, Tayabia; Qamar, Shamsul

2015-01-01

The accurate modeling of various features in high energy astrophysical scenarios requires the solution of the Einstein equations together with those of special relativistic hydrodynamics (SRHD). Such models are more complicated than the non-relativistic ones due to the nonlinear relations between the conserved and state variables. A high-resolution shock-capturing central upwind scheme is implemented to solve the given set of equations. The proposed technique uses the precise information of local propagation speeds to avoid the excessive numerical diffusion. The second order accuracy of the scheme is obtained with the use of MUSCL-type initial reconstruction and Runge-Kutta time stepping method. After a discussion of the equations solved and of the techniques employed, a series of one and two-dimensional test problems are carried out. To validate the method and assess its accuracy, the staggered central and the kinetic flux-vector splitting schemes are also applied to the same model. The scheme is robust and efficient. Its results are comparable to those obtained from the sophisticated algorithms, even in the case of highly relativistic two-dimensional test problems. PMID:26070067

A homotopy method for solving Riccati equations on a shared memory parallel computer

International Nuclear Information System (INIS)

Zigic, D.; Watson, L.T.; Collins, E.G. Jr.; Davis, L.D.

1993-01-01

Although there are numerous algorithms for solving Riccati equations, there still remains a need for algorithms which can operate efficiently on large problems and on parallel machines. This paper gives a new homotopy-based algorithm for solving Riccati equations on a shared memory parallel computer. The central part of the algorithm is the computation of the kernel of the Jacobian matrix, which is essential for the corrector iterations along the homotopy zero curve. Using a Schur decomposition the tensor product structure of various matrices can be efficiently exploited. The algorithm allows for efficient parallelization on shared memory machines

A framework for simultaneous aerodynamic design optimization in the presence of chaos

Energy Technology Data Exchange (ETDEWEB)

Günther, Stefanie, E-mail: stefanie.guenther@scicomp.uni-kl.de [TU Kaiserslautern, Chair for Scientific Computing, Paul-Ehrlich-Straße 34, 67663 Kaiserslautern (Germany); Gauger, Nicolas R. [TU Kaiserslautern, Chair for Scientific Computing, Paul-Ehrlich-Straße 34, 67663 Kaiserslautern (Germany); Wang, Qiqi [Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 77 Massachusetts Avenue, Cambridge, MA 02139 (United States)

2017-01-01

Integrating existing solvers for unsteady partial differential equations into a simultaneous optimization method is challenging due to the forward-in-time information propagation of classical time-stepping methods. This paper applies the simultaneous single-step one-shot optimization method to a reformulated unsteady constraint that allows for both forward- and backward-in-time information propagation. Especially in the presence of chaotic and turbulent flow, solving the initial value problem simultaneously with the optimization problem often scales poorly with the time domain length. The new formulation relaxes the initial condition and instead solves a least squares problem for the discrete partial differential equations. This enables efficient one-shot optimization that is independent of the time domain length, even in the presence of chaos.

Adomian decomposition method for solving the telegraph equation in charged particle transport

International Nuclear Information System (INIS)

Abdou, M.A.

2005-01-01

In this paper, the analysis for the telegraph equation in case of isotropic small angle scattering from the Boltzmann transport equation for charged particle is presented. The Adomian decomposition is used to solve the telegraph equation. By means of MAPLE the Adomian polynomials of obtained series (ADM) solution have been calculated. The behaviour of the distribution function are shown graphically. The results reported in this article provide further evidence of the usefulness of Adomain decomposition for obtaining solution of linear and nonlinear problems

The discontinuous finite element method for solving Eigenvalue problems of transport equations

International Nuclear Information System (INIS)

Yang, Shulin; Wang, Ruihong

2011-01-01

In this paper, the multigroup transport equations for solving the eigenvalues λ and K_e_f_f under two dimensional cylindrical coordinate are discussed. Aimed at the equations, the discretizing way combining discontinuous finite element method (DFE) with discrete ordinate method (SN) is developed, and the iterative algorithms and steps are studied. The numerical results show that the algorithms are efficient. (author)

The H-N method for solving linear transport equation: theory and application

International Nuclear Information System (INIS)

Kaskas, A.; Gulecyuz, M.C.; Tezcan, C.

2002-01-01

The system of singular integral equation which is obtained from the integro-differential form of the linear transport equation as a result of Placzec lemma is solved. Application are given using the exit distributions and the infinite medium Green's function. The same theoretical results are also obtained with the use of the singular eigenfunction of the method of elementary solutions

Simultaneous-equations Analysis in Regional Science and Economic Geography

DEFF Research Database (Denmark)

Mitze, Timo; Stephan, Andreas

This paper provides an overview over simultaneous equation models (SEM) in the context of analyses based on regional data. We describe various modelling approaches and highlight close link of SEMs to theory and also comment on the advantages and disadvantages of SEMs.We present selected empirical...

A novel method to solve functional differential equations

International Nuclear Information System (INIS)

Tapia, V.

1990-01-01

A method to solve differential equations containing the variational operator as the derivation operation is presented. They are called variational differential equations (VDE). The solution to a VDE should be a function containing the derivatives, with respect to the base space coordinates, of the fields up to a generic order s: a s-th-order function. The variational operator doubles the order of the function on which it acts. Therefore, in order to make compatible the orders of the different terms appearing in a VDE, the solution should be a function containing the derivatives of the fields at all orders. But this takes us again back to the functional methods. In order to avoid this, one must restrict the considerations, in the case of second-order VDEs, to the space of s-th-order functions on which the variational operator acts transitively. These functions have been characterized for a one-dimensional base space for the first- and second-order cases. These functions turn out to be polynomial in the highest-order derivatives of the fields with functions of the lower-order derivatives as coefficients. Then VDEs reduce to a system of coupled partial differential equations for the coefficients above mentioned. The importance of the method lies on the fact that the solutions to VDEs are in a one-to-one correspondence with the solutions of functional differential equations. The previous method finds direct applications in quantum field theory, where the Schroedinger equation plays a central role. Since the Schroedinger equation is reduced to a system of coupled partial differential equations, this provides a nonperturbative scheme for quantum field theory. As an example, the massless scalar field is considered

Optimal Homotopy Asymptotic Method for Solving the Linear Fredholm Integral Equations of the First Kind

Directory of Open Access Journals (Sweden)

Mohammad Almousa

2013-01-01

Full Text Available The aim of this study is to present the use of a semi analytical method called the optimal homotopy asymptotic method (OHAM for solving the linear Fredholm integral equations of the first kind. Three examples are discussed to show the ability of the method to solve the linear Fredholm integral equations of the first kind. The results indicated that the method is very effective and simple.

A toolbox to solve coupled systems of differential and difference equations

International Nuclear Information System (INIS)

Ablinger, Jakob; Schneider, Carsten; Bluemlein, Johannes; Freitas, Abilio de

2016-01-01

We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do not request special choices of bases. Here we assume that the desired solution has a power series representation and we seek for the coefficients in closed form. In particular, if the coefficients depend on a small parameter ε (the dimensional parameter), we assume that the coefficients themselves can be expanded in formal Laurent series w.r.t. ε and we try to compute the first terms in closed form. More precisely, we have a decision algorithm which solves the following problem: if the terms can be represented by an indefinite nested hypergeometric sum expression (covering as special cases the harmonic sums, cyclotomic sums, generalized harmonic sums or nested binomial sums), then we can calculate them. If the algorithm fails, we obtain a proof that the terms cannot be represented by the class of indefinite nested hypergeometric sum expressions. Internally, this problem is reduced by holonomic closure properties to solving a coupled system of linear difference equations. The underlying method in this setting relies on decoupling algorithms, difference ring algorithms and recurrence solving. We demonstrate by a concrete example how this algorithm can be applied with the new Mathematica package SolveCoupledSystem which is based on the packages Sigma, HarmonicSums and OreSys. In all applications the representation in x-space is obtained as an iterated integral representation over general alphabets, generalizing Poincare iterated integrals.

A toolbox to solve coupled systems of differential and difference equations

Energy Technology Data Exchange (ETDEWEB)

Ablinger, Jakob; Schneider, Carsten [Linz Univ. (Austria). Research Inst. for Symbolic Computation (RISC); Bluemlein, Johannes; Freitas, Abilio de [DESY Zeuthen (Germany)

2016-01-15

We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do not request special choices of bases. Here we assume that the desired solution has a power series representation and we seek for the coefficients in closed form. In particular, if the coefficients depend on a small parameter ε (the dimensional parameter), we assume that the coefficients themselves can be expanded in formal Laurent series w.r.t. ε and we try to compute the first terms in closed form. More precisely, we have a decision algorithm which solves the following problem: if the terms can be represented by an indefinite nested hypergeometric sum expression (covering as special cases the harmonic sums, cyclotomic sums, generalized harmonic sums or nested binomial sums), then we can calculate them. If the algorithm fails, we obtain a proof that the terms cannot be represented by the class of indefinite nested hypergeometric sum expressions. Internally, this problem is reduced by holonomic closure properties to solving a coupled system of linear difference equations. The underlying method in this setting relies on decoupling algorithms, difference ring algorithms and recurrence solving. We demonstrate by a concrete example how this algorithm can be applied with the new Mathematica package SolveCoupledSystem which is based on the packages Sigma, HarmonicSums and OreSys. In all applications the representation in x-space is obtained as an iterated integral representation over general alphabets, generalizing Poincare iterated integrals.

The ATOMFT integrator - Using Taylor series to solve ordinary differential equations

Science.gov (United States)

Berryman, Kenneth W.; Stanford, Richard H.; Breckheimer, Peter J.

1988-01-01

This paper discusses the application of ATOMFT, an integration package based on Taylor series solution with a sophisticated user interface. ATOMFT has the capabilities to allow the implementation of user defined functions and the solution of stiff and algebraic equations. Detailed examples, including the solutions to several astrodynamics problems, are presented. Comparisons with its predecessor ATOMCC and other modern integrators indicate that ATOMFT is a fast, accurate, and easy method to use to solve many differential equation problems.

Application of the Generalized Differential Quadrature Method in Solving Burgers' Equations

International Nuclear Information System (INIS)

Mokhtari, R.; Toodar, A. Samadi; Chegini, N.G.

2011-01-01

The aim of this paper is to obtain numerical solutions of the one-dimensional, two-dimensional and coupled Burgers' equations through the generalized differential quadrature method (GDQM). The polynomial-based differential quadrature (PDQ) method is employed and the obtained system of ordinary differential equations is solved via the total variation diminishing Runge-Kutta (TVD-RK) method. The numerical solutions are satisfactorily coincident with the exact solutions. The method can compete against the methods applied in the literature. (general)

A Simple Derivation of Kepler's Laws without Solving Differential Equations

Science.gov (United States)

Provost, J.-P.; Bracco, C.

2009-01-01

Proceeding like Newton with a discrete time approach of motion and a geometrical representation of velocity and acceleration, we obtain Kepler's laws without solving differential equations. The difficult part of Newton's work, when it calls for non-trivial properties of ellipses, is avoided by the introduction of polar coordinates. Then a simple…

Using packaged software for solving two differential equation problems that arise in plasma physics

International Nuclear Information System (INIS)

Gaffney, P.W.

1980-01-01

Experience in using packaged numerical software for solving two related problems that arise in Plasma physics is described. These problems are (i) the solution of the reduced resistive MHD equations and (ii) the solution of the Grad-Shafranov equation

Scilab software as an alternative low-cost computing in solving the linear equations problem

Science.gov (United States)

Agus, Fahrul; Haviluddin

2017-02-01

Numerical computation packages are widely used both in teaching and research. These packages consist of license (proprietary) and open source software (non-proprietary). One of the reasons to use the package is a complexity of mathematics function (i.e., linear problems). Also, number of variables in a linear or non-linear function has been increased. The aim of this paper was to reflect on key aspects related to the method, didactics and creative praxis in the teaching of linear equations in higher education. If implemented, it could be contribute to a better learning in mathematics area (i.e., solving simultaneous linear equations) that essential for future engineers. The focus of this study was to introduce an additional numerical computation package of Scilab as an alternative low-cost computing programming. In this paper, Scilab software was proposed some activities that related to the mathematical models. In this experiment, four numerical methods such as Gaussian Elimination, Gauss-Jordan, Inverse Matrix, and Lower-Upper Decomposition (LU) have been implemented. The results of this study showed that a routine or procedure in numerical methods have been created and explored by using Scilab procedures. Then, the routine of numerical method that could be as a teaching material course has exploited.

Time-Dependent Heat Conduction Problems Solved by an Integral-Equation Approach

International Nuclear Information System (INIS)

Oberaigner, E.R.; Leindl, M.; Antretter, T.

2010-01-01

Full text: A classical task of mathematical physics is the formulation and solution of a time dependent thermoelastic problem. In this work we develop an algorithm for solving the time-dependent heat conduction equation c p ρ∂ t T-kT, ii =0 in an analytical, exact fashion for a two-component domain. By the Green's function approach the formal solution of the problem is obtained. As an intermediate result an integral-equation for the temperature history at the domain interface is formulated which can be solved analytically. This method is applied to a classical engineering problem, i.e. to a special case of a Stefan-Problem. The Green's function approach in conjunction with the integral-equation method is very useful in cases were strong discontinuities or jumps occur. The initial conditions and the system parameters of the investigated problem give rise to two jumps in the temperature field. Purely numerical solutions are obtained by using the FEM (finite element method) and the FDM (finite difference method) and compared with the analytical approach. At the domain boundary the analytical solution and the FEM-solution are in good agreement, but the FDM results show a signicant smearing effect. (author)

Equations of motion of test particles for solving the spin-dependent Boltzmann–Vlasov equation

Energy Technology Data Exchange (ETDEWEB)

Xia, Yin [Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800 (China); University of Chinese Academy of Science, Beijing 100049 (China); Xu, Jun, E-mail: xujun@sinap.ac.cn [Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800 (China); Li, Bao-An [Department of Physics and Astronomy, Texas A& M University-Commerce, Commerce, TX 75429-3011 (United States); Department of Applied Physics, Xi' an Jiao Tong University, Xi' an 710049 (China); Shen, Wen-Qing [Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800 (China)

2016-08-10

A consistent derivation of the equations of motion (EOMs) of test particles for solving the spin-dependent Boltzmann–Vlasov equation is presented. The resulting EOMs in phase space are similar to the canonical equations in Hamiltonian dynamics, and the EOM of spin is the same as that in the Heisenburg picture of quantum mechanics. Considering further the quantum nature of spin and choosing the direction of total angular momentum in heavy-ion reactions as a reference of measuring nucleon spin, the EOMs of spin-up and spin-down nucleons are given separately. The key elements affecting the spin dynamics in heavy-ion collisions are identified. The resulting EOMs provide a solid foundation for using the test-particle approach in studying spin dynamics in heavy-ion collisions at intermediate energies. Future comparisons of model simulations with experimental data will help to constrain the poorly known in-medium nucleon spin–orbit coupling relevant for understanding properties of rare isotopes and their astrophysical impacts.

New numerical method for solving the solute transport equation

International Nuclear Information System (INIS)

Ross, B.; Koplik, C.M.

1978-01-01

The solute transport equation can be solved numerically by approximating the water flow field by a network of stream tubes and using a Green's function solution within each stream tube. Compared to previous methods, this approach permits greater computational efficiency and easier representation of small discontinuities, and the results are easier to interpret physically. The method has been used to study hypothetical sites for disposal of high-level radioactive waste

Theory of Perturbed Equilibria for Solving the Grad-Shafranov Equation

International Nuclear Information System (INIS)

Pletzer, A.; Zakharov, L.E.

1999-01-01

The theory of perturbed magnetohydrodynamic equilibria is presented for different formulations of the tokamak equilibrium problem. For numerical codes, it gives an explicit Newton scheme for solving the Grad-Shafranov equation subject to different constraints. The problem of stability of axisymmetric modes is shown to be a particular case of the equilibrium perturbation theory

Insights into the School Mathematics Tradition from Solving Linear Equations

Science.gov (United States)

Buchbinder, Orly; Chazan, Daniel; Fleming, Elizabeth

2015-01-01

In this article, we explore how the solving of linear equations is represented in English­-language algebra text books from the early nineteenth century when schooling was becoming institutionalized, and then survey contemporary teachers. In the text books, we identify the increasing presence of a prescribed order of steps (a canonical method) for…

Solving Nonlinear Fractional Differential Equation by Generalized Mittag-Leffler Function Method

Science.gov (United States)

Arafa, A. A. M.; Rida, S. Z.; Mohammadein, A. A.; Ali, H. M.

2013-06-01

In this paper, we use Mittag—Leffler function method for solving some nonlinear fractional differential equations. A new solution is constructed in power series. The fractional derivatives are described by Caputo's sense. To illustrate the reliability of the method, some examples are provided.

A pertinent approach to solve nonlinear fuzzy integro-differential equations.

Science.gov (United States)

Narayanamoorthy, S; Sathiyapriya, S P

2016-01-01

Fuzzy integro-differential equations is one of the important parts of fuzzy analysis theory that holds theoretical as well as applicable values in analytical dynamics and so an appropriate computational algorithm to solve them is in essence. In this article, we use parametric forms of fuzzy numbers and suggest an applicable approach for solving nonlinear fuzzy integro-differential equations using homotopy perturbation method. A clear and detailed description of the proposed method is provided. Our main objective is to illustrate that the construction of appropriate convex homotopy in a proper way leads to highly accurate solutions with less computational work. The efficiency of the approximation technique is expressed via stability and convergence analysis so as to guarantee the efficiency and performance of the methodology. Numerical examples are demonstrated to verify the convergence and it reveals the validity of the presented numerical technique. Numerical results are tabulated and examined by comparing the obtained approximate solutions with the known exact solutions. Graphical representations of the exact and acquired approximate fuzzy solutions clarify the accuracy of the approach.

A Fortran program (RELAX3D) to solve the 3 dimensional Poisson (Laplace) equation

International Nuclear Information System (INIS)

Houtman, H.; Kost, C.J.

1983-09-01

RELAX3D is an efficient, user friendly, interactive FORTRAN program which solves the Poisson (Laplace) equation Λ 2 =p for a general 3 dimensional geometry consisting of Dirichlet and Neumann boundaries approximated to lie on a regular 3 dimensional mesh. The finite difference equations at these nodes are solved using a successive point-iterative over-relaxation method. A menu of commands, supplemented by HELP facility, controls the dynamic loading of the subroutine describing the problem case, the iterations to converge to a solution, and the contour plotting of any desired slices, etc

Solving Differential Equations Analytically. Elementary Differential Equations. Modules and Monographs in Undergraduate Mathematics and Its Applications Project. UMAP Unit 335.

Science.gov (United States)

Goldston, J. W.

This unit introduces analytic solutions of ordinary differential equations. The objective is to enable the student to decide whether a given function solves a given differential equation. Examples of problems from biology and chemistry are covered. Problem sets, quizzes, and a model exam are included, and answers to all items are provided. The…

Imaginary Time Step Method to Solve the Dirac Equation with Nonlocal Potential

International Nuclear Information System (INIS)

Zhang Ying; Liang Haozhao; Meng Jie

2009-01-01

The imaginary time step (ITS) method is applied to solve the Dirac equation with nonlocal potentials in coordinate space. Taking the nucleus 12 C as an example, even with nonlocal potentials, the direct ITS evolution for the Dirac equation still meets the disaster of the Dirac sea. However, following the recipe in our former investigation, the disaster can be avoided by the ITS evolution for the corresponding Schroedinger-like equation without localization, which gives the convergent results exactly the same with those obtained iteratively by the shooting method with localized effective potentials.

A rapid numerical method for solving Serre-Green-Naghdi equations describing long free surface gravity waves

Science.gov (United States)

Favrie, N.; Gavrilyuk, S.

2017-07-01

A new numerical method for solving the Serre-Green-Naghdi (SGN) equations describing dispersive waves on shallow water is proposed. From the mathematical point of view, the SGN equations are the Euler-Lagrange equations for a ‘master’ lagrangian submitted to a differential constraint which is the mass conservation law. One major numerical challenge in solving the SGN equations is the resolution of an elliptic problem at each time instant. This is the most time-consuming part of the numerical method. The idea is to replace the ‘master’ lagrangian by a one-parameter family of ‘augmented’ lagrangians, depending on a greater number of variables, for which the corresponding Euler-Lagrange equations are hyperbolic. In such an approach, the ‘master’ lagrangian is recovered by the augmented lagrangian in some limit (for example, when the corresponding parameter is large). The choice of such a family of augmented lagrangians is proposed and discussed. The corresponding hyperbolic system is numerically solved by a Godunov type method. Numerical solutions are compared with exact solutions to the SGN equations. It appears that the computational time in solving the hyperbolic system is much lower than in the case where the elliptic operator is inverted. The new method is applied, in particular, to the study of ‘Favre waves’ representing non-stationary undular bores produced after reflection of the fluid flow with a free surface at an immobile wall.

Solving Second-Order Ordinary Differential Equations without Using Complex Numbers

Science.gov (United States)

Kougias, Ioannis E.

2009-01-01

Ordinary differential equations (ODEs) is a subject with a wide range of applications and the need of introducing it to students often arises in the last year of high school, as well as in the early stages of tertiary education. The usual methods of solving second-order ODEs with constant coefficients, among others, rely upon the use of complex…

On the Efficiency of Algorithms for Solving Hartree–Fock and Kohn–Sham Response Equations

DEFF Research Database (Denmark)

Kauczor, Joanna; Jørgensen, Poul; Norman, Patrick

2011-01-01

The response equations as occurring in the Hartree–Fock, multiconfigurational self-consistent field, and Kohn–Sham density functional theory have identical matrix structures. The algorithms that are used for solving these equations are discussed, and new algorithms are proposed where trial vectors...

The Convergence Study of the Homotopy Analysis Method for Solving Nonlinear Volterra-Fredholm Integrodifferential Equations

Directory of Open Access Journals (Sweden)

Behzad Ghanbari

2014-01-01

Full Text Available We aim to study the convergence of the homotopy analysis method (HAM in short for solving special nonlinear Volterra-Fredholm integrodifferential equations. The sufficient condition for the convergence of the method is briefly addressed. Some illustrative examples are also presented to demonstrate the validity and applicability of the technique. Comparison of the obtained results HAM with exact solution shows that the method is reliable and capable of providing analytic treatment for solving such equations.

New Efficient Fourth Order Method for Solving Nonlinear Equations

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Farooq Ahmad

2013-12-01

Full Text Available In a paper [Appl. Math. Comput., 188 (2 (2007 1587--1591], authors have suggested and analyzed a method for solving nonlinear equations. In the present work, we modified this method by using the finite difference scheme, which has a quintic convergence. We have compared this modified Halley method with some other iterative of fifth-orders convergence methods, which shows that this new method having convergence of fourth order, is efficient.

A parallel algorithm for solving the integral form of the discrete ordinates equations

International Nuclear Information System (INIS)

Zerr, R. J.; Azmy, Y. Y.

2009-01-01

The integral form of the discrete ordinates equations involves a system of equations that has a large, dense coefficient matrix. The serial construction methodology is presented and properties that affect the execution times to construct and solve the system are evaluated. Two approaches for massively parallel implementation of the solution algorithm are proposed and the current results of one of these are presented. The system of equations May be solved using two parallel solvers-block Jacobi and conjugate gradient. Results indicate that both methods can reduce overall wall-clock time for execution. The conjugate gradient solver exhibits better performance to compete with the traditional source iteration technique in terms of execution time and scalability. The parallel conjugate gradient method is synchronous, hence it does not increase the number of iterations for convergence compared to serial execution, and the efficiency of the algorithm demonstrates an apparent asymptotic decline. (authors)

New extended (G'/G)-expansion method to solve nonlinear evolution equation: the (3 + 1)-dimensional potential-YTSF equation.

Science.gov (United States)

Roshid, Harun-Or-; Akbar, M Ali; Alam, Md Nur; Hoque, Md Fazlul; Rahman, Nizhum

2014-01-01

In this article, a new extended (G'/G) -expansion method has been proposed for constructing more general exact traveling wave solutions of nonlinear evolution equations with the aid of symbolic computation. In order to illustrate the validity and effectiveness of the method, we pick the (3 + 1)-dimensional potential-YTSF equation. As a result, abundant new and more general exact solutions have been achieved of this equation. It has been shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in applied mathematics, engineering and mathematical physics.

Numerical evaluation of the intensity transport equation for well-known wavefronts and intensity distributions

Science.gov (United States)

Campos-García, Manuel; Granados-Agustín, Fermín.; Cornejo-Rodríguez, Alejandro; Estrada-Molina, Amilcar; Avendaño-Alejo, Maximino; Moreno-Oliva, Víctor Iván.

2013-11-01

In order to obtain a clearer interpretation of the Intensity Transport Equation (ITE), in this work, we propose an algorithm to solve it for some particular wavefronts and its corresponding intensity distributions. By simulating intensity distributions in some planes, the ITE is turns into a Poisson equation with Neumann boundary conditions. The Poisson equation is solved by means of the iterative algorithm SOR (Simultaneous Over-Relaxation).

A highly accurate finite-difference method with minimum dispersion error for solving the Helmholtz equation

KAUST Repository

Wu, Zedong

2018-04-05

Numerical simulation of the acoustic wave equation in either isotropic or anisotropic media is crucial to seismic modeling, imaging and inversion. Actually, it represents the core computation cost of these highly advanced seismic processing methods. However, the conventional finite-difference method suffers from severe numerical dispersion errors and S-wave artifacts when solving the acoustic wave equation for anisotropic media. We propose a method to obtain the finite-difference coefficients by comparing its numerical dispersion with the exact form. We find the optimal finite difference coefficients that share the dispersion characteristics of the exact equation with minimal dispersion error. The method is extended to solve the acoustic wave equation in transversely isotropic (TI) media without S-wave artifacts. Numerical examples show that the method is is highly accurate and efficient.

Solving Kepler's equation using implicit functions

Science.gov (United States)

Mortari, Daniele; Elipe, Antonio

2014-01-01

A new approach to solve Kepler's equation based on the use of implicit functions is proposed here. First, new upper and lower bounds are derived for two ranges of mean anomaly. These upper and lower bounds initialize a two-step procedure involving the solution of two implicit functions. These two implicit functions, which are non-rational (polynomial) Bézier functions, can be linear or quadratic, depending on the derivatives of the initial bound values. These are new initial bounds that have been compared and proven more accurate than Serafin's bounds. The procedure reaches machine error accuracy with no more that one quadratic and one linear iterations, experienced in the "tough range", where the eccentricity is close to one and the mean anomaly to zero. The proposed method is particularly suitable for space-based applications with limited computational capability.

Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets

Directory of Open Access Journals (Sweden)

Ai-Min Yang

2013-01-01

Full Text Available We proposed a local fractional series expansion method to solve the wave and diffusion equations on Cantor sets. Some examples are given to illustrate the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations within the local fractional derivatives.

Modeling Blazar Spectra by Solving an Electron Transport Equation

Science.gov (United States)

Lewis, Tiffany; Finke, Justin; Becker, Peter A.

2018-01-01

Blazars are luminous active galaxies across the entire electromagnetic spectrum, but the spectral formation mechanisms, especially the particle acceleration, in these sources are not well understood. We develop a new theoretical model for simulating blazar spectra using a self-consistent electron number distribution. Specifically, we solve the particle transport equation considering shock acceleration, adiabatic expansion, stochastic acceleration due to MHD waves, Bohm diffusive particle escape, synchrotron radiation, and Compton radiation, where we implement the full Compton cross-section for seed photons from the accretion disk, the dust torus, and 26 individual broad lines. We used a modified Runge-Kutta method to solve the 2nd order equation, including development of a new mathematical method for normalizing stiff steady-state ordinary differential equations. We show that our self-consistent, transport-based blazar model can qualitatively fit the IR through Fermi g-ray data for 3C 279, with a single-zone, leptonic configuration. We use the solution for the electron distribution to calculate multi-wavelength SED spectra for 3C 279. We calculate the particle and magnetic field energy densities, which suggest that the emitting region is not always in equipartition (a common assumption), but sometimes matter dominated. The stratified broad line region (based on ratios in quasar reverberation mapping, and thus adding no free parameters) improves our estimate of the location of the emitting region, increasing it by ~5x. Our model provides a novel view into the physics at play in blazar jets, especially the relative strength of the shock and stochastic acceleration, where our model is well suited to distinguish between these processes, and we find that the latter tends to dominate.

Solving the transport equation with quadratic finite elements: Theory and applications

International Nuclear Information System (INIS)

Ferguson, J.M.

1997-01-01

At the 4th Joint Conference on Computational Mathematics, the author presented a paper introducing a new quadratic finite element scheme (QFEM) for solving the transport equation. In the ensuing year the author has obtained considerable experience in the application of this method, including solution of eigenvalue problems, transmission problems, and solution of the adjoint form of the equation as well as the usual forward solution. He will present detailed results, and will also discuss other refinements of his transport codes, particularly for 3-dimensional problems on rectilinear and non-rectilinear grids

Recent advances in marching-on-in-time schemes for solving time domain volume integral equations

KAUST Repository

Sayed, Sadeed Bin; Ulku, Huseyin Arda; Bagci, Hakan

2015-01-01

Transient electromagnetic field interactions on inhomogeneous penetrable scatterers can be analyzed by solving time domain volume integral equations (TDVIEs). TDVIEs are constructed by setting the summation of the incident and scattered field intensities to the total field intensity on the volumetric support of the scatterer. The unknown can be the field intensity or flux/current density. Representing the total field intensity in terms of the unknown using the relevant constitutive relation and the scattered field intensity in terms of the spatiotemporal convolution of the unknown with the Green function yield the final form of the TDVIE. The unknown is expanded in terms of local spatial and temporal basis functions. Inserting this expansion into the TDVIE and testing the resulting equation at discrete times yield a system of equations that is solved by the marching on-in-time (MOT) scheme. At each time step, a smaller system of equations, termed MOT system is solved for the coefficients of the expansion. The right-hand side of this system consists of the tested incident field and discretized spatio-temporal convolution of the unknown samples computed at the previous time steps with the Green function.

Recent advances in marching-on-in-time schemes for solving time domain volume integral equations

KAUST Repository

Sayed, Sadeed Bin

2015-05-16

Transient electromagnetic field interactions on inhomogeneous penetrable scatterers can be analyzed by solving time domain volume integral equations (TDVIEs). TDVIEs are constructed by setting the summation of the incident and scattered field intensities to the total field intensity on the volumetric support of the scatterer. The unknown can be the field intensity or flux/current density. Representing the total field intensity in terms of the unknown using the relevant constitutive relation and the scattered field intensity in terms of the spatiotemporal convolution of the unknown with the Green function yield the final form of the TDVIE. The unknown is expanded in terms of local spatial and temporal basis functions. Inserting this expansion into the TDVIE and testing the resulting equation at discrete times yield a system of equations that is solved by the marching on-in-time (MOT) scheme. At each time step, a smaller system of equations, termed MOT system is solved for the coefficients of the expansion. The right-hand side of this system consists of the tested incident field and discretized spatio-temporal convolution of the unknown samples computed at the previous time steps with the Green function.

Numerical method for solving linear Fredholm fuzzy integral equations of the second kind

Energy Technology Data Exchange (ETDEWEB)

Abbasbandy, S. [Department of Mathematics, Imam Khomeini International University, P.O. Box 288, Ghazvin 34194 (Iran, Islamic Republic of)]. E-mail: saeid@abbasbandy.com; Babolian, E. [Faculty of Mathematical Sciences and Computer Engineering, Teacher Training University, Tehran 15618 (Iran, Islamic Republic of); Alavi, M. [Department of Mathematics, Arak Branch, Islamic Azad University, Arak 38135 (Iran, Islamic Republic of)

2007-01-15

In this paper we use parametric form of fuzzy number and convert a linear fuzzy Fredholm integral equation to two linear system of integral equation of the second kind in crisp case. We can use one of the numerical method such as Nystrom and find the approximation solution of the system and hence obtain an approximation for fuzzy solution of the linear fuzzy Fredholm integral equations of the second kind. The proposed method is illustrated by solving some numerical examples.

Performance and Difficulties of Students in Formulating and Solving Quadratic Equations with One Unknown

Science.gov (United States)

Didis, Makbule Gozde; Erbas, Ayhan Kursat

2015-01-01

This study attempts to investigate the performance of tenth-grade students in solving quadratic equations with one unknown, using symbolic equation and word-problem representations. The participants were 217 tenth-grade students, from three different public high schools. Data was collected through an open-ended questionnaire comprising eight…

Solving Some Special Cases of Monomial Ratio Equations Appearing Frequently in Physical and Engineering Problems

Directory of Open Access Journals (Sweden)

Enrique Castillo

2016-01-01

Full Text Available We first show that monomial ratio equations are not only very common in Physics and Engineering, but the natural type of equations in many practical problems. More precisely, in the case of models involving scale variables if the used formulas are not of this type they are not physically valid. The consequence is that when estimating the model parameters we are faced with systems of monomial ratio equations that are nonlinear and difficult to solve. In this paper, we provide an original algorithm to obtain the unique solutions of systems of equations made of linear combinations of monomial ratios whose coefficient matrix has a proper null space with low dimension that permits solving the problem in a simple way. Finally, we illustrate the proposed methods by their application to two practical problems from the hydraulic and structural fields.

A Method for Solving the Voltage and Torque Equations of the Split-Phase Induction Machines

Directory of Open Access Journals (Sweden)

G. A. Olarinoye

2013-06-01

Full Text Available Single phase induction machines have been the subject of many researches in recent times. The voltage and torque equations which describe the dynamic characteristics of these machines have been quoted in many papers, including the papers that present the simulation results of these model equations. The way and manner in which these equations are solved is not common in literature. This paper presents a detailed procedure of how these equations are to be solved with respect to the splitphase induction machine which is one of the different types of the single phase induction machines available in the market. In addition, these equations have been used to simulate the start-up response of the split phase induction motor on no-load. The free acceleration characteristics of the motor voltages, currents and electromagnetic torque have been plotted and discussed. The simulation results presented include the instantaneous torque-speed characteristics of the Split phase Induction machine. A block diagram of the method for the solution of the machine equations has also been presented.

A universal concept based on cellular neural networks for ultrafast and flexible solving of differential equations.

Science.gov (United States)

Chedjou, Jean Chamberlain; Kyamakya, Kyandoghere

2015-04-01

This paper develops and validates a comprehensive and universally applicable computational concept for solving nonlinear differential equations (NDEs) through a neurocomputing concept based on cellular neural networks (CNNs). High-precision, stability, convergence, and lowest-possible memory requirements are ensured by the CNN processor architecture. A significant challenge solved in this paper is that all these cited computing features are ensured in all system-states (regular or chaotic ones) and in all bifurcation conditions that may be experienced by NDEs.One particular quintessence of this paper is to develop and demonstrate a solver concept that shows and ensures that CNN processors (realized either in hardware or in software) are universal solvers of NDE models. The solving logic or algorithm of given NDEs (possible examples are: Duffing, Mathieu, Van der Pol, Jerk, Chua, Rössler, Lorenz, Burgers, and the transport equations) through a CNN processor system is provided by a set of templates that are computed by our comprehensive templates calculation technique that we call nonlinear adaptive optimization. This paper is therefore a significant contribution and represents a cutting-edge real-time computational engineering approach, especially while considering the various scientific and engineering applications of this ultrafast, energy-and-memory-efficient, and high-precise NDE solver concept. For illustration purposes, three NDE models are demonstratively solved, and related CNN templates are derived and used: the periodically excited Duffing equation, the Mathieu equation, and the transport equation.

The spectral transform as a tool for solving nonlinear discrete evolution equations

International Nuclear Information System (INIS)

Levi, D.

1979-01-01

In this contribution we study nonlinear differential difference equations which became important to the description of an increasing number of problems in natural science. Difference equations arise for instance in the study of electrical networks, in statistical problems, in queueing problems, in ecological problems, as computer models for differential equations and as models for wave excitation in plasma or vibrations of particles in an anharmonic lattice. We shall first review the passages necessary to solve linear discrete evolution equations by the discrete Fourier transfrom, then, starting from the Zakharov-Shabat discretized eigenvalue, problem, we shall introduce the spectral transform. In the following part we obtain the correlation between the evolution of the potentials and scattering data through the Wronskian technique, giving at the same time many other properties as, for example, the Baecklund transformations. Finally we recover some of the important equations belonging to this class of nonlinear discrete evolution equations and extend the method to equations with n-dependent coefficients. (HJ)

Hermite Functional Link Neural Network for Solving the Van der Pol-Duffing Oscillator Equation.

Science.gov (United States)

Mall, Susmita; Chakraverty, S

2016-08-01

Hermite polynomial-based functional link artificial neural network (FLANN) is proposed here to solve the Van der Pol-Duffing oscillator equation. A single-layer hermite neural network (HeNN) model is used, where a hidden layer is replaced by expansion block of input pattern using Hermite orthogonal polynomials. A feedforward neural network model with the unsupervised error backpropagation principle is used for modifying the network parameters and minimizing the computed error function. The Van der Pol-Duffing and Duffing oscillator equations may not be solved exactly. Here, approximate solutions of these types of equations have been obtained by applying the HeNN model for the first time. Three mathematical example problems and two real-life application problems of Van der Pol-Duffing oscillator equation, extracting the features of early mechanical failure signal and weak signal detection problems, are solved using the proposed HeNN method. HeNN approximate solutions have been compared with results obtained by the well known Runge-Kutta method. Computed results are depicted in term of graphs. After training the HeNN model, we may use it as a black box to get numerical results at any arbitrary point in the domain. Thus, the proposed HeNN method is efficient. The results reveal that this method is reliable and can be applied to other nonlinear problems too.

Guiding brine shrimp through mazes by solving reaction diffusion equations

Science.gov (United States)

Singal, Krishma; Fenton, Flavio

Excitable systems driven by reaction diffusion equations have been shown to not only find solutions to mazes but to also to find the shortest path between the beginning and the end of the maze. In this talk we describe how we can use the Fitzhugh-Nagumo model, a generic model for excitable media, to solve a maze by varying the basin of attraction of its two fixed points. We demonstrate how two dimensional mazes are solved numerically using a Java Applet and then accelerated to run in real time by using graphic processors (GPUs). An application of this work is shown by guiding phototactic brine shrimp through a maze solved by the algorithm. Once the path is obtained, an Arduino directs the shrimp through the maze using lights from LEDs placed at the floor of the Maze. This method running in real time could be eventually used for guiding robots and cars through traffic.

Deterministic methods to solve the integral transport equation in neutronic

International Nuclear Information System (INIS)

Warin, X.

1993-11-01

We present a synthesis of the methods used to solve the integral transport equation in neutronic. This formulation is above all used to compute solutions in 2D in heterogeneous assemblies. Three kinds of methods are described: - the collision probability method; - the interface current method; - the current coupling collision probability method. These methods don't seem to be the most effective in 3D. (author). 9 figs

On Newton-Kantorovich Method for Solving the Nonlinear Operator Equation

Directory of Open Access Journals (Sweden)

Hameed Husam Hameed

2015-01-01

Full Text Available We develop the Newton-Kantorovich method to solve the system of 2×2 nonlinear Volterra integral equations where the unknown function is in logarithmic form. A new majorant function is introduced which leads to the increment of the convergence interval. The existence and uniqueness of approximate solution are proved and a numerical example is provided to show the validation of the method.

A mixed Fourier–Galerkin–finite-volume method to solve the fluid dynamics equations in cylindrical geometries

International Nuclear Information System (INIS)

Núñez, Jóse; Ramos, Eduardo; Lopez, Juan M

2012-01-01

We describe a hybrid method based on the combined use of the Fourier Galerkin and finite-volume techniques to solve the fluid dynamics equations in cylindrical geometries. A Fourier expansion is used in the angular direction, partially translating the problem to the Fourier space and then solving the resulting equations using a finite-volume technique. We also describe an algorithm required to solve the coupled mass and momentum conservation equations similar to a pressure-correction SIMPLE method that is adapted for the present formulation. Using the Fourier–Galerkin method for the azimuthal direction has two advantages. Firstly, it has a high-order approximation of the partial derivatives in the angular direction, and secondly, it naturally satisfies the azimuthal periodic boundary conditions. Also, using the finite-volume method in the r and z directions allows one to handle boundary conditions with discontinuities in those directions. It is important to remark that with this method, the resulting linear system of equations are band-diagonal, leading to fast and efficient solvers. The benefits of the mixed method are illustrated with example problems. (paper)

Numerics made easy: solving the Navier-Stokes equation for arbitrary channel cross-sections using Microsoft Excel.

Science.gov (United States)

Richter, Christiane; Kotz, Frederik; Giselbrecht, Stefan; Helmer, Dorothea; Rapp, Bastian E

2016-06-01

The fluid mechanics of microfluidics is distinctively simpler than the fluid mechanics of macroscopic systems. In macroscopic systems effects such as non-laminar flow, convection, gravity etc. need to be accounted for all of which can usually be neglected in microfluidic systems. Still, there exists only a very limited selection of channel cross-sections for which the Navier-Stokes equation for pressure-driven Poiseuille flow can be solved analytically. From these equations, velocity profiles as well as flow rates can be calculated. However, whenever a cross-section is not highly symmetric (rectangular, elliptical or circular) the Navier-Stokes equation can usually not be solved analytically. In all of these cases, numerical methods are required. However, in many instances it is not necessary to turn to complex numerical solver packages for deriving, e.g., the velocity profile of a more complex microfluidic channel cross-section. In this paper, a simple spreadsheet analysis tool (here: Microsoft Excel) will be used to implement a simple numerical scheme which allows solving the Navier-Stokes equation for arbitrary channel cross-sections.

Domain decomposition method for solving the neutron diffusion equation

International Nuclear Information System (INIS)

Coulomb, F.

1989-03-01

The aim of this work is to study methods for solving the neutron diffusion equation; we are interested in methods based on a classical finite element discretization and well suited for use on parallel computers. Domain decomposition methods seem to answer this preoccupation. This study deals with a decomposition of the domain. A theoretical study is carried out for Lagrange finite elements and some examples are given; in the case of mixed dual finite elements, the study is based on examples [fr

A simple derivation of Kepler's laws without solving differential equations

International Nuclear Information System (INIS)

Provost, J-P; Bracco, C

2009-01-01

Proceeding like Newton with a discrete time approach of motion and a geometrical representation of velocity and acceleration, we obtain Kepler's laws without solving differential equations. The difficult part of Newton's work, when it calls for non-trivial properties of ellipses, is avoided by the introduction of polar coordinates. Then a simple reconsideration of Newton's figure naturally leads to an explicit expression of the velocity and to the equation of the trajectory. This derivation, which can be fully apprehended by undergraduates or by secondary school teachers (who might use it with their pupils), can be considered as a first application of mechanical concepts to a physical problem of great historical and pedagogical interest

Set of difference spitting schemes for solving the Navier-Stokes incompressible equations in natural variables

International Nuclear Information System (INIS)

Koleshko, S.B.

1989-01-01

A three-parametric set of difference schemes is suggested to solve Navier-Stokes equations with the use of the relaxation form of the continuity equation. The initial equations are stated for time increments. Use is made of splitting the operator into one-dimensional forms that reduce calculations to scalar factorizations. Calculated results for steady- and unsteady-state flows in a cavity are presented

ADI as a preconditioning for solving the convection-diffusion equation

International Nuclear Information System (INIS)

Chin, R.C.Y.; Manteuffel, T.A.; De Pillis, J.

1984-01-01

The authors examine a splitting of the operator obtained from a steady convection-diffusion equation with variable coefficients in which the convection term dominates. The operator is split into a dominant and subdominant parts consistent with the inherent directional property of the partial differential equation. The equations involving the dominant parts should be easily solved. The authors accelerate the convergence of this splitting or the outer iteration by a Chebyshev semi-iterative method. When the dominant part has constant coefficients, it can be easily solved using alternating direction implicit (ADI) methods. This is called the inner iteration. The optimal parameters for a stationary two-parameter ADI method are obtained when the eigenvalues become complex. This corresponds to either the horizontal or the vertical half-grid Reynolds number larger than unity. The Chebyshev semi-iterative method is used to accelerate the convergence of the inner ADI iteration. A two-fold increase in speed is obtained when the ADI iteration matrix has real eigenvalues, and the increase is less significant when the eigenvalues are complex. If either the horizontal or the vertical half-grid Reynolds number is equal to one, the spectral radius of the optimal ADI iterative matrix is zero. However, a high degree of nilpotency impairs rapid convergence. This problem is removed by introducing a more implicit iterative method called ADI/Gauss-Seidel (ADI/GS). ADI/GS resolves the nilpotency and, thus, converges more rapidly for half-grid Reynolds number near 1. Finally, these methods are compared with several well-known schemes on test problems. 23 references, 5 figures

A Four-Stage Fifth-Order Trigonometrically Fitted Semi-Implicit Hybrid Method for Solving Second-Order Delay Differential Equations

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Sufia Zulfa Ahmad

2016-01-01

Full Text Available We derived a two-step, four-stage, and fifth-order semi-implicit hybrid method which can be used for solving special second-order ordinary differential equations. The method is then trigonometrically fitted so that it is suitable for solving problems which are oscillatory in nature. The methods are then used for solving oscillatory delay differential equations. Numerical results clearly show the efficiency of the new method when compared to the existing explicit and implicit methods in the scientific literature.

Formulae of differentiation for solving differential equations with complex-valued random coefficients

International Nuclear Information System (INIS)

Kim, Ki Hong; Lee, Dong Hun

1999-01-01

Generalizing the work of Shapiro and Loginov, we derive new formulae of differentiation useful for solving differential equations with complex-valued random coefficients. We apply the formulae to the quantum-mechanical problem of noninteracting electrons moving in a correlated random potential in one dimension

An Efficient Numerical Approach for Solving Nonlinear Coupled Hyperbolic Partial Differential Equations with Nonlocal Conditions

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A. H. Bhrawy

2014-01-01

Full Text Available One of the most important advantages of collocation method is the possibility of dealing with nonlinear partial differential equations (PDEs as well as PDEs with variable coefficients. A numerical solution based on a Jacobi collocation method is extended to solve nonlinear coupled hyperbolic PDEs with variable coefficients subject to initial-boundary nonlocal conservation conditions. This approach, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled hyperbolic PDEs with variable coefficients to a system of nonlinear ordinary differential equation which is far easier to solve. In fact, we deal with initial-boundary coupled hyperbolic PDEs with variable coefficients as well as initial-nonlocal conditions. Using triangular, soliton, and exponential-triangular solutions as exact solutions, the obtained results show that the proposed numerical algorithm is efficient and very accurate.

Simultaneous spectrophotometric determination of tungsten and molybdenum with dithiol

International Nuclear Information System (INIS)

Navale, A.S.

1987-01-01

Toluene-3,4-dithiol is a very sensitive reagent for the spectrophotometric determination of tungsten and molybdenum. Since the absorption spectra of the dithiol complexes of these two elements overlap, a separation of the two elements is carried out. This leads to time consuming extraction procedures. Measuring the absorption of the mixed complexes at two wavelengths and solving a set of simultaneous equations is also not favorable because a lot of time and effort is required for solving the simultaneous equations for each sample. A faster and simpler method is described here for the simultaneous determination of the two elements. The method is based on measurement of absorbance of the mixed complexes at three pre-selected wavelengths and simple calculations involving the absorbance differences. The criteria for selecting the three wavelengths and the theory are described. Application of the method for the determination of tungsten and molybdenum in ore samples is presented. The method is applicable to any similar system consisting of two interfering components. 4 figures, 2 tables, 6 refs. (author)

Solving inverse problems for biological models using the collage method for differential equations.

Science.gov (United States)

Capasso, V; Kunze, H E; La Torre, D; Vrscay, E R

2013-07-01

In the first part of this paper we show how inverse problems for differential equations can be solved using the so-called collage method. Inverse problems can be solved by minimizing the collage distance in an appropriate metric space. We then provide several numerical examples in mathematical biology. We consider applications of this approach to the following areas: population dynamics, mRNA and protein concentration, bacteria and amoeba cells interaction, tumor growth.

Derivative free Davidon-Fletcher-Powell (DFP) for solving symmetric systems of nonlinear equations

Science.gov (United States)

Mamat, M.; Dauda, M. K.; Mohamed, M. A. bin; Waziri, M. Y.; Mohamad, F. S.; Abdullah, H.

2018-03-01

Research from the work of engineers, economist, modelling, industry, computing, and scientist are mostly nonlinear equations in nature. Numerical solution to such systems is widely applied in those areas of mathematics. Over the years, there has been significant theoretical study to develop methods for solving such systems, despite these efforts, unfortunately the methods developed do have deficiency. In a contribution to solve systems of the form F(x) = 0, x ∈ Rn , a derivative free method via the classical Davidon-Fletcher-Powell (DFP) update is presented. This is achieved by simply approximating the inverse Hessian matrix with {Q}k+1-1 to θkI. The modified method satisfied the descent condition and possess local superlinear convergence properties. Interestingly, without computing any derivative, the proposed method never fail to converge throughout the numerical experiments. The output is based on number of iterations and CPU time, different initial starting points were used on a solve 40 benchmark test problems. With the aid of the squared norm merit function and derivative-free line search technique, the approach yield a method of solving symmetric systems of nonlinear equations that is capable of significantly reducing the CPU time and number of iteration, as compared to its counterparts. A comparison between the proposed method and classical DFP update were made and found that the proposed methodis the top performer and outperformed the existing method in almost all the cases. In terms of number of iterations, out of the 40 problems solved, the proposed method solved 38 successfully, (95%) while classical DFP solved 2 problems (i.e. 05%). In terms of CPU time, the proposed method solved 29 out of the 40 problems given, (i.e.72.5%) successfully whereas classical DFP solves 11 (27.5%). The method is valid in terms of derivation, reliable in terms of number of iterations and accurate in terms of CPU time. Thus, suitable and achived the objective.

Solving Coupled Gross--Pitaevskii Equations on a Cluster of PlayStation 3 Computers

Science.gov (United States)

Edwards, Mark; Heward, Jeffrey; Clark, C. W.

2009-05-01

At Georgia Southern University we have constructed an 8+1--node cluster of Sony PlayStation 3 (PS3) computers with the intention of using this computing resource to solve problems related to the behavior of ultra--cold atoms in general with a particular emphasis on studying bose--bose and bose--fermi mixtures confined in optical lattices. As a first project that uses this computing resource, we have implemented a parallel solver of the coupled time--dependent, one--dimensional Gross--Pitaevskii (TDGP) equations. These equations govern the behavior of dual-- species bosonic mixtures. We chose the split--operator/FFT to solve the coupled 1D TDGP equations. The fast Fourier transform component of this solver can be readily parallelized on the PS3 cpu known as the Cell Broadband Engine (CellBE). Each CellBE chip contains a single 64--bit PowerPC Processor Element known as the PPE and eight ``Synergistic Processor Element'' identified as the SPE's. We report on this algorithm and compare its performance to a non--parallel solver as applied to modeling evaporative cooling in dual--species bosonic mixtures.

Simultaneous baldness and cosmic baldness and the Kottler spacetime

Science.gov (United States)

Faraoni, Valerio; Cardini, Adriana M.; Chung, Wen-Jian

2018-01-01

The uniqueness of the Kottler/Schwarzschild-de Sitter solution (KSdS) of the vacuum Einstein equations with positive cosmological constant is discussed and certain putative alternatives are shown to either solve different equations or to be the KSdS solution in disguise. A simultaneous no-hair and cosmic no-hair theorem for the KSdS geometry in the presence of an imperfect fluid is proved.

Application of discontinuous Galerkin method for solving a compressible five-equation two-phase flow model

Science.gov (United States)

Saleem, M. Rehan; Ali, Ishtiaq; Qamar, Shamsul

2018-03-01

In this article, a reduced five-equation two-phase flow model is numerically investigated. The formulation of the model is based on the conservation and energy exchange laws. The model is non-conservative and the governing equations contain two equations for the mass conservation, one for the over all momentum and one for the total energy. The fifth equation is the energy equation for one of the two phases that includes a source term on the right hand side for incorporating energy exchange between the two fluids in the form of mechanical and thermodynamical works. A Runge-Kutta discontinuous Galerkin finite element method is applied to solve the model equations. The main attractive features of the proposed method include its formal higher order accuracy, its nonlinear stability, its ability to handle complicated geometries, and its ability to capture sharp discontinuities or strong gradients in the solutions without producing spurious oscillations. The proposed method is robust and well suited for large-scale time-dependent computational problems. Several case studies of two-phase flows are presented. For validation and comparison of the results, the same model equations are also solved by using a staggered central scheme. It was found that discontinuous Galerkin scheme produces better results as compared to the staggered central scheme.

Free-complement local-Schrödinger-equation method for solving the Schrödinger equation of atoms and molecules: Basic theories and features

Science.gov (United States)

Nakatsuji, Hiroshi; Nakashima, Hiroyuki

2015-02-01

The free-complement (FC) method is a general method for solving the Schrödinger equation (SE): The produced wave function has the potentially exact structure as the solution of the Schrödinger equation. The variables included are determined either by using the variational principle (FC-VP) or by imposing the local Schrödinger equations (FC-LSE) at the chosen set of the sampling points. The latter method, referred to as the local Schrödinger equation (LSE) method, is integral-free and therefore applicable to any atom and molecule. The purpose of this paper is to formulate the basic theories of the LSE method and explain their basic features. First, we formulate three variants of the LSE method, the AB, HS, and HTQ methods, and explain their properties. Then, the natures of the LSE methods are clarified in some detail using the simple examples of the hydrogen atom and the Hooke's atom. Finally, the ideas obtained in this study are applied to solving the SE of the helium atom highly accurately with the FC-LSE method. The results are very encouraging: we could get the world's most accurate energy of the helium atom within the sampling-type methodologies, which is comparable to those obtained with the FC-VP method. Thus, the FC-LSE method is an easy and yet a powerful integral-free method for solving the Schrödinger equation of general atoms and molecules.

Gabor Wave Packet Method to Solve Plasma Wave Equations

International Nuclear Information System (INIS)

Pletzer, A.; Phillips, C.K.; Smithe, D.N.

2003-01-01

A numerical method for solving plasma wave equations arising in the context of mode conversion between the fast magnetosonic and the slow (e.g ion Bernstein) wave is presented. The numerical algorithm relies on the expansion of the solution in Gaussian wave packets known as Gabor functions, which have good resolution properties in both real and Fourier space. The wave packets are ideally suited to capture both the large and small wavelength features that characterize mode conversion problems. The accuracy of the scheme is compared with a standard finite element approach

Improving Teaching Quality and Problem Solving Ability through Contextual Teaching and Learning in Differential Equations: A Lesson Study Approach

Science.gov (United States)

Khotimah, Rita Pramujiyanti; Masduki

2016-01-01

Differential equations is a branch of mathematics which is closely related to mathematical modeling that arises in real-world problems. Problem solving ability is an essential component to solve contextual problem of differential equations properly. The purposes of this study are to describe contextual teaching and learning (CTL) model in…

How students process equations in solving quantitative synthesis problems? Role of mathematical complexity in students’ mathematical performance

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Bashirah Ibrahim

2017-10-01

Full Text Available We examine students’ mathematical performance on quantitative “synthesis problems” with varying mathematical complexity. Synthesis problems are tasks comprising multiple concepts typically taught in different chapters. Mathematical performance refers to the formulation, combination, and simplification of equations. Generally speaking, formulation and combination of equations require conceptual reasoning; simplification of equations requires manipulation of equations as computational tools. Mathematical complexity is operationally defined by the number and the type of equations to be manipulated concurrently due to the number of unknowns in each equation. We use two types of synthesis problems, namely, sequential and simultaneous tasks. Sequential synthesis tasks require a chronological application of pertinent concepts, and simultaneous synthesis tasks require a concurrent application of the pertinent concepts. A total of 179 physics major students from a second year mechanics course participated in the study. Data were collected from written tasks and individual interviews. Results show that mathematical complexity negatively influences the students’ mathematical performance on both types of synthesis problems. However, for the sequential synthesis tasks, it interferes only with the students’ simplification of equations. For the simultaneous synthesis tasks, mathematical complexity additionally impedes the students’ formulation and combination of equations. Several reasons may explain this difference, including the students’ different approaches to the two types of synthesis problems, cognitive load, and the variation of mathematical complexity within each synthesis type.

Optimal Homotopy Asymptotic Method for Solving System of Fredholm Integral Equations

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Bahman Ghazanfari

2013-08-01

Full Text Available In this paper, optimal homotopy asymptotic method (OHAM is applied to solve system of Fredholm integral equations. The effectiveness of optimal homotopy asymptotic method is presented. This method provides easy tools to control the convergence region of approximating solution series wherever necessary. The results of OHAM are compared with homotopy perturbation method (HPM and Taylor series expansion method (TSEM.

Integrator Performance Analysis In Solving Stiff Differential Equation System

International Nuclear Information System (INIS)

B, Alhadi; Basaruddin, T.

2001-01-01

In this paper we discuss the four-stage index-2 singly diagonally implicit Runge-Kutta method, which is used to solve stiff ordinary differential equations (SODE). Stiff problems require a method where step size is not restricted by the method's stability. We desire SDIRK to be A-stable that has no stability restrictions when solving y'= λy with Reλ>0 and h>0, so by choosing suitable stability function we can determine appropriate constant g) to formulate SDIRK integrator to solve SODE. We select the second stage of the internal stage as embedded method to perform low order estimate for error predictor. The strategy for choosing the step size is adopted from the strategy proposed by Hall(1996:6). And the algorithm that is developed in this paper is implemented using MATLAB 5.3, which is running on Window's 95 environment. Our performance measurement's local truncation error accuracy, and efficiency were evaluated by statistical results of sum of steps, sum of calling functions, average of Newton iterations and elapsed times.As the results, our numerical experiment show that SDIRK is unconditionally stable. By using Hall's step size strategy, the method can be implemented efficiently, provided that suitable parameters are used

A Compact Numerical Implementation for Solving Stokes Equations Using Matrix-vector Operations

KAUST Repository

Zhang, Tao; Salama, Amgad; Sun, Shuyu; Zhong, Hua

2015-01-01

In this work, a numerical scheme is implemented to solve Stokes equations based on cell-centered finite difference over staggered grid. In this scheme, all the difference operations have been vectorized thereby eliminating loops. This is particularly important when using programming languages that require interpretations, e.g., MATLAB and Python. Using this scheme, the execution time becomes significantly smaller compared with non-vectorized operations and also become comparable with those languages that require no repeated interpretations like FORTRAN, C, etc. This technique has also been applied to Navier-Stokes equations under laminar flow conditions.

A Compact Numerical Implementation for Solving Stokes Equations Using Matrix-vector Operations

KAUST Repository

Zhang, Tao

2015-06-01

In this work, a numerical scheme is implemented to solve Stokes equations based on cell-centered finite difference over staggered grid. In this scheme, all the difference operations have been vectorized thereby eliminating loops. This is particularly important when using programming languages that require interpretations, e.g., MATLAB and Python. Using this scheme, the execution time becomes significantly smaller compared with non-vectorized operations and also become comparable with those languages that require no repeated interpretations like FORTRAN, C, etc. This technique has also been applied to Navier-Stokes equations under laminar flow conditions.

Effective quadrature formula in solving linear integro-differential equations of order two

Science.gov (United States)

Eshkuvatov, Z. K.; Kammuji, M.; Long, N. M. A. Nik; Yunus, Arif A. M.

2017-08-01

In this note, we solve general form of Fredholm-Volterra integro-differential equations (IDEs) of order 2 with boundary condition approximately and show that proposed method is effective and reliable. Initially, IDEs is reduced into integral equation of the third kind by using standard integration techniques and identity between multiple and single integrals then truncated Legendre series are used to estimate the unknown function. For the kernel integrals, we have applied Gauss-Legendre quadrature formula and collocation points are chosen as the roots of the Legendre polynomials. Finally, reduce the integral equations of the third kind into the system of algebraic equations and Gaussian elimination method is applied to get approximate solutions. Numerical examples and comparisons with other methods reveal that the proposed method is very effective and dominated others in many cases. General theory of existence of the solution is also discussed.

Navier-Stokes equations by the finite element method

International Nuclear Information System (INIS)

Portella, P.E.

1984-01-01

A computer program to solve the Navier-Stokes equations by using the Finite Element Method is implemented. The solutions variables investigated are stream-function/vorticity in the steady case and velocity/pressure in the steady state and transient cases. For steady state flow the equations are solved simultaneously by the Newton-Raphson method. For the time dependent formulation, a fractional step method is employed to discretize in time and artificial viscosity is used to preclude spurious oscilations in the solution. The element used is the three node triangle. Some numerical examples are presented and comparisons are made with applications already existent. (Author) [pt

Performance of a parallel algorithm for solving the neutron diffusion equation on the hypercube

International Nuclear Information System (INIS)

Kirk, B.L.; Azmy, Y.Y.

1989-01-01

The one-group, steady state neutron diffusion equation in two- dimensional Cartesian geometry is solved using the nodal method technique. By decoupling sets of equations representing the neutron current continuity along the length of rows and columns of computational cells a new iterative algorithm is derived that is more suitable to solving large practical problems. This algorithm is highly parallelizable and is implemented on the Intel iPSC/2 hypercube in three versions which differ essentially in the total size of communicated data. Even though speedup was achieved, the efficiency is very low when many processors are used leading to the conclusion that the hypercube is not as well suited for this algorithm as shared memory machines. 10 refs., 1 fig., 3 tabs

Method for the Direct Solve of the Many-Body Schrödinger Wave Equation

Science.gov (United States)

Jerke, Jonathan; Tymczak, C. J.; Poirier, Bill

We report on theoretical and computational developments towards a computationally efficient direct solve of the many-body Schrödinger wave equation for electronic systems. This methodology relies on two recent developments pioneered by the authors: 1) the development of a Cardinal Sine basis for electronic structure calculations; and 2) the development of a highly efficient and compact representation of multidimensional functions using the Canonical tensor rank representation developed by Belykin et. al. which we have adapted to electronic structure problems. We then show several relevant examples of the utility and accuracy of this methodology, scaling with system size, and relevant convergence issues of the methodology. Method for the Direct Solve of the Many-Body Schrödinger Wave Equation.

A Proposed Method for Solving Fuzzy System of Linear Equations

Directory of Open Access Journals (Sweden)

Reza Kargar

2014-01-01

Full Text Available This paper proposes a new method for solving fuzzy system of linear equations with crisp coefficients matrix and fuzzy or interval right hand side. Some conditions for the existence of a fuzzy or interval solution of m×n linear system are derived and also a practical algorithm is introduced in detail. The method is based on linear programming problem. Finally the applicability of the proposed method is illustrated by some numerical examples.

Solving Ordinary Differential Equations

Science.gov (United States)

Krogh, F. T.

1987-01-01

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving the Fluid Pressure Poisson Equation Using Multigrid-Evaluation and Improvements.

Science.gov (United States)

Dick, Christian; Rogowsky, Marcus; Westermann, Rudiger

2016-11-01

In many numerical simulations of fluids governed by the incompressible Navier-Stokes equations, the pressure Poisson equation needs to be solved to enforce mass conservation. Multigrid solvers show excellent convergence in simple scenarios, yet they can converge slowly in domains where physically separated regions are combined at coarser scales. Moreover, existing multigrid solvers are tailored to specific discretizations of the pressure Poisson equation, and they cannot easily be adapted to other discretizations. In this paper we analyze the convergence properties of existing multigrid solvers for the pressure Poisson equation in different simulation domains, and we show how to further improve the multigrid convergence rate by using a graph-based extension to determine the coarse grid hierarchy. The proposed multigrid solver is generic in that it can be applied to different kinds of discretizations of the pressure Poisson equation, by using solely the specification of the simulation domain and pre-assembled computational stencils. We analyze the proposed solver in combination with finite difference and finite volume discretizations of the pressure Poisson equation. Our evaluations show that, despite the common assumption, multigrid schemes can exploit their potential even in the most complicated simulation scenarios, yet this behavior is obtained at the price of higher memory consumption.

New Approaches for Solving Fokker Planck Equation on Cantor Sets within Local Fractional Operators

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Hassan Kamil Jassim

2015-01-01

Full Text Available We discuss new approaches to handling Fokker Planck equation on Cantor sets within local fractional operators by using the local fractional Laplace decomposition and Laplace variational iteration methods based on the local fractional calculus. The new approaches maintain the efficiency and accuracy of the analytical methods for solving local fractional differential equations. Illustrative examples are given to show the accuracy and reliable results.

Solving nonlinear, High-order partial differential equations using a high-performance isogeometric analysis framework

KAUST Repository

Cortes, Adriano Mauricio; Vignal, Philippe; Sarmiento, Adel; Garcí a, Daniel O.; Collier, Nathan; Dalcin, Lisandro; Calo, Victor M.

2014-01-01

In this paper we present PetIGA, a high-performance implementation of Isogeometric Analysis built on top of PETSc. We show its use in solving nonlinear and time-dependent problems, such as phase-field models, by taking advantage of the high-continuity of the basis functions granted by the isogeometric framework. In this work, we focus on the Cahn-Hilliard equation and the phase-field crystal equation.

Solving the Einstein constraint equations on multi-block triangulations using finite element methods

Energy Technology Data Exchange (ETDEWEB)

Korobkin, Oleg; Pazos, Enrique [Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803 (United States); Aksoylu, Burak [Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803 (United States); Holst, Michael [Department of Mathematics, University of California at San Diego 9500 Gilman Drive La Jolla, CA 92093-0112 (United States); Tiglio, Manuel [Department of Physics, University of Maryland, College Park, MD 20742 (United States)

2009-07-21

In order to generate initial data for nonlinear relativistic simulations, one needs to solve the Einstein constraints, which can be cast into a coupled set of nonlinear elliptic equations. Here we present an approach for solving these equations on three-dimensional multi-block domains using finite element methods. We illustrate our approach on a simple example of Brill wave initial data, with the constraints reducing to a single linear elliptic equation for the conformal factor psi. We use quadratic Lagrange elements on semi-structured simplicial meshes, obtained by triangulation of multi-block grids. In the case of uniform refinement the scheme is superconvergent at most mesh vertices, due to local symmetry of the finite element basis with respect to local spatial inversions. We show that in the superconvergent case subsequent unstructured mesh refinements do not improve the quality of our initial data. As proof of concept that this approach is feasible for generating multi-block initial data in three dimensions, after constructing the initial data we evolve them in time using a high-order finite-differencing multi-block approach and extract the gravitational waves from the numerical solution.

Solving the Einstein constraint equations on multi-block triangulations using finite element methods

International Nuclear Information System (INIS)

Korobkin, Oleg; Pazos, Enrique; Aksoylu, Burak; Holst, Michael; Tiglio, Manuel

2009-01-01

In order to generate initial data for nonlinear relativistic simulations, one needs to solve the Einstein constraints, which can be cast into a coupled set of nonlinear elliptic equations. Here we present an approach for solving these equations on three-dimensional multi-block domains using finite element methods. We illustrate our approach on a simple example of Brill wave initial data, with the constraints reducing to a single linear elliptic equation for the conformal factor ψ. We use quadratic Lagrange elements on semi-structured simplicial meshes, obtained by triangulation of multi-block grids. In the case of uniform refinement the scheme is superconvergent at most mesh vertices, due to local symmetry of the finite element basis with respect to local spatial inversions. We show that in the superconvergent case subsequent unstructured mesh refinements do not improve the quality of our initial data. As proof of concept that this approach is feasible for generating multi-block initial data in three dimensions, after constructing the initial data we evolve them in time using a high-order finite-differencing multi-block approach and extract the gravitational waves from the numerical solution.

A rational function based scheme for solving advection equation

International Nuclear Information System (INIS)

Xiao, Feng; Yabe, Takashi.

1995-07-01

A numerical scheme for solving advection equations is presented. The scheme is derived from a rational interpolation function. Some properties of the scheme with respect to convex-concave preserving and monotone preserving are discussed. We find that the scheme is attractive in surpressinging overshoots and undershoots even in the vicinities of discontinuity. The scheme can also be easily swicthed as the CIP (Cubic interpolated Pseudo-Particle) method to get a third-order accuracy in smooth region. Numbers of numerical tests are carried out to show the non-oscillatory and less diffusive nature of the scheme. (author)

A meshless method for solving two-dimensional variable-order time fractional advection-diffusion equation

Science.gov (United States)

Tayebi, A.; Shekari, Y.; Heydari, M. H.

2017-07-01

Several physical phenomena such as transformation of pollutants, energy, particles and many others can be described by the well-known convection-diffusion equation which is a combination of the diffusion and advection equations. In this paper, this equation is generalized with the concept of variable-order fractional derivatives. The generalized equation is called variable-order time fractional advection-diffusion equation (V-OTFA-DE). An accurate and robust meshless method based on the moving least squares (MLS) approximation and the finite difference scheme is proposed for its numerical solution on two-dimensional (2-D) arbitrary domains. In the time domain, the finite difference technique with a θ-weighted scheme and in the space domain, the MLS approximation are employed to obtain appropriate semi-discrete solutions. Since the newly developed method is a meshless approach, it does not require any background mesh structure to obtain semi-discrete solutions of the problem under consideration, and the numerical solutions are constructed entirely based on a set of scattered nodes. The proposed method is validated in solving three different examples including two benchmark problems and an applied problem of pollutant distribution in the atmosphere. In all such cases, the obtained results show that the proposed method is very accurate and robust. Moreover, a remarkable property so-called positive scheme for the proposed method is observed in solving concentration transport phenomena.

Solving large sets of coupled equations iteratively by vector processing on the CYBER 205 computer

International Nuclear Information System (INIS)

Tolsma, L.D.

1985-01-01

The set of coupled linear second-order differential equations which has to be solved for the quantum-mechanical description of inelastic scattering of atomic and nuclear particles can be rewritten as an equivalent set of coupled integral equations. When some type of functions is used as piecewise analytic reference solutions, the integrals that arise in this set can be evaluated analytically. The set of integral equations can be solved iteratively. For the results mentioned an inward-outward iteration scheme has been applied. A concept of vectorization of coupled-channel Fortran programs, based on this integral method, is presented for the use on the Cyber 205 computer. It turns out that, for two heavy ion nuclear scattering test cases, this vector algorithm gives an overall speed-up of about a factor of 2 to 3 compared to a highly optimized scalar algorithm for a one vector pipeline computer

Performance prediction of gas turbines by solving a system of non-linear equations

Energy Technology Data Exchange (ETDEWEB)

Kaikko, J

1998-09-01

This study presents a novel method for implementing the performance prediction of gas turbines from the component models. It is based on solving the non-linear set of equations that corresponds to the process equations, and the mass and energy balances for the engine. General models have been presented for determining the steady state operation of single components. Single and multiple shad arrangements have been examined with consideration also being given to heat regeneration and intercooling. Emphasis has been placed upon axial gas turbines of an industrial scale. Applying the models requires no information of the structural dimensions of the gas turbines. On comparison with the commonly applied component matching procedures, this method incorporates several advantages. The application of the models for providing results is facilitated as less attention needs to be paid to calculation sequences and routines. Solving the set of equations is based on zeroing co-ordinate functions that are directly derived from the modelling equations. Therefore, controlling the accuracy of the results is easy. This method gives more freedom for the selection of the modelling parameters since, unlike for the matching procedures, exchanging these criteria does not itself affect the algorithms. Implicit relationships between the variables are of no significance, thus increasing the freedom for the modelling equations as well. The mathematical models developed in this thesis will provide facilities to optimise the operation of any major gas turbine configuration with respect to the desired process parameters. The computational methods used in this study may also be adapted to any other modelling problems arising in industry. (orig.) 36 refs.

Application of Local Fractional Series Expansion Method to Solve Klein-Gordon Equations on Cantor Sets

Directory of Open Access Journals (Sweden)

Ai-Min Yang

2014-01-01

Full Text Available We use the local fractional series expansion method to solve the Klein-Gordon equations on Cantor sets within the local fractional derivatives. The analytical solutions within the nondifferential terms are discussed. The obtained results show the simplicity and efficiency of the present technique with application to the problems of the liner differential equations on Cantor sets.

On a numereeical method for solving the Faddv integral equation without deformation of contour

International Nuclear Information System (INIS)

Belyaev, V.O.; Moller, K.

1976-01-01

A numerical method is proposed for solving the Faddeev equation for separable potentials at positive total energy. The method is based on the fact that after applying a simple interpolation procedure the logarithmic singularities in the kernel of the integral equation can be extracted in the same way as usually the pole singularity is extracted. The method has been applied to calculate the eigenvalues of the Faddeev kernel

Solving the Vlasov equation in two spatial dimensions with the Schrödinger method

Science.gov (United States)

Kopp, Michael; Vattis, Kyriakos; Skordis, Constantinos

2017-12-01

We demonstrate that the Vlasov equation describing collisionless self-gravitating matter may be solved with the so-called Schrödinger method (ScM). With the ScM, one solves the Schrödinger-Poisson system of equations for a complex wave function in d dimensions, rather than the Vlasov equation for a 2 d -dimensional phase space density. The ScM also allows calculating the d -dimensional cumulants directly through quasilocal manipulations of the wave function, avoiding the complexity of 2 d -dimensional phase space. We perform for the first time a quantitative comparison of the ScM and a conventional Vlasov solver in d =2 dimensions. Our numerical tests were carried out using two types of cold cosmological initial conditions: the classic collapse of a sine wave and those of a Gaussian random field as commonly used in cosmological cold dark matter N-body simulations. We compare the first three cumulants, that is, the density, velocity and velocity dispersion, to those obtained by solving the Vlasov equation using the publicly available code ColDICE. We find excellent qualitative and quantitative agreement between these codes, demonstrating the feasibility and advantages of the ScM as an alternative to N-body simulations. We discuss, the emergence of effective vorticity in the ScM through the winding number around the points where the wave function vanishes. As an application we evaluate the background pressure induced by the non-linearity of large scale structure formation, thereby estimating the magnitude of cosmological backreaction. We find that it is negligibly small and has time dependence and magnitude compatible with expectations from the effective field theory of large scale structure.

Comparison of numerical approaches to solve a Poincare-covariant Faddeev equation

International Nuclear Information System (INIS)

Alkofer, R.; Eichmann, G.; Krassnigg, A.; Schwinzerl, M.

2006-01-01

Full text: The quark core of Baryons can be described with the help of the numerical solution of the Poincare-Faddeev equation. Hereby the used elements, as e.g. the quark propagator are taken from non-perturbative studies of Landau gauge QCD. Different numerical approaches to solve in this way the relativistic three quark problem are compared and benchmarked results for the efficiency of different algorithms are presented. (author)

Who Solved the Bernoulli Differential Equation and How Did They Do It?

Science.gov (United States)

Parker, Adam E.

2013-01-01

The Bernoulli brothers, Jacob and Johann, and Leibniz: Any of these might have been first to solve what is called the Bernoulli differential equation. We explore their ideas and the chronology of their work, finding out, among other things, that variation of parameters was used in 1697, 78 years before 1775, when Lagrange introduced it in general.

Solving the RPA eigenvalue equation in real-space

CERN Document Server

Muta, A; Hashimoto, Y; Yabana, K

2002-01-01

We present a computational method to solve the RPA eigenvalue equation employing a uniform grid representation in three-dimensional Cartesian coordinates. The conjugate gradient method is used for this purpose as an interactive method for a generalized eigenvalue problem. No construction of unoccupied orbitals is required in the procedure. We expect this method to be useful for systems lacking spatial symmetry to calculate accurate eigenvalues and transition matrix elements of a few low-lying excitations. Some applications are presented to demonstrate the feasibility of the method, considering the simplified mean-field model as an example of a nuclear physics system and the electronic excitations in molecules with time-dependent density functional theory as an example of an electronic system. (author)

Simultaneous exact controllability for Maxwell equations and for a second-order hyperbolic system

Directory of Open Access Journals (Sweden)

Boris V. Kapitonov

2010-02-01

Full Text Available We present a result on "simultaneous" exact controllability for two models that describe two hyperbolic dynamics. One is the system of Maxwell equations and the other a vector-wave equation with a pressure term. We obtain the main result using modified multipliers in order to generate a necessary observability estimate which allow us to use the Hilbert Uniqueness Method (HUM introduced by Lions.

Full information estimations of a system of simultaneous equations with error component structure

OpenAIRE

Balestra, Pietro; Krishnakumar, Jaya

1987-01-01

In this paper we develop full information methods for estimating the parameters of a system of simultaneous equations with error component struc-ture and establish relationships between the various structural estimat

A Regularized Approach for Solving Magnetic Differential Equations and a Revised Iterative Equilibrium Algorithm

International Nuclear Information System (INIS)

Hudson, S.R.

2010-01-01

A method for approximately solving magnetic differential equations is described. The approach is to include a small diffusion term to the equation, which regularizes the linear operator to be inverted. The extra term allows a 'source-correction' term to be defined, which is generally required in order to satisfy the solvability conditions. The approach is described in the context of computing the pressure and parallel currents in the iterative approach for computing magnetohydrodynamic equilibria.

Solving delay differential equations in S-ADAPT by method of steps.

Science.gov (United States)

Bauer, Robert J; Mo, Gary; Krzyzanski, Wojciech

2013-09-01

S-ADAPT is a version of the ADAPT program that contains additional simulation and optimization abilities such as parametric population analysis. S-ADAPT utilizes LSODA to solve ordinary differential equations (ODEs), an algorithm designed for large dimension non-stiff and stiff problems. However, S-ADAPT does not have a solver for delay differential equations (DDEs). Our objective was to implement in S-ADAPT a DDE solver using the methods of steps. The method of steps allows one to solve virtually any DDE system by transforming it to an ODE system. The solver was validated for scalar linear DDEs with one delay and bolus and infusion inputs for which explicit analytic solutions were derived. Solutions of nonlinear DDE problems coded in S-ADAPT were validated by comparing them with ones obtained by the MATLAB DDE solver dde23. The estimation of parameters was tested on the MATLB simulated population pharmacodynamics data. The comparison of S-ADAPT generated solutions for DDE problems with the explicit solutions as well as MATLAB produced solutions which agreed to at least 7 significant digits. The population parameter estimates from using importance sampling expectation-maximization in S-ADAPT agreed with ones used to generate the data. Published by Elsevier Ireland Ltd.

Solving differential equations for Feynman integrals by expansions near singular points

Science.gov (United States)

Lee, Roman N.; Smirnov, Alexander V.; Smirnov, Vladimir A.

2018-03-01

We describe a strategy to solve differential equations for Feynman integrals by powers series expansions near singular points and to obtain high precision results for the corresponding master integrals. We consider Feynman integrals with two scales, i.e. non-trivially depending on one variable. The corresponding algorithm is oriented at situations where canonical form of the differential equations is impossible. We provide a computer code constructed with the help of our algorithm for a simple example of four-loop generalized sunset integrals with three equal non-zero masses and two zero masses. Our code gives values of the master integrals at any given point on the real axis with a required accuracy and a given order of expansion in the regularization parameter ɛ.

Homotopy perturbation method with Laplace Transform (LT-HPM) for solving Lane-Emden type differential equations (LETDEs).

Science.gov (United States)

Tripathi, Rajnee; Mishra, Hradyesh Kumar

2016-01-01

In this communication, we describe the Homotopy Perturbation Method with Laplace Transform (LT-HPM), which is used to solve the Lane-Emden type differential equations. It's very difficult to solve numerically the Lane-Emden types of the differential equation. Here we implemented this method for two linear homogeneous, two linear nonhomogeneous, and four nonlinear homogeneous Lane-Emden type differential equations and use their appropriate comparisons with exact solutions. In the current study, some examples are better than other existing methods with their nearer results in the form of power series. The Laplace transform used to accelerate the convergence of power series and the results are shown in the tables and graphs which have good agreement with the other existing method in the literature. The results show that LT-HPM is very effective and easy to implement.

FMCEIR: a Monte Carlo program for solving the stationary neutron and gamma transport equation

International Nuclear Information System (INIS)

Taormina, A.

1978-05-01

FMCEIR is a three-dimensional Monte Carlo program for solving the stationary neutron and gamma transport equation. It is used to study the problem of neutron and gamma streaming in the GCFR and HHT reactor channels. (G.T.H.)

Derivation of the physical equations solved in the inertial confinement stability code DOC. Informal report

International Nuclear Information System (INIS)

Scannapieco, A.J.; Cranfill, C.W.

1978-11-01

There now exists an inertial confinement stability code called DOC, which runs as a postprocessor. DOC (a code that has evolved from a previous code, PANSY) is a spherical harmonic linear stability code that integrates, in time, a set of Lagrangian perturbation equations. Effects due to real equations of state, asymmetric energy deposition, thermal conduction, shock propagation, and a time-dependent zeroth-order state are handled in the code. We present here a detailed derivation of the physical equations that are solved in the code

Derivation of the physical equations solved in the inertial confinement stability code DOC. Informal report

Energy Technology Data Exchange (ETDEWEB)

Scannapieco, A.J.; Cranfill, C.W.

1978-11-01

There now exists an inertial confinement stability code called DOC, which runs as a postprocessor. DOC (a code that has evolved from a previous code, PANSY) is a spherical harmonic linear stability code that integrates, in time, a set of Lagrangian perturbation equations. Effects due to real equations of state, asymmetric energy deposition, thermal conduction, shock propagation, and a time-dependent zeroth-order state are handled in the code. We present here a detailed derivation of the physical equations that are solved in the code.

Simultaneous viscous-inviscid coupling via transpiration

International Nuclear Information System (INIS)

Yiu, K.F.C.; Giles, M.B.

1995-01-01

In viscous-inviscid coupling analysis, the direct coupling technique and the inverse coupling technique are commonly adopted. However, stability and convergence of the algorithms derived are usually very unsatisfactory. Here, by using the transpiration technique to simulate the effect of the displacement thickness, a new simultaneous coupling method is derived. The integral boundary layer equations and the full potential equation are chosen to be the viscous-inviscid coupled system. After discretization, the Newton-Raphson technique is proposed to solve the coupled nonlinear system. Several numerical results are used to demonstrate the accuracy and efficiency of the proposed method. 15 refs., 23 figs

Neural network error correction for solving coupled ordinary differential equations

Science.gov (United States)

Shelton, R. O.; Darsey, J. A.; Sumpter, B. G.; Noid, D. W.

1992-01-01

A neural network is presented to learn errors generated by a numerical algorithm for solving coupled nonlinear differential equations. The method is based on using a neural network to correctly learn the error generated by, for example, Runge-Kutta on a model molecular dynamics (MD) problem. The neural network programs used in this study were developed by NASA. Comparisons are made for training the neural network using backpropagation and a new method which was found to converge with fewer iterations. The neural net programs, the MD model and the calculations are discussed.

AIREK-MOD, Time Dependent Reactor Kinetics with Feedback Differential Equation

International Nuclear Information System (INIS)

Tamagnini, C.

1984-01-01

1 - Nature of physical problem solved: Solves the reactor kinetic equations with respect to time. A standard form for the reactivity behaviour has been introduced in which the reactivity is given by the sum of a polynomial, sine, cosine and exponential expansion. Tabular form is also included. The presence of feedback differential equations in which the dependence on variables different from the considered one is considered enables many heat-exchange problems to be dealt with. 2 - Method of solution: The method employed for the solution of the differential equations is the one developed by E.R. Cohen (Geneva Conference, 1958). 3 - Restrictions on the complexity of the problem: The maximum number of differential equations that can be solved simultaneously is 50. Within this limitation there may be n delayed neutron groups (n less than or equal to 25), on m other linear feedback equations (n+m less than or equal to 49). CDC 1604 version was offered by EIR (Institut Federal de Recherches en matiere de reacteurs, Switzerland)

Numerical solution of multi groups point kinetic equations by simulink toolbox of Matlab software

International Nuclear Information System (INIS)

Hadad, K.; Mohamadi, A.; Sabet, H.; Ayobian, N.; Khani, M.

2004-01-01

The simulink toolbox of Matlab Software was employed to solve the point kinetics equation with six group delayed neutrons. The method of Adams-Bash ford showed a good convergence in solving the system of simultaneous equations and the obtained results showed good agreements with other numerical schemes. The flexibility of the package in changing the system parameters and the user friendly interface makes this approach a reliable educational package in revealing the affects of reactivity changes on power incursions

Solving Witten's string field theory using the butterfly state

International Nuclear Information System (INIS)

Okawa, Yuji

2004-01-01

We solve the equation of motion of Witten's cubic open string field theory in a series expansion using the regulated butterfly state. The expansion parameter is given by the regularization parameter of the butterfly state, which can be taken to be arbitrarily small. Unlike the case of level truncation, the equation of motion can be solved for an arbitrary component of the Fock space up to a positive power of the expansion parameter. The energy density of the solution is well defined and remains finite even in the singular butterfly limit, and it gives approximately 68% of the D25-brane tension for the solution at the leading order. Moreover, it simultaneously solves the equation of motion of vacuum string field theory, providing support for the conjecture at this order. We further improve our ansatz by taking into account next-to-leading terms, and find two numerical solutions which give approximately 88% and 109%, respectively, of the D25-brane tension for the energy density. These values are interestingly close to those by level truncation at level 2 without gauge fixing studied by Rastelli and Zwiebach and by Ellwood and Taylor

Solving differential equations with unknown constitutive relations as recurrent neural networks

Energy Technology Data Exchange (ETDEWEB)

Hagge, Tobias J.; Stinis, Panagiotis; Yeung, Enoch H.; Tartakovsky, Alexandre M.

2017-12-08

We solve a system of ordinary differential equations with an unknown functional form of a sink (reaction rate) term. We assume that the measurements (time series) of state variables are partially available, and use a recurrent neural network to “learn” the reaction rate from this data. This is achieved by including discretized ordinary differential equations as part of a recurrent neural network training problem. We extend TensorFlow’s recurrent neural network architecture to create a simple but scalable and effective solver for the unknown functions, and apply it to a fedbatch bioreactor simulation problem. Use of techniques from recent deep learning literature enables training of functions with behavior manifesting over thousands of time steps. Our networks are structurally similar to recurrent neural networks, but differ in purpose, and require modified training strategies.

A new modified conjugate gradient coefficient for solving system of linear equations

Science.gov (United States)

Hajar, N.; ‘Aini, N.; Shapiee, N.; Abidin, Z. Z.; Khadijah, W.; Rivaie, M.; Mamat, M.

2017-09-01

Conjugate gradient (CG) method is an evolution of computational method in solving unconstrained optimization problems. This approach is easy to implement due to its simplicity and has been proven to be effective in solving real-life application. Although this field has received copious amount of attentions in recent years, some of the new approaches of CG algorithm cannot surpass the efficiency of the previous versions. Therefore, in this paper, a new CG coefficient which retains the sufficient descent and global convergence properties of the original CG methods is proposed. This new CG is tested on a set of test functions under exact line search. Its performance is then compared to that of some of the well-known previous CG methods based on number of iterations and CPU time. The results show that the new CG algorithm has the best efficiency amongst all the methods tested. This paper also includes an application of the new CG algorithm for solving large system of linear equations

An efficient computer based wavelets approximation method to solve Fuzzy boundary value differential equations

Science.gov (United States)

Alam Khan, Najeeb; Razzaq, Oyoon Abdul

2016-03-01

In the present work a wavelets approximation method is employed to solve fuzzy boundary value differential equations (FBVDEs). Essentially, a truncated Legendre wavelets series together with the Legendre wavelets operational matrix of derivative are utilized to convert FB- VDE into a simple computational problem by reducing it into a system of fuzzy algebraic linear equations. The capability of scheme is investigated on second order FB- VDE considered under generalized H-differentiability. Solutions are represented graphically showing competency and accuracy of this method.

The hyperspherical-harmonics expansion method and the integral-equation approach to solving the few-body problem in momentum space

International Nuclear Information System (INIS)

Liu, F.-Q.; Lim, T.K.

1988-01-01

The Faddeev and Faddeev-Yakubovsky equations for three- and four-body systems are solved by applying the hyperspherical-harmonics expansion to them in momentum space. This coupling of two popular approaches to the few-body problem together with the use of the so-called Raynal-Revai transformation, which relates hyperspherical functions, allows the few-body equations to be written as one-dimensional coupled integral equations. Numerical solutions for these are achieved through standard matrix methods; these are made straightforward, because a second transformation renders potential multipoles easily calculable. For sample potentials and a restricted size of matrix in each case, the binding energies extracted match those previously obtained in solving the Schroedinger equation through the hyperspherical-harmonics expansion in coordinate space. 9 refs

A fast semi-discrete Kansa method to solve the two-dimensional spatiotemporal fractional diffusion equation

Science.gov (United States)

Sun, HongGuang; Liu, Xiaoting; Zhang, Yong; Pang, Guofei; Garrard, Rhiannon

2017-09-01

Fractional-order diffusion equations (FDEs) extend classical diffusion equations by quantifying anomalous diffusion frequently observed in heterogeneous media. Real-world diffusion can be multi-dimensional, requiring efficient numerical solvers that can handle long-term memory embedded in mass transport. To address this challenge, a semi-discrete Kansa method is developed to approximate the two-dimensional spatiotemporal FDE, where the Kansa approach first discretizes the FDE, then the Gauss-Jacobi quadrature rule solves the corresponding matrix, and finally the Mittag-Leffler function provides an analytical solution for the resultant time-fractional ordinary differential equation. Numerical experiments are then conducted to check how the accuracy and convergence rate of the numerical solution are affected by the distribution mode and number of spatial discretization nodes. Applications further show that the numerical method can efficiently solve two-dimensional spatiotemporal FDE models with either a continuous or discrete mixing measure. Hence this study provides an efficient and fast computational method for modeling super-diffusive, sub-diffusive, and mixed diffusive processes in large, two-dimensional domains with irregular shapes.

Solving Nonlinear Coupled Differential Equations

Science.gov (United States)

Mitchell, L.; David, J.

1986-01-01

Harmonic balance method developed to obtain approximate steady-state solutions for nonlinear coupled ordinary differential equations. Method usable with transfer matrices commonly used to analyze shaft systems. Solution to nonlinear equation, with periodic forcing function represented as sum of series similar to Fourier series but with form of terms suggested by equation itself.

Alternating-direction implicit numerical solution of the time-dependent, three-dimensional, single fluid, resistive magnetohydrodynamic equations

Energy Technology Data Exchange (ETDEWEB)

Finan, C.H. III

1980-12-01

Resistive magnetohydrodynamics (MHD) is described by a set of eight coupled, nonlinear, three-dimensional, time-dependent, partial differential equations. A computer code, IMP (Implicit MHD Program), has been developed to solve these equations numerically by the method of finite differences on an Eulerian mesh. In this model, the equations are expressed in orthogonal curvilinear coordinates, making the code applicable to a variety of coordinate systems. The Douglas-Gunn algorithm for Alternating-Direction Implicit (ADI) temporal advancement is used to avoid the limitations in timestep size imposed by explicit methods. The equations are solved simultaneously to avoid syncronization errors.

A nearly orthogonal 2D grid system in solving the shallow water equations in the head bay of Bengal

International Nuclear Information System (INIS)

Roy, G.D. . E.mail: roy_gd@hotmail.com; Hussain, Farzana . E.mail: farzana@sust.edu

2001-11-01

A typical nearly orthogonal grid system is considered to solve the shallow water equations along the head bay of Bengal. A pencil of straight lines at uniform angular distance through a suitable origin, O at the mean sea level (MSL), are considered as a system of grid lines. A system of concentric and uniformly distributed ellipses with center at O is considered as the other system of grid lines. In order to solve the shallow water equations numerically, a system of transformations is applied so that the grid system in the transformed domain becomes a rectangular one. Shallow water equations are solved using appropriate initial and boundary conditions to estimate the water level due to tide and surge. The typical grid system is found to be suitable in incorporating the bending of the coastline and the island boundaries accurately in the numerical scheme along the coast of Bangladesh. (author)

Modifying a numerical algorithm for solving the matrix equation X + AX T B = C

Science.gov (United States)

Vorontsov, Yu. O.

2013-06-01

Certain modifications are proposed for a numerical algorithm solving the matrix equation X + AX T B = C. By keeping the intermediate results in storage and repeatedly using them, it is possible to reduce the total complexity of the algorithm from O( n 4) to O( n 3) arithmetic operations.

An inherently parallel method for solving discretized diffusion equations

International Nuclear Information System (INIS)

Eccleston, B.R.; Palmer, T.S.

1999-01-01

A Monte Carlo approach to solving linear systems of equations is being investigated in the context of the solution of discretized diffusion equations. While the technique was originally devised decades ago, changes in computer architectures (namely, massively parallel machines) have driven the authors to revisit this technique. There are a number of potential advantages to this approach: (1) Analog Monte Carlo techniques are inherently parallel; this is not necessarily true to today's more advanced linear equation solvers (multigrid, conjugate gradient, etc.); (2) Some forms of this technique are adaptive in that they allow the user to specify locations in the problem where resolution is of particular importance and to concentrate the work at those locations; and (3) These techniques permit the solution of very large systems of equations in that matrix elements need not be stored. The user could trade calculational speed for storage if elements of the matrix are calculated on the fly. The goal of this study is to compare the parallel performance of Monte Carlo linear solvers to that of a more traditional parallelized linear solver. The authors observe the linear speedup that they expect from the Monte Carlo algorithm, given that there is no domain decomposition to cause significant communication overhead. Overall, PETSc outperforms the Monte Carlo solver for the test problem. The PETSc parallel performance improves with larger numbers of unknowns for a given number of processors. Parallel performance of the Monte Carlo technique is independent of the size of the matrix and the number of processes. They are investigating modifications to the scheme to accommodate matrix problems with positive off-diagonal elements. They are also currently coding an on-the-fly version of the algorithm to investigate the solution of very large linear systems

A Fibonacci collocation method for solving a class of Fredholm–Volterra integral equations in two-dimensional spaces

Directory of Open Access Journals (Sweden)

Farshid Mirzaee

2014-06-01

Full Text Available In this paper, we present a numerical method for solving two-dimensional Fredholm–Volterra integral equations (F-VIE. The method reduces the solution of these integral equations to the solution of a linear system of algebraic equations. The existence and uniqueness of the solution and error analysis of proposed method are discussed. The method is computationally very simple and attractive. Finally, numerical examples illustrate the efficiency and accuracy of the method.

GPELab, a Matlab toolbox to solve Gross-Pitaevskii equations II: Dynamics and stochastic simulations

Science.gov (United States)

Antoine, Xavier; Duboscq, Romain

2015-08-01

GPELab is a free Matlab toolbox for modeling and numerically solving large classes of systems of Gross-Pitaevskii equations that arise in the physics of Bose-Einstein condensates. The aim of this second paper, which follows (Antoine and Duboscq, 2014), is to first present the various pseudospectral schemes available in GPELab for computing the deterministic and stochastic nonlinear dynamics of Gross-Pitaevskii equations (Antoine, et al., 2013). Next, the corresponding GPELab functions are explained in detail. Finally, some numerical examples are provided to show how the code works for the complex dynamics of BEC problems.

An Operational Matrix Technique for Solving Variable Order Fractional Differential-Integral Equation Based on the Second Kind of Chebyshev Polynomials

Directory of Open Access Journals (Sweden)

Jianping Liu

2016-01-01

Full Text Available An operational matrix technique is proposed to solve variable order fractional differential-integral equation based on the second kind of Chebyshev polynomials in this paper. The differential operational matrix and integral operational matrix are derived based on the second kind of Chebyshev polynomials. Using two types of operational matrixes, the original equation is transformed into the arithmetic product of several dependent matrixes, which can be viewed as an algebraic system after adopting the collocation points. Further, numerical solution of original equation is obtained by solving the algebraic system. Finally, several examples show that the numerical algorithm is computationally efficient.

Advantages and Disadvantages of Using MATLAB/ode45 for Solving Differential Equations in Engineering Applications

OpenAIRE

Waleed K. Ahmed

2013-01-01

The present paper demonstrates the route used for solving differential equations for the engineering applications at UAEU. Usually students at the Engineering Requirements Unit (ERU) stage of the Faculty of Engineering at the UAEU must enroll in a course of Differential Equations and Engineering Applications (MATH 2210) as a prerequisite for the subsequent stages of their study. Mainly, one of the objectives of this course is that the students practice MATLAB software package during the cours...

Moving mesh generation with a sequential approach for solving PDEs

DEFF Research Database (Denmark)

In moving mesh methods, physical PDEs and a mesh equation derived from equidistribution of an error metrics (so-called the monitor function) are simultaneously solved and meshes are dynamically concentrated on steep regions (Lim et al., 2001). However, the simultaneous solution procedure...... a simple and robust moving mesh algorithm in one or multidimension. In this study, we propose a sequential solution procedure including two separate parts: prediction step to obtain an approximate solution to a next time level (integration of physical PDEs) and regriding step at the next time level (mesh...... generation and solution interpolation). Convection terms, which appear in physical PDEs and a mesh equation, are discretized by a WENO (Weighted Essentially Non-Oscillatory) scheme under the consrvative form. This sequential approach is to keep the advantages of robustness and simplicity for the static...

Projective-Dual Method for Solving Systems of Linear Equations with Nonnegative Variables

Science.gov (United States)

Ganin, B. V.; Golikov, A. I.; Evtushenko, Yu. G.

2018-02-01

In order to solve an underdetermined system of linear equations with nonnegative variables, the projection of a given point onto its solutions set is sought. The dual of this problem—the problem of unconstrained maximization of a piecewise-quadratic function—is solved by Newton's method. The problem of unconstrained optimization dual of the regularized problem of finding the projection onto the solution set of the system is considered. A connection of duality theory and Newton's method with some known algorithms of projecting onto a standard simplex is shown. On the example of taking into account the specifics of the constraints of the transport linear programming problem, the possibility to increase the efficiency of calculating the generalized Hessian matrix is demonstrated. Some examples of numerical calculations using MATLAB are presented.

A novel technique to solve nonlinear higher-index Hessenberg differential-algebraic equations by Adomian decomposition method.

Science.gov (United States)

Benhammouda, Brahim

2016-01-01

Since 1980, the Adomian decomposition method (ADM) has been extensively used as a simple powerful tool that applies directly to solve different kinds of nonlinear equations including functional, differential, integro-differential and algebraic equations. However, for differential-algebraic equations (DAEs) the ADM is applied only in four earlier works. There, the DAEs are first pre-processed by some transformations like index reductions before applying the ADM. The drawback of such transformations is that they can involve complex algorithms, can be computationally expensive and may lead to non-physical solutions. The purpose of this paper is to propose a novel technique that applies the ADM directly to solve a class of nonlinear higher-index Hessenberg DAEs systems efficiently. The main advantage of this technique is that; firstly it avoids complex transformations like index reductions and leads to a simple general algorithm. Secondly, it reduces the computational work by solving only linear algebraic systems with a constant coefficient matrix at each iteration, except for the first iteration where the algebraic system is nonlinear (if the DAE is nonlinear with respect to the algebraic variable). To demonstrate the effectiveness of the proposed technique, we apply it to a nonlinear index-three Hessenberg DAEs system with nonlinear algebraic constraints. This technique is straightforward and can be programmed in Maple or Mathematica to simulate real application problems.

Solving the generalized Langevin equation with the algebraically correlated noise

International Nuclear Information System (INIS)

Srokowski, T.; Ploszajczak, M.

1997-01-01

The Langevin equation with the memory kernel is solved. The stochastic force possesses algebraic correlations, proportional to 1/t. The velocity autocorrelation function and related quantities characterizing transport properties are calculated at the assumption that the system is in the thermal equilibrium. Stochastic trajectories are simulated numerically, using the kangaroo process as a noise generator. Results of this simulation resemble Levy walks with divergent moments of the velocity distribution. The motion of a Brownian particle is considered both without any external potential and in the harmonic oscillator field, in particular the escape from a potential well. The results are compared with memory-free calculations for the Brownian particle. (author)

Development and adjustment of programs for solving systems of linear equations

International Nuclear Information System (INIS)

Fujimura, Toichiro

1978-03-01

Programs for solving the systems of linear equations have been adjusted and developed in expanding the scientific subroutine library SSL. The principal programs adjusted are based on the congruent method, method of product form of the inverse, orthogonal method, Crout's method for sparse system, and acceleration of iterative methods. The programs developed are based on the escalator method, direct parallel residue method and block tridiagonal method for band system. Described are usage of the programs developed and their future improvement. FORTRAN lists with simple examples in tests of the programs are also given. (auth.)

EDEF: a program for solving the neutron diffusion equation using microcomputers

International Nuclear Information System (INIS)

Fernandes, A.; Maiorino, J.R.

1990-01-01

This work presents the development of a program to solve the two-group two-dimensional diffusion equation (with a buckling option to simulate axial leakage) applying the finite element method. It has been developed to microcomputers compatibles to the IBM-PC. Among the facilities of the program, we can mention the simplicity to represent two-dimensional complex domains, the input through a pre-processor and the output in which the fluxes are presented graphically. The program also calculates the multiplication factor, the peaking factor and the power distribution. (author) [pt

Modeling digital pulse waveforms by solving one-dimensional Navier-stokes equations.

Science.gov (United States)

Fedotov, Aleksandr A; Akulova, Anna S; Akulov, Sergey A

2016-08-01

Mathematical modeling for composition distal arterial pulse wave in the blood vessels of the upper limbs was considered. Formation of distal arterial pulse wave is represented as a composition of forward and reflected pulse waves propagating along the arterial vessels. The formal analogy between pulse waves propagation along the human arterial system and the propagation of electrical oscillations in electrical transmission lines with distributed parameters was proposed. Dependencies of pulse wave propagation along the human arterial system were obtained by solving the one-dimensional Navier-Stokes equations for a few special cases.

Auxiliary equation method for solving nonlinear partial differential equations

International Nuclear Information System (INIS)

Sirendaoreji,; Jiong, Sun

2003-01-01

By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation

An optimal iterative algorithm to solve Cauchy problem for Laplace equation

KAUST Repository

Majeed, Muhammad Usman

2015-05-25

An optimal mean square error minimizer algorithm is developed to solve severely ill-posed Cauchy problem for Laplace equation on an annulus domain. The mathematical problem is presented as a first order state space-like system and an optimal iterative algorithm is developed that minimizes the mean square error in states. Finite difference discretization schemes are used to discretize first order system. After numerical discretization algorithm equations are derived taking inspiration from Kalman filter however using one of the space variables as a time-like variable. Given Dirichlet and Neumann boundary conditions are used on the Cauchy data boundary and fictitious points are introduced on the unknown solution boundary. The algorithm is run for a number of iterations using the solution of previous iteration as a guess for the next one. The method developed happens to be highly robust to noise in Cauchy data and numerically efficient results are illustrated.

A higher-order conservation element solution element method for solving hyperbolic differential equations on unstructured meshes

Science.gov (United States)

Bilyeu, David

This dissertation presents an extension of the Conservation Element Solution Element (CESE) method from second- to higher-order accuracy. The new method retains the favorable characteristics of the original second-order CESE scheme, including (i) the use of the space-time integral equation for conservation laws, (ii) a compact mesh stencil, (iii) the scheme will remain stable up to a CFL number of unity, (iv) a fully explicit, time-marching integration scheme, (v) true multidimensionality without using directional splitting, and (vi) the ability to handle two- and three-dimensional geometries by using unstructured meshes. This algorithm has been thoroughly tested in one, two and three spatial dimensions and has been shown to obtain the desired order of accuracy for solving both linear and non-linear hyperbolic partial differential equations. The scheme has also shown its ability to accurately resolve discontinuities in the solutions. Higher order unstructured methods such as the Discontinuous Galerkin (DG) method and the Spectral Volume (SV) methods have been developed for one-, two- and three-dimensional application. Although these schemes have seen extensive development and use, certain drawbacks of these methods have been well documented. For example, the explicit versions of these two methods have very stringent stability criteria. This stability criteria requires that the time step be reduced as the order of the solver increases, for a given simulation on a given mesh. The research presented in this dissertation builds upon the work of Chang, who developed a fourth-order CESE scheme to solve a scalar one-dimensional hyperbolic partial differential equation. The completed research has resulted in two key deliverables. The first is a detailed derivation of a high-order CESE methods on unstructured meshes for solving the conservation laws in two- and three-dimensional spaces. The second is the code implementation of these numerical methods in a computer code. For

Integration of Chandrasekhar's integral equation

International Nuclear Information System (INIS)

Tanaka, Tasuku

2003-01-01

We solve Chandrasekhar's integration equation for radiative transfer in the plane-parallel atmosphere by iterative integration. The primary thrust in radiative transfer has been to solve the forward problem, i.e., to evaluate the radiance, given the optical thickness and the scattering phase function. In the area of satellite remote sensing, our problem is the inverse problem: to retrieve the surface reflectance and the optical thickness of the atmosphere from the radiance measured by satellites. In order to retrieve the optical thickness and the surface reflectance from the radiance at the top-of-the atmosphere (TOA), we should express the radiance at TOA 'explicitly' in the optical thickness and the surface reflectance. Chandrasekhar formalized radiative transfer in the plane-parallel atmosphere in a simultaneous integral equation, and he obtained the second approximation. Since then no higher approximation has been reported. In this paper, we obtain the third approximation of the scattering function. We integrate functions derived from the second approximation in the integral interval from 1 to ∞ of the inverse of the cos of zenith angles. We can obtain the indefinite integral rather easily in the form of a series expansion. However, the integrals at the upper limit, ∞, are not yet known to us. We can assess the converged values of those series expansions at ∞ through calculus. For integration, we choose coupling pairs to avoid unnecessary terms in the outcome of integral and discover that the simultaneous integral equation can be deduced to the mere integral equation. Through algebraic calculation, we obtain the third approximation as a polynomial of the third degree in the atmospheric optical thickness

A NUMERICAL SCHEME FOR SPECIAL RELATIVISTIC RADIATION MAGNETOHYDRODYNAMICS BASED ON SOLVING THE TIME-DEPENDENT RADIATIVE TRANSFER EQUATION

Energy Technology Data Exchange (ETDEWEB)

Ohsuga, Ken; Takahashi, Hiroyuki R. [National Astronomical Observatory of Japan, Osawa, Mitaka, Tokyo 181-8588 (Japan)

2016-02-20

We develop a numerical scheme for solving the equations of fully special relativistic, radiation magnetohydrodynamics (MHDs), in which the frequency-integrated, time-dependent radiation transfer equation is solved to calculate the specific intensity. The radiation energy density, the radiation flux, and the radiation stress tensor are obtained by the angular quadrature of the intensity. In the present method, conservation of total mass, momentum, and energy of the radiation magnetofluids is guaranteed. We treat not only the isotropic scattering but also the Thomson scattering. The numerical method of MHDs is the same as that of our previous work. The advection terms are explicitly solved, and the source terms, which describe the gas–radiation interaction, are implicitly integrated. Our code is suitable for massive parallel computing. We present that our code shows reasonable results in some numerical tests for propagating radiation and radiation hydrodynamics. Particularly, the correct solution is given even in the optically very thin or moderately thin regimes, and the special relativistic effects are nicely reproduced.

Generalized Path Analysis and Generalized Simultaneous Equations Model for Recursive Systems with Responses of Mixed Types

Science.gov (United States)

Tsai, Tien-Lung; Shau, Wen-Yi; Hu, Fu-Chang

2006-01-01

This article generalizes linear path analysis (PA) and simultaneous equations models (SiEM) to deal with mixed responses of different types in a recursive or triangular system. An efficient instrumental variable (IV) method for estimating the structural coefficients of a 2-equation partially recursive generalized path analysis (GPA) model and…

USING SIMULTANEOUS EQUATIONS MODELS TO ANALYZE THE CAUSES OF CORRUPTION AND ITS IMPLICATIONS

Directory of Open Access Journals (Sweden)

TUDOREL ANDREI

2011-04-01

Full Text Available For a country that is in the process of integration into EU structures, reducing the corruption is a good sign for attracting foreign investment and developing the economic environment. The paper estimates the parameters of a simultaneous equation model based on data sets obtained at a sample of employees in public administration. Statistical sample consist in 407 people and the maximum allowable error was estimated at ± 2.5%. For the effective development of the statistical questionnaire we identified major themes of public administration that are directly related to the problem of corruption: managing the institution, the civil service, transparency in the system, the decentralization process, causes and effects of corruption and the quality of the reform in the public administration. Based on the questionnaire we defined primary and secondary variables that have been used to define the model with simultaneous equations. For the variables in the model they have been divided into endogenous and exogenous.

A Generalized FDM for solving the Poisson's Equation on 3D Irregular Domains

Directory of Open Access Journals (Sweden)

J. Izadian

2014-01-01

Full Text Available In this paper a new method for solving the Poisson's equation with Dirichlet conditions on irregular domains is presented. For this purpose a generalized finite differences method is applied for numerical differentiation on irregular meshes. Three examples on cylindrical and spherical domains are considered. The numerical results are compared with analytical solution. These results show the performance and efficiency of the proposed method.

Mathematical Modeling and Simulation of SWRO Process Based on Simultaneous Method

Directory of Open Access Journals (Sweden)

Aipeng Jiang

2014-01-01

Full Text Available Reverse osmosis (RO technique is one of the most efficient ways for seawater desalination to solve the shortage of freshwater. For prediction and analysis of the performance of seawater reverse osmosis (SWRO process, an accurate and detailed model based on the solution-diffusion and mass transfer theory is established. Since the accurate formulation of the model includes many differential equations and strong nonlinear equations (differential and algebraic equations, DAEs, to solve the problem efficiently, the simultaneous method through orthogonal collocation on finite elements and large scale solver were used to obtain the solutions. The model was fully discretized into NLP (nonlinear programming with large scale variables and equations, and then the NLP was solved by large scale solver of IPOPT. Validation of the formulated model and solution method is verified by case study on a SWRO plant. Then simulation and analysis are carried out to demonstrate the performance of reverse osmosis process; operational conditions such as feed pressure and feed flow rate as well as feed temperature are also analyzed. This work is of significant meaning for the detailed understanding of RO process and future energy saving through operational optimization.

SOLVING NONLINEAR KLEIN-GORDON EQUATION WITH A QUADRATIC NONLINEAR TERM USING HOMOTOPY ANALYSIS METHOD

Directory of Open Access Journals (Sweden)

H. Jafari

2010-07-01

Full Text Available In this paper, nonlinear Klein-Gordon equation with quadratic term is solved by means of an analytic technique, namely the Homotopy analysis method (HAM.Comparisons are made between the Adomian decomposition method (ADM, the exact solution and homotopy analysis method. The results reveal that the proposed method is very effective and simple.

Predicting Eight Grade Students' Equation Solving Performances via Concepts of Variable and Equality

Science.gov (United States)

Ertekin, Erhan

2017-01-01

This study focused on how two algebraic concepts- equality and variable- predicted 8th grade students' equation solving performance. In this study, predictive design as a correlational research design was used. Randomly selected 407 eight-grade students who were from the central districts of a city in the central region of Turkey participated in…

Development of C++ Application Program for Solving Quadratic Equation in Elementary School in Nigeria

Science.gov (United States)

Bandele, Samuel Oye; Adekunle, Adeyemi Suraju

2015-01-01

The study was conducted to design, develop and test a c++ application program CAP-QUAD for solving quadratic equation in elementary school in Nigeria. The package was developed in c++ using object-oriented programming language, other computer program that were also utilized during the development process is DevC++ compiler, it was used for…

A boundary value approach for solving three-dimensional elliptic and hyperbolic partial differential equations.

Science.gov (United States)

Biala, T A; Jator, S N

2015-01-01

In this article, the boundary value method is applied to solve three dimensional elliptic and hyperbolic partial differential equations. The partial derivatives with respect to two of the spatial variables (y, z) are discretized using finite difference approximations to obtain a large system of ordinary differential equations (ODEs) in the third spatial variable (x). Using interpolation and collocation techniques, a continuous scheme is developed and used to obtain discrete methods which are applied via the Block unification approach to obtain approximations to the resulting large system of ODEs. Several test problems are investigated to elucidate the solution process.

An Adaptive Observer-Based Algorithm for Solving Inverse Source Problem for the Wave Equation

KAUST Repository

Asiri, Sharefa M.; Zayane, Chadia; Laleg-Kirati, Taous-Meriem

2015-01-01

Observers are well known in control theory. Originally designed to estimate the hidden states of dynamical systems given some measurements, the observers scope has been recently extended to the estimation of some unknowns, for systems governed by partial differential equations. In this paper, observers are used to solve inverse source problem for a one-dimensional wave equation. An adaptive observer is designed to estimate the state and source components for a fully discretized system. The effectiveness of the algorithm is emphasized in noise-free and noisy cases and an insight on the impact of measurements’ size and location is provided.

An Adaptive Observer-Based Algorithm for Solving Inverse Source Problem for the Wave Equation

KAUST Repository

Asiri, Sharefa M.

2015-08-31

Observers are well known in control theory. Originally designed to estimate the hidden states of dynamical systems given some measurements, the observers scope has been recently extended to the estimation of some unknowns, for systems governed by partial differential equations. In this paper, observers are used to solve inverse source problem for a one-dimensional wave equation. An adaptive observer is designed to estimate the state and source components for a fully discretized system. The effectiveness of the algorithm is emphasized in noise-free and noisy cases and an insight on the impact of measurements’ size and location is provided.

Killing vector fields in three dimensions: a method to solve massive gravity field equations

Energy Technology Data Exchange (ETDEWEB)

Guerses, Metin, E-mail: gurses@fen.bilkent.edu.t [Department of Mathematics, Faculty of Sciences, Bilkent University, 06800 Ankara (Turkey)

2010-10-21

Killing vector fields in three dimensions play an important role in the construction of the related spacetime geometry. In this work we show that when a three-dimensional geometry admits a Killing vector field then the Ricci tensor of the geometry is determined in terms of the Killing vector field and its scalars. In this way we can generate all products and covariant derivatives at any order of the Ricci tensor. Using this property we give ways to solve the field equations of topologically massive gravity (TMG) and new massive gravity (NMG) introduced recently. In particular when the scalars of the Killing vector field (timelike, spacelike and null cases) are constants then all three-dimensional symmetric tensors of the geometry, the Ricci and Einstein tensors, their covariant derivatives at all orders, and their products of all orders are completely determined by the Killing vector field and the metric. Hence, the corresponding three-dimensional metrics are strong candidates for solving all higher derivative gravitational field equations in three dimensions.

Solving Linear Differential Equations

NARCIS (Netherlands)

Nguyen, K.A.; Put, M. van der

2010-01-01

The theme of this paper is to 'solve' an absolutely irreducible differential module explicitly in terms of modules of lower dimension and finite extensions of the differential field K. Representations of semi-simple Lie algebras and differential Galo is theory are the main tools. The results extend

A Time Marching Scheme for Solving Volume Integral Equations on Nonlinear Scatterers

KAUST Repository

Bagci, Hakan

2015-01-07

Transient electromagnetic field interactions on inhomogeneous penetrable scatterers can be analyzed by solving time domain volume integral equations (TDVIEs). TDVIEs are oftentimes solved using marchingon-in-time (MOT) schemes. Unlike finite difference and finite element schemes, MOT-TDVIE solvers require discretization of only the scatterers, do not call for artificial absorbing boundary conditions, and are more robust to numerical phase dispersion. On the other hand, their computational cost is high, they suffer from late-time instabilities, and their implicit nature makes incorporation of nonlinear constitutive relations more difficult. Development of plane-wave time-domain (PWTD) and FFT-based schemes has significantly reduced the computational cost of the MOT-TDVIE solvers. Additionally, latetime instability problem has been alleviated for all practical purposes with the development of accurate integration schemes and specially designed temporal basis functions. Addressing the third challenge is the topic of this presentation. I will talk about an explicit MOT scheme developed for solving the TDVIE on scatterers with nonlinear material properties. The proposed scheme separately discretizes the TDVIE and the nonlinear constitutive relation between electric field intensity and flux density. The unknown field intensity and flux density are expanded using half and full Schaubert-Wilton-Glisson (SWG) basis functions in space and polynomial temporal interpolators in time. The resulting coupled system of the discretized TDVIE and constitutive relation is integrated in time using an explicit P E(CE) m scheme to yield the unknown expansion coefficients. Explicitness of time marching allows for straightforward incorporation of the nonlinearity as a function evaluation on the right hand side of the coupled system of equations. Consequently, the resulting MOT scheme does not call for a Newton-like nonlinear solver. Numerical examples, which demonstrate the applicability

A Time Marching Scheme for Solving Volume Integral Equations on Nonlinear Scatterers

KAUST Repository

Bagci, Hakan

2015-01-01

Transient electromagnetic field interactions on inhomogeneous penetrable scatterers can be analyzed by solving time domain volume integral equations (TDVIEs). TDVIEs are oftentimes solved using marchingon-in-time (MOT) schemes. Unlike finite difference and finite element schemes, MOT-TDVIE solvers require discretization of only the scatterers, do not call for artificial absorbing boundary conditions, and are more robust to numerical phase dispersion. On the other hand, their computational cost is high, they suffer from late-time instabilities, and their implicit nature makes incorporation of nonlinear constitutive relations more difficult. Development of plane-wave time-domain (PWTD) and FFT-based schemes has significantly reduced the computational cost of the MOT-TDVIE solvers. Additionally, latetime instability problem has been alleviated for all practical purposes with the development of accurate integration schemes and specially designed temporal basis functions. Addressing the third challenge is the topic of this presentation. I will talk about an explicit MOT scheme developed for solving the TDVIE on scatterers with nonlinear material properties. The proposed scheme separately discretizes the TDVIE and the nonlinear constitutive relation between electric field intensity and flux density. The unknown field intensity and flux density are expanded using half and full Schaubert-Wilton-Glisson (SWG) basis functions in space and polynomial temporal interpolators in time. The resulting coupled system of the discretized TDVIE and constitutive relation is integrated in time using an explicit P E(CE) m scheme to yield the unknown expansion coefficients. Explicitness of time marching allows for straightforward incorporation of the nonlinearity as a function evaluation on the right hand side of the coupled system of equations. Consequently, the resulting MOT scheme does not call for a Newton-like nonlinear solver. Numerical examples, which demonstrate the applicability

Operational matrices with respect to Hermite polynomials and their applications in solving linear dierential equations with variable coecients

Directory of Open Access Journals (Sweden)

A. Aminataei

2014-05-01

Full Text Available In this paper, a new and ecient approach is applied for numerical approximation of the linear dierential equations with variable coecients based on operational matrices with respect to Hermite polynomials. Explicit formulae which express the Hermite expansioncoecients for the moments of derivatives of any dierentiable function in terms of the original expansion coecients of the function itself are given in the matrix form. The mainimportance of this scheme is that using this approach reduces solving the linear dierentialequations to solve a system of linear algebraic equations, thus greatly simplifying the problem. In addition, two experiments are given to demonstrate the validity and applicability of the method

A numerical spectral approach to solve the dislocation density transport equation

International Nuclear Information System (INIS)

Djaka, K S; Taupin, V; Berbenni, S; Fressengeas, C

2015-01-01

A numerical spectral approach is developed to solve in a fast, stable and accurate fashion, the quasi-linear hyperbolic transport equation governing the spatio-temporal evolution of the dislocation density tensor in the mechanics of dislocation fields. The approach relies on using the Fast Fourier Transform algorithm. Low-pass spectral filters are employed to control both the high frequency Gibbs oscillations inherent to the Fourier method and the fast-growing numerical instabilities resulting from the hyperbolic nature of the transport equation. The numerical scheme is validated by comparison with an exact solution in the 1D case corresponding to dislocation dipole annihilation. The expansion and annihilation of dislocation loops in 2D and 3D settings are also produced and compared with finite element approximations. The spectral solutions are shown to be stable, more accurate for low Courant numbers and much less computation time-consuming than the finite element technique based on an explicit Galerkin-least squares scheme. (paper)

On an adaptive time stepping strategy for solving nonlinear diffusion equations

International Nuclear Information System (INIS)

Chen, K.; Baines, M.J.; Sweby, P.K.

1993-01-01

A new time step selection procedure is proposed for solving non- linear diffusion equations. It has been implemented in the ASWR finite element code of Lorenz and Svoboda [10] for 2D semiconductor process modelling diffusion equations. The strategy is based on equi-distributing the local truncation errors of the numerical scheme. The use of B-splines for interpolation (as well as for the trial space) results in a banded and diagonally dominant matrix. The approximate inverse of such a matrix can be provided to a high degree of accuracy by another banded matrix, which in turn can be used to work out the approximate finite difference scheme corresponding to the ASWR finite element method, and further to calculate estimates of the local truncation errors of the numerical scheme. Numerical experiments on six full simulation problems arising in semiconductor process modelling have been carried out. Results show that our proposed strategy is more efficient and better conserves the total mass. 18 refs., 6 figs., 2 tabs

From stochastic processes to numerical methods: A new scheme for solving reaction subdiffusion fractional partial differential equations

Energy Technology Data Exchange (ETDEWEB)

Angstmann, C.N.; Donnelly, I.C. [School of Mathematics and Statistics, UNSW Australia, Sydney NSW 2052 (Australia); Henry, B.I., E-mail: B.Henry@unsw.edu.au [School of Mathematics and Statistics, UNSW Australia, Sydney NSW 2052 (Australia); Jacobs, B.A. [School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050 (South Africa); DST–NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) (South Africa); Langlands, T.A.M. [Department of Mathematics and Computing, University of Southern Queensland, Toowoomba QLD 4350 (Australia); Nichols, J.A. [School of Mathematics and Statistics, UNSW Australia, Sydney NSW 2052 (Australia)

2016-02-15

We have introduced a new explicit numerical method, based on a discrete stochastic process, for solving a class of fractional partial differential equations that model reaction subdiffusion. The scheme is derived from the master equations for the evolution of the probability density of a sum of discrete time random walks. We show that the diffusion limit of the master equations recovers the fractional partial differential equation of interest. This limiting procedure guarantees the consistency of the numerical scheme. The positivity of the solution and stability results are simply obtained, provided that the underlying process is well posed. We also show that the method can be applied to standard reaction–diffusion equations. This work highlights the broader applicability of using discrete stochastic processes to provide numerical schemes for partial differential equations, including fractional partial differential equations.

Highly efficient parallel direct solver for solving dense complex matrix equations from method of moments

Directory of Open Access Journals (Sweden)

Yan Chen

2017-03-01

Full Text Available Based on the vectorised and cache optimised kernel, a parallel lower upper decomposition with a novel communication avoiding pivoting scheme is developed to solve dense complex matrix equations generated by the method of moments. The fine-grain data rearrangement and assembler instructions are adopted to reduce memory accessing times and improve CPU cache utilisation, which also facilitate vectorisation of the code. Through grouping processes in a binary tree, a parallel pivoting scheme is designed to optimise the communication pattern and thus reduces the solving time of the proposed solver. Two large electromagnetic radiation problems are solved on two supercomputers, respectively, and the numerical results demonstrate that the proposed method outperforms those in open source and commercial libraries.

Numerical method for solving the three-dimensional time-dependent neutron diffusion equation

International Nuclear Information System (INIS)

Khaled, S.M.; Szatmary, Z.

2005-01-01

A numerical time-implicit method has been developed for solving the coupled three-dimensional time-dependent multi-group neutron diffusion and delayed neutron precursor equations. The numerical stability of the implicit computation scheme and the convergence of the iterative associated processes have been evaluated. The computational scheme requires the solution of large linear systems at each time step. For this purpose, the point over-relaxation Gauss-Seidel method was chosen. A new scheme was introduced instead of the usual source iteration scheme. (author)

Numerical methods to solve the two-dimensional heat conduction equation

International Nuclear Information System (INIS)

Santos, R.S. dos.

1981-09-01

A class of numerical methods, called 'Hopscotch Algorithms', was used to solve the heat conduction equation in cylindrical geometry. Using a time dependent heat source, the temperature versus time behaviour of cylindric rod was analysed. Numerical simulation was used to study the stability and the convergence of each different method. Another test had the temperature specified on the outer surface as boundary condition. The various Hopscotch methods analysed exhibit differing degrees of accuracy, few of them being so accurate as the ADE method, but requiring more computational operations than the later, were observed. Finally, compared with the so called ODD-EVEN method, two other Hopscotch methods, are more time consuming. (Author) [pt

Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations

Directory of Open Access Journals (Sweden)

Huiping Cao

2014-01-01

Full Text Available Schubert’s method is an extension of Broyden’s method for solving sparse nonlinear equations, which can preserve the zero-nonzero structure defined by the sparse Jacobian matrix and can retain many good properties of Broyden’s method. In particular, Schubert’s method has been proved to be locally and q-superlinearly convergent. In this paper, we globalize Schubert’s method by using a nonmonotone line search. Under appropriate conditions, we show that the proposed algorithm converges globally and superlinearly. Some preliminary numerical experiments are presented, which demonstrate that our algorithm is effective for large-scale problems.

Solving equations by topological methods

Directory of Open Access Journals (Sweden)

Lech Górniewicz

2005-01-01

Full Text Available In this paper we survey most important results from topological fixed point theory which can be directly applied to differential equations. Some new formulations are presented. We believe that our article will be useful for analysts applying topological fixed point theory in nonlinear analysis and in differential equations.

An iterative algorithm for solving the multidimensional neutron diffusion nodal method equations on parallel computers

International Nuclear Information System (INIS)

Kirk, B.L.; Azmy, Y.Y.

1992-01-01

In this paper the one-group, steady-state neutron diffusion equation in two-dimensional Cartesian geometry is solved using the nodal integral method. The discrete variable equations comprise loosely coupled sets of equations representing the nodal balance of neutrons, as well as neutron current continuity along rows or columns of computational cells. An iterative algorithm that is more suitable for solving large problems concurrently is derived based on the decomposition of the spatial domain and is accelerated using successive overrelaxation. This algorithm is very well suited for parallel computers, especially since the spatial domain decomposition occurs naturally, so that the number of iterations required for convergence does not depend on the number of processors participating in the calculation. Implementation of the authors' algorithm on the Intel iPSC/2 hypercube and Sequent Balance 8000 parallel computer is presented, and measured speedup and efficiency for test problems are reported. The results suggest that the efficiency of the hypercube quickly deteriorates when many processors are used, while the Sequent Balance retains very high efficiency for a comparable number of participating processors. This leads to the conjecture that message-passing parallel computers are not as well suited for this algorithm as shared-memory machines

The Use of Generalized Laguerre Polynomials in Spectral Methods for Solving Fractional Delay Differential Equations.

Science.gov (United States)

Khader, M M

2013-10-01

In this paper, an efficient numerical method for solving the fractional delay differential equations (FDDEs) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based on the derived approximate formula of the Laguerre polynomials. The properties of Laguerre polynomials are utilized to reduce FDDEs to a linear or nonlinear system of algebraic equations. Special attention is given to study the error and the convergence analysis of the proposed method. Several numerical examples are provided to confirm that the proposed method is in excellent agreement with the exact solution.

The meshless local Petrov-Galerkin method based on moving Kriging interpolation for solving the time fractional Navier-Stokes equations.

Science.gov (United States)

Thamareerat, N; Luadsong, A; Aschariyaphotha, N

2016-01-01

In this paper, we present a numerical scheme used to solve the nonlinear time fractional Navier-Stokes equations in two dimensions. We first employ the meshless local Petrov-Galerkin (MLPG) method based on a local weak formulation to form the system of discretized equations and then we will approximate the time fractional derivative interpreted in the sense of Caputo by a simple quadrature formula. The moving Kriging interpolation which possesses the Kronecker delta property is applied to construct shape functions. This research aims to extend and develop further the applicability of the truly MLPG method to the generalized incompressible Navier-Stokes equations. Two numerical examples are provided to illustrate the accuracy and efficiency of the proposed algorithm. Very good agreement between the numerically and analytically computed solutions can be observed in the verification. The present MLPG method has proved its efficiency and reliability for solving the two-dimensional time fractional Navier-Stokes equations arising in fluid dynamics as well as several other problems in science and engineering.

Study of a method to solve the one speed, three dimensional transport equation using the finite element method and the associated Legendre function

International Nuclear Information System (INIS)

Fernandes, A.

1991-01-01

A method to solve three dimensional neutron transport equation and it is based on the original work suggested by J.K. Fletcher (42, 43). The angular dependence of the flux is approximated by associated Legendre functions and the finite element method is applied to the space components is presented. When the angular flux, the scattering cross section and the neutrons source are expanded in associated Legendre functions, the first order neutron transport equation is reduced to a coupled set of second order diffusion like equations. These equations are solved in an iterative way by the finite element method to the moments. (author)

DIAGRAM SOLVE THE USE OF SIMULINK BLOCK DIAGRAM TO SOLVE MATHEMA THEMATICAL CONTROL EQU MATHEMATICAL MODELS AND CONTROL EQUATIONS

Directory of Open Access Journals (Sweden)

N.M. Ghasem

2003-12-01

Full Text Available In this paper, the simulink block diagram is used to solve a model consists of a set of ordinary differential and algebraic equations to control the temperature inside a simple stirred tank heater. The flexibility of simulink block diagram gives students a better understanding of the control systems. The simulink also allows solution of mathematical models and easy visualization of the system variables. A polyethylene fluidized bed reactor is considered as an industrial example and the effect of the Proportional, Integral and Derivative control policy is presented for comparison.

Solving large-scale sparse eigenvalue problems and linear systems of equations for accelerator modeling

International Nuclear Information System (INIS)

Gene Golub; Kwok Ko

2009-01-01

The solutions of sparse eigenvalue problems and linear systems constitute one of the key computational kernels in the discretization of partial differential equations for the modeling of linear accelerators. The computational challenges faced by existing techniques for solving those sparse eigenvalue problems and linear systems call for continuing research to improve on the algorithms so that ever increasing problem size as required by the physics application can be tackled. Under the support of this award, the filter algorithm for solving large sparse eigenvalue problems was developed at Stanford to address the computational difficulties in the previous methods with the goal to enable accelerator simulations on then the world largest unclassified supercomputer at NERSC for this class of problems. Specifically, a new method, the Hemitian skew-Hemitian splitting method, was proposed and researched as an improved method for solving linear systems with non-Hermitian positive definite and semidefinite matrices.

Mixed and mixed-hybrid elements for the diffusion equation

International Nuclear Information System (INIS)

Coulomb, F.; Fedon-Magnaud, C.

1987-04-01

To solve the diffusion equation, one often uses a Lagrangian finite element method. We want to introduce the mixed elements which allow a simultaneous approximation of the same order for the flux and its gradient. Though the linear systems are not positive definite, it is possible to make them so by eliminating some of the unknowns

Newton's method for solving a quadratic matrix equation with special coefficient matrices

International Nuclear Information System (INIS)

Seo, Sang-Hyup; Seo, Jong Hyun; Kim, Hyun-Min

2014-01-01

We consider the iterative method for solving a quadratic matrix equation with special coefficient matrices which arises in the quasi-birth-death problem. In this paper, we show that the elementwise minimal positive solvents to quadratic matrix equations can be obtained using Newton's method. We also prove that the convergence rate of the Newton iteration is quadratic if the Fréchet derivative at the elementwise minimal positive solvent is nonsingular. However, if the Fréchet derivative is singular, the convergence rate is at least linear. Numerical experiments of the convergence rate are given.(This is summarized a paper which is to appear in Honam Mathematical Journal.)

Recent symbolic summation methods to solve coupled systems of differential and difference equations

International Nuclear Information System (INIS)

Schneider, Carsten; Bluemlein, Johannes; Freitas, Abilio de

2014-07-01

We outline a new algorithm to solve coupled systems of differential equations in one continuous variable x (resp. coupled difference equations in one discrete variable N) depending on a small parameter ε: given such a system and given sufficiently many initial values, we can determine the first coefficients of the Laurent-series solutions in ε if they are expressible in terms of indefinite nested sums and products. This systematic approach is based on symbolic summation algorithms in the context of difference rings/fields and uncoupling algorithms. The proposed method gives rise to new interesting applications in connection with integration by parts (IBP) methods. As an illustrative example, we will demonstrate how one can calculate the ε-expansion of a ladder graph with 6 massive fermion lines.

Recent symbolic summation methods to solve coupled systems of differential and difference equations

Energy Technology Data Exchange (ETDEWEB)

Schneider, Carsten [Johannes Kepler Univ., Linz (Austria). Research Inst. for Symbolic Computation (RISC); Bluemlein, Johannes; Freitas, Abilio de [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany)

2014-07-15

We outline a new algorithm to solve coupled systems of differential equations in one continuous variable x (resp. coupled difference equations in one discrete variable N) depending on a small parameter ε: given such a system and given sufficiently many initial values, we can determine the first coefficients of the Laurent-series solutions in ε if they are expressible in terms of indefinite nested sums and products. This systematic approach is based on symbolic summation algorithms in the context of difference rings/fields and uncoupling algorithms. The proposed method gives rise to new interesting applications in connection with integration by parts (IBP) methods. As an illustrative example, we will demonstrate how one can calculate the ε-expansion of a ladder graph with 6 massive fermion lines.

Comparison of two Ssub(infinity) methods for solving the neutron transport equation

International Nuclear Information System (INIS)

Mennig, J.; Brandt, D.; Haelg, W.

1978-01-01

A semianalytic method (S 0 sub(infinity)) is presented for solving the monoenergetic multi-region transport equation. This method is compared with results from S 1 sub(infinity)-theory given in the literature. Application of S 1 sub(infinity)-theory to reactor shields may lead to negative neutron fluxes and to flux oscillations. These unphysical effects are completely avoided by the new method. Numerical results demonstrate the limitations of S 1 sub(infinity) and confirm the numerical stability of (S 0 sub(infinity)). (Auth.)

He's homotopy perturbation method for solving systems of Volterra integral equations of the second kind

International Nuclear Information System (INIS)

Biazar, J.; Ghazvini, H.

2009-01-01

In this paper, the He's homotopy perturbation method is applied to solve systems of Volterra integral equations of the second kind. Some examples are presented to illustrate the ability of the method for linear and non-linear such systems. The results reveal that the method is very effective and simple.

Two split cell numerical methods for solving 2-D non-equilibrium radiation transport equations

International Nuclear Information System (INIS)

Feng Tinggui

2004-11-01

Two numerically positive methods, the step characteristic integral method and subcell balance method, for solving radiative transfer equations on quadrilateral grids are presented. Numerical examples shows that the schemes presented are feasible on non-rectangle grid computation, and that the computing results by the schemes presented are comparative to that by the discrete ordinate diamond scheme on rectangle grid. (author)

Propel: A Discontinuous-Galerkin Finite Element Code for Solving the Reacting Navier-Stokes Equations

Science.gov (United States)

Johnson, Ryan; Kercher, Andrew; Schwer, Douglas; Corrigan, Andrew; Kailasanath, Kazhikathra

2017-11-01

This presentation focuses on the development of a Discontinuous Galerkin (DG) method for application to chemically reacting flows. The in-house code, called Propel, was developed by the Laboratory of Computational Physics and Fluid Dynamics at the Naval Research Laboratory. It was designed specifically for developing advanced multi-dimensional algorithms to run efficiently on new and innovative architectures such as GPUs. For these results, Propel solves for convection and diffusion simultaneously with detailed transport and thermodynamics. Chemistry is currently solved in a time-split approach using Strang-splitting with finite element DG time integration of chemical source terms. Results presented here show canonical unsteady reacting flow cases, such as co-flow and splitter plate, and we report performance for higher order DG on CPU and GPUs.

New numerical approximation for solving fractional delay differential equations of variable order using artificial neural networks

Science.gov (United States)

Zúñiga-Aguilar, C. J.; Coronel-Escamilla, A.; Gómez-Aguilar, J. F.; Alvarado-Martínez, V. M.; Romero-Ugalde, H. M.

2018-02-01

In this paper, we approximate the solution of fractional differential equations with delay using a new approach based on artificial neural networks. We consider fractional differential equations of variable order with the Mittag-Leffler kernel in the Liouville-Caputo sense. With this new neural network approach, an approximate solution of the fractional delay differential equation is obtained. Synaptic weights are optimized using the Levenberg-Marquardt algorithm. The neural network effectiveness and applicability were validated by solving different types of fractional delay differential equations, linear systems with delay, nonlinear systems with delay and a system of differential equations, for instance, the Newton-Leipnik oscillator. The solution of the neural network was compared with the analytical solutions and the numerical simulations obtained through the Adams-Bashforth-Moulton method. To show the effectiveness of the proposed neural network, different performance indices were calculated.

Algorithms to solve coupled systems of differential equations in terms of power series

International Nuclear Information System (INIS)

Ablinger, Jakob; Schneider, Carsten

2016-08-01

Using integration by parts relations, Feynman integrals can be represented in terms of coupled systems of differential equations. In the following we suppose that the unknown Feynman integrals can be given in power series representations, and that sufficiently many initial values of the integrals are given. Then there exist algorithms that decide constructively if the coefficients of their power series representations can be given within the class of nested sums over hypergeometric products. In this article we work out the calculation steps that solve this problem. First, we present a successful tactic that has been applied recently to challenging problems coming from massive 3-loop Feynman integrals. Here our main tool is to solve scalar linear recurrences within the class of nested sums over hypergeometric products. Second, we will present a new variation of this tactic which relies on more involved summation technologies but succeeds in reducing the problem to solve scalar recurrences with lower recurrence orders. The article works out the different challenges of this new tactic and demonstrates how they can be treated efficiently with our existing summation technologies.

Improved Quasi-Newton method via PSB update for solving systems of nonlinear equations

Science.gov (United States)

Mamat, Mustafa; Dauda, M. K.; Waziri, M. Y.; Ahmad, Fadhilah; Mohamad, Fatma Susilawati

2016-10-01

The Newton method has some shortcomings which includes computation of the Jacobian matrix which may be difficult or even impossible to compute and solving the Newton system in every iteration. Also, the common setback with some quasi-Newton methods is that they need to compute and store an n × n matrix at each iteration, this is computationally costly for large scale problems. To overcome such drawbacks, an improved Method for solving systems of nonlinear equations via PSB (Powell-Symmetric-Broyden) update is proposed. In the proposed method, the approximate Jacobian inverse Hk of PSB is updated and its efficiency has improved thereby require low memory storage, hence the main aim of this paper. The preliminary numerical results show that the proposed method is practically efficient when applied on some benchmark problems.

Fast numerical method for solving the three-dimensional Stokes' equations in the presence of suspended particles

International Nuclear Information System (INIS)

Fogelson, A.L.; Peskin, C.S.

1988-01-01

A new fast numerical method for solving the three-dimensional Stokes' equations in the presence of suspended particles is presented. The fluid dynamics equations are solved on a lattice. A particle is represented by a set of points each of which moves at the local fluid velocity and is not constrained to lie on the lattice. These points are coupled by forces which resist deformation of the particle. These forces contribute to the force density in the Stokes' equations. As a result, a single set of fluid dynamics equations holds at all points of the domain and there are no internal boundaries. Particles size, shape, and deformability may be prescribed. Computational work increases only linearly with the number of particles, so large numbers (500--1000) of particles may be studied efficiently. The numerical method involves implicit calculation of the particle forces by minimizing an energy function and solution of a finite-difference approximation to the Stokes' equations using the Fourier--Toeplitz method. The numerical method has been implemented to run on all CRAY computers: the implementation exploits the CRAY's vectorized arithmetic, and on machines with insufficient central memory, it performs efficient disk I/O while storing most of the data on disk. Applications of the method to sedimentation of one-, two-, and many-particle systems are described. Trajectories and settling speeds for two-particle sedimentation, and settling speed for multiparticle sedimentation from initial distributions on a cubic lattice or at random give good quantitative agreement with existing theories. copyright 1988 Academic Press, Inc

Solving the neutron diffusion equation on combinatorial geometry computational cells for reactor physics calculations

International Nuclear Information System (INIS)

Azmy, Y. Y.

2004-01-01

An approach is developed for solving the neutron diffusion equation on combinatorial geometry computational cells, that is computational cells composed by combinatorial operations involving simple-shaped component cells. The only constraint on the component cells from which the combinatorial cells are assembled is that they possess a legitimate discretization of the underlying diffusion equation. We use the Finite Difference (FD) approximation of the x, y-geometry diffusion equation in this work. Performing the same combinatorial operations involved in composing the combinatorial cell on these discrete-variable equations yields equations that employ new discrete variables defined only on the combinatorial cell's volume and faces. The only approximation involved in this process, beyond the truncation error committed in discretizing the diffusion equation over each component cell, is a consistent-order Legendre series expansion. Preliminary results for simple configurations establish the accuracy of the solution to the combinatorial geometry solution compared to straight FD as the system dimensions decrease. Furthermore numerical results validate the consistent Legendre-series expansion order by illustrating the second order accuracy of the combinatorial geometry solution, the same as standard FD. Nevertheless the magnitude of the error for the new approach is larger than FD's since it incorporates the additional truncated series approximation. (authors)

Results of numerically solving an integral equation for a two-fermion system

International Nuclear Information System (INIS)

Skachkov, N.B.; Solov'eva, T.M.

2003-01-01

A two-particle system is described by integral equations whose kernels are dependent on the total energy of the system. Such equations can be reduced to an eigenvalue problem featuring an eigenvalue-dependent operator. This nonlinear eigenvalue problem is solved by means of an iterative procedure developed by the present authors. The energy spectra of a two-fermion system formed by particles of identical masses are obtained for two cases, that where the total spin of the system is equal to zero and that where the total spin of the system is equal to unity. The splitting of the ground-state levels of positronium and dimuonium, the frequency of the transition from the ground state of orthopositronium to its first excited state, and the probabilities of parapositronium and paradimuonium decays are computed. The results obtained in this way are found to be in good agreement with experimental data

Markovian Monte Carlo program EvolFMC v.2 for solving QCD evolution equations

Science.gov (United States)

Jadach, S.; Płaczek, W.; Skrzypek, M.; Stokłosa, P.

2010-02-01

We present the program EvolFMC v.2 that solves the evolution equations in QCD for the parton momentum distributions by means of the Monte Carlo technique based on the Markovian process. The program solves the DGLAP-type evolution as well as modified-DGLAP ones. In both cases the evolution can be performed in the LO or NLO approximation. The quarks are treated as massless. The overall technical precision of the code has been established at 5×10. This way, for the first time ever, we demonstrate that with the Monte Carlo method one can solve the evolution equations with precision comparable to the other numerical methods. New version program summaryProgram title: EvolFMC v.2 Catalogue identifier: AEFN_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEFN_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including binary test data, etc.: 66 456 (7407 lines of C++ code) No. of bytes in distributed program, including test data, etc.: 412 752 Distribution format: tar.gz Programming language: C++ Computer: PC, Mac Operating system: Linux, Mac OS X RAM: Less than 256 MB Classification: 11.5 External routines: ROOT ( http://root.cern.ch/drupal/) Nature of problem: Solution of the QCD evolution equations for the parton momentum distributions of the DGLAP- and modified-DGLAP-type in the LO and NLO approximations. Solution method: Monte Carlo simulation of the Markovian process of a multiple emission of partons. Restrictions:Limited to the case of massless partons. Implemented in the LO and NLO approximations only. Weighted events only. Unusual features: Modified-DGLAP evolutions included up to the NLO level. Additional comments: Technical precision established at 5×10. Running time: For the 10 6 events at 100 GeV: DGLAP NLO: 27s; C-type modified DGLAP NLO: 150s (MacBook Pro with Mac OS X v.10

Improved algorithm for solving nonlinear parabolized stability equations

International Nuclear Information System (INIS)

Zhao Lei; Zhang Cun-bo; Liu Jian-xin; Luo Ji-sheng

2016-01-01

Due to its high computational efficiency and ability to consider nonparallel and nonlinear effects, nonlinear parabolized stability equations (NPSE) approach has been widely used to study the stability and transition mechanisms. However, it often diverges in hypersonic boundary layers when the amplitude of disturbance reaches a certain level. In this study, an improved algorithm for solving NPSE is developed. In this algorithm, the mean flow distortion is included into the linear operator instead of into the nonlinear forcing terms in NPSE. An under-relaxation factor for computing the nonlinear terms is introduced during the iteration process to guarantee the robustness of the algorithm. Two case studies, the nonlinear development of stationary crossflow vortices and the fundamental resonance of the second mode disturbance in hypersonic boundary layers, are presented to validate the proposed algorithm for NPSE. Results from direct numerical simulation (DNS) are regarded as the baseline for comparison. Good agreement can be found between the proposed algorithm and DNS, which indicates the great potential of the proposed method on studying the crossflow and streamwise instability in hypersonic boundary layers. (paper)

A Discrete-Time Recurrent Neural Network for Solving Rank-Deficient Matrix Equations With an Application to Output Regulation of Linear Systems.

Science.gov (United States)

Liu, Tao; Huang, Jie

2017-04-17

This paper presents a discrete-time recurrent neural network approach to solving systems of linear equations with two features. First, the system of linear equations may not have a unique solution. Second, the system matrix is not known precisely, but a sequence of matrices that converges to the unknown system matrix exponentially is known. The problem is motivated from solving the output regulation problem for linear systems. Thus, an application of our main result leads to an online solution to the output regulation problem for linear systems.

Gender, Space, and the Location Changes of Jobs and People : A Spatial Simultaneous Equations Analysis

NARCIS (Netherlands)

Hoogstra, Gerke J.

This article summarizes a spatial econometric analysis of local population and employment growth in the Netherlands, with specific reference to impacts of gender and space. The simultaneous equations model used distinguishes between population- and gender-specific employment groups, and includes

Algorithm for solving the linear Cauchy problem for large systems of ordinary differential equations with the use of parallel computations

Energy Technology Data Exchange (ETDEWEB)

Moryakov, A. V., E-mail: sailor@orc.ru [National Research Centre Kurchatov Institute (Russian Federation)

2016-12-15

An algorithm for solving the linear Cauchy problem for large systems of ordinary differential equations is presented. The algorithm for systems of first-order differential equations is implemented in the EDELWEISS code with the possibility of parallel computations on supercomputers employing the MPI (Message Passing Interface) standard for the data exchange between parallel processes. The solution is represented by a series of orthogonal polynomials on the interval [0, 1]. The algorithm is characterized by simplicity and the possibility to solve nonlinear problems with a correction of the operator in accordance with the solution obtained in the previous iterative process.

A parallel algorithm for solving the multidimensional within-group discrete ordinates equations with spatial domain decomposition - 104

International Nuclear Information System (INIS)

Zerr, R.J.; Azmy, Y.Y.

2010-01-01

A spatial domain decomposition with a parallel block Jacobi solution algorithm has been developed based on the integral transport matrix formulation of the discrete ordinates approximation for solving the within-group transport equation. The new methodology abandons the typical source iteration scheme and solves directly for the fully converged scalar flux. Four matrix operators are constructed based upon the integral form of the discrete ordinates equations. A single differential mesh sweep is performed to construct these operators. The method is parallelized by decomposing the problem domain into several smaller sub-domains, each treated as an independent problem. The scalar flux of each sub-domain is solved exactly given incoming angular flux boundary conditions. Sub-domain boundary conditions are updated iteratively, and convergence is achieved when the scalar flux error in all cells meets a pre-specified convergence criterion. The method has been implemented in a computer code that was then employed for strong scaling studies of the algorithm's parallel performance via a fixed-size problem in tests ranging from one domain up to one cell per sub-domain. Results indicate that the best parallel performance compared to source iterations occurs for optically thick, highly scattering problems, the variety that is most difficult for the traditional SI scheme to solve. Moreover, the minimum execution time occurs when each sub-domain contains a total of four cells. (authors)

A parallel algorithm for solving linear equations arising from one-dimensional network problems

International Nuclear Information System (INIS)

Mesina, G.L.

1991-01-01

One-dimensional (1-D) network problems, such as those arising from 1- D fluid simulations and electrical circuitry, produce systems of sparse linear equations which are nearly tridiagonal and contain a few non-zero entries outside the tridiagonal. Most direct solution techniques for such problems either do not take advantage of the special structure of the matrix or do not fully utilize parallel computer architectures. We describe a new parallel direct linear equation solution algorithm, called TRBR, which is especially designed to take advantage of this structure on MIMD shared memory machines. The new method belongs to a family of methods which split the coefficient matrix into the sum of a tridiagonal matrix T and a matrix comprised of the remaining coefficients R. Efficient tridiagonal methods are used to algebraically simplify the linear system. A smaller auxiliary subsystem is created and solved and its solution is used to calculate the solution of the original system. The newly devised BR method solves the subsystem. The serial and parallel operation counts are given for the new method and related earlier methods. TRBR is shown to have the smallest operation count in this class of direct methods. Numerical results are given. Although the algorithm is designed for one-dimensional networks, it has been applied successfully to three-dimensional problems as well. 20 refs., 2 figs., 4 tabs

Chemical Equation Balancing.

Science.gov (United States)

Blakley, G. R.

1982-01-01

Reviews mathematical techniques for solving systems of homogeneous linear equations and demonstrates that the algebraic method of balancing chemical equations is a matter of solving a system of homogeneous linear equations. FORTRAN programs using this matrix method to chemical equation balancing are available from the author. (JN)

LUCKY-TD code for solving the time-dependent transport equation with the use of parallel computations

Energy Technology Data Exchange (ETDEWEB)

Moryakov, A. V., E-mail: sailor@orc.ru [National Research Centre Kurchatov Institute (Russian Federation)

2016-12-15

An algorithm for solving the time-dependent transport equation in the P{sub m}S{sub n} group approximation with the use of parallel computations is presented. The algorithm is implemented in the LUCKY-TD code for supercomputers employing the MPI standard for the data exchange between parallel processes.

Finite element method for solving Kohn-Sham equations based on self-adaptive tetrahedral mesh

International Nuclear Information System (INIS)

Zhang Dier; Shen Lihua; Zhou Aihui; Gong Xingao

2008-01-01

A finite element (FE) method with self-adaptive mesh-refinement technique is developed for solving the density functional Kohn-Sham equations. The FE method adopts local piecewise polynomials basis functions, which produces sparsely structured matrices of Hamiltonian. The method is well suitable for parallel implementation without using Fourier transform. In addition, the self-adaptive mesh-refinement technique can control the computational accuracy and efficiency with optimal mesh density in different regions

The orthogonal gradients method: A radial basis functions method for solving partial differential equations on arbitrary surfaces

KAUST Repository

Piret, Cé cile

2012-01-01

Much work has been done on reconstructing arbitrary surfaces using the radial basis function (RBF) method, but one can hardly find any work done on the use of RBFs to solve partial differential equations (PDEs) on arbitrary surfaces. In this paper

Numerical method for solving integral equations of neutron transport. II

International Nuclear Information System (INIS)

Loyalka, S.K.; Tsai, R.W.

1975-01-01

In a recent paper it was pointed out that the weakly singular integral equations of neutron transport can be quite conveniently solved by a method based on subtraction of singularity. This previous paper was devoted entirely to the consideration of simple one-dimensional isotropic-scattering and one-group problems. The present paper constitutes interesting extensions of the previous work in that in addition to a typical two-group anisotropic-scattering albedo problem in the slab geometry, the method is also applied to an isotropic-scattering problem in the x-y geometry. These results are compared with discrete S/sub N/ (ANISN or TWOTRAN-II) results, and for the problems considered here, the proposed method is found to be quite effective. Thus, the method appears to hold considerable potential for future applications. (auth)

Solving Fully Fuzzy Linear System of Equations in General Form

Directory of Open Access Journals (Sweden)

A. Yousefzadeh

2012-06-01

Full Text Available In this work, we propose an approach for computing the positive solution of a fully fuzzy linear system where the coefficient matrix is a fuzzy $nimes n$ matrix. To do this, we use arithmetic operations on fuzzy numbers that introduced by Kaffman in and convert the fully fuzzy linear system into two $nimes n$ and $2nimes 2n$ crisp linear systems. If the solutions of these linear systems don't satisfy in positive fuzzy solution condition, we introduce the constrained least squares problem to obtain optimal fuzzy vector solution by applying the ranking function in given fully fuzzy linear system. Using our proposed method, the fully fuzzy linear system of equations always has a solution. Finally, we illustrate the efficiency of proposed method by solving some numerical examples.

NUMERICAL TECHNIQUES TO SOLVE CONDENSATIONAL AND DISSOLUTIONAL GROWTH EQUATIONS WHEN GROWTH IS COUPLED TO REVERSIBLE REACTIONS (R823186)

Science.gov (United States)

Noniterative, unconditionally stable numerical techniques for solving condensational anddissolutional growth equations are given. Growth solutions are compared to Gear-code solutions forthree cases when growth is coupled to reversible equilibrium chemistry. In all cases, ...

Construction of Interval Wavelet Based on Restricted Variational Principle and Its Application for Solving Differential Equations

OpenAIRE

Mei, Shu-Li; Lv, Hong-Liang; Ma, Qin

2008-01-01

Based on restricted variational principle, a novel method for interval wavelet construction is proposed. For the excellent local property of quasi-Shannon wavelet, its interval wavelet is constructed, and then applied to solve ordinary differential equations. Parameter choices for the interval wavelet method are discussed and its numerical performance is demonstrated.

Diagrams benefit symbolic problem-solving.

Science.gov (United States)

Chu, Junyi; Rittle-Johnson, Bethany; Fyfe, Emily R

2017-06-01

The format of a mathematics problem often influences students' problem-solving performance. For example, providing diagrams in conjunction with story problems can benefit students' understanding, choice of strategy, and accuracy on story problems. However, it remains unclear whether providing diagrams in conjunction with symbolic equations can benefit problem-solving performance as well. We tested the impact of diagram presence on students' performance on algebra equation problems to determine whether diagrams increase problem-solving success. We also examined the influence of item- and student-level factors to test the robustness of the diagram effect. We worked with 61 seventh-grade students who had received 2 months of pre-algebra instruction. Students participated in an experimenter-led classroom session. Using a within-subjects design, students solved algebra problems in two matched formats (equation and equation-with-diagram). The presence of diagrams increased equation-solving accuracy and the use of informal strategies. This diagram benefit was independent of student ability and item complexity. The benefits of diagrams found previously for story problems generalized to symbolic problems. The findings are consistent with cognitive models of problem-solving and suggest that diagrams may be a useful additional representation of symbolic problems. © 2017 The British Psychological Society.

A Divide-and-Conquer Approach for Solving Fuzzy Max-Archimedean t-Norm Relational Equations

Directory of Open Access Journals (Sweden)

Jun-Lin Lin

2014-01-01

Full Text Available A system of fuzzy relational equations with the max-Archimedean t-norm composition was considered. The relevant literature indicated that this problem can be reduced to the problem of finding all the irredundant coverings of a binary matrix. A divide-and-conquer approach is proposed to solve this problem and, subsequently, to solve the original problem. This approach was used to analyze the binary matrix and then decompose the matrix into several submatrices such that the irredundant coverings of the original matrix could be constructed using the irredundant coverings of each of these submatrices. This step was performed recursively for each of these submatrices to obtain the irredundant coverings. Finally, once all the irredundant coverings of the original matrix were found, they were easily converted into the minimal solutions of the fuzzy relational equations. Experiments on binary matrices, with the number of irredundant coverings ranging from 24 to 9680, were also performed. The results indicated that, for test matrices that could initially be partitioned into more than one submatrix, this approach reduced the execution time by more than three orders of magnitude. For the other test matrices, this approach was still useful because certain submatrices could be partitioned into more than one submatrix.

A 3D nodal mixed dual method for nuclear reactor kinetics with improved quasistatic model and a semi-implicit scheme to solve the precursor equations

International Nuclear Information System (INIS)

Dahmani, M.; Baudron, A.M.; Lautard, J.J.; Erradi, L.

2001-01-01

The mixed dual nodal method MINOS is used to solve the reactor kinetics equations with improved quasistatic IQS model and the θ method is used to solve the precursor equations. The speed of calculation which is the main advantage of the MINOS method and the possibility to use the large time step for shape flux calculation permitted by the IQS method, allow us to reduce considerably the computing time. The IQS/MINOS method is implemented in CRONOS 3D reactor code. Numerical tests on different transient benchmarks show that the results obtained with the IQS/MINOS method and the direct numerical method used to solve the kinetics equations, are very close and the total computing time is largely reduced

Local Fractional Laplace Variational Iteration Method for Solving Linear Partial Differential Equations with Local Fractional Derivative

Directory of Open Access Journals (Sweden)

Ai-Min Yang

2014-01-01

Full Text Available The local fractional Laplace variational iteration method was applied to solve the linear local fractional partial differential equations. The local fractional Laplace variational iteration method is coupled by the local fractional variational iteration method and Laplace transform. The nondifferentiable approximate solutions are obtained and their graphs are also shown.

Unbiased minimum variance estimator of a matrix exponential function. Application to Boltzmann/Bateman coupled equations solving

International Nuclear Information System (INIS)

Dumonteil, E.; Diop, C. M.

2009-01-01

This paper derives an unbiased minimum variance estimator (UMVE) of a matrix exponential function of a normal wean. The result is then used to propose a reference scheme to solve Boltzmann/Bateman coupled equations, thanks to Monte Carlo transport codes. The last section will present numerical results on a simple example. (authors)

Improved algorithm for solving nonlinear parabolized stability equations

Science.gov (United States)

Zhao, Lei; Zhang, Cun-bo; Liu, Jian-xin; Luo, Ji-sheng

2016-08-01

Due to its high computational efficiency and ability to consider nonparallel and nonlinear effects, nonlinear parabolized stability equations (NPSE) approach has been widely used to study the stability and transition mechanisms. However, it often diverges in hypersonic boundary layers when the amplitude of disturbance reaches a certain level. In this study, an improved algorithm for solving NPSE is developed. In this algorithm, the mean flow distortion is included into the linear operator instead of into the nonlinear forcing terms in NPSE. An under-relaxation factor for computing the nonlinear terms is introduced during the iteration process to guarantee the robustness of the algorithm. Two case studies, the nonlinear development of stationary crossflow vortices and the fundamental resonance of the second mode disturbance in hypersonic boundary layers, are presented to validate the proposed algorithm for NPSE. Results from direct numerical simulation (DNS) are regarded as the baseline for comparison. Good agreement can be found between the proposed algorithm and DNS, which indicates the great potential of the proposed method on studying the crossflow and streamwise instability in hypersonic boundary layers. Project supported by the National Natural Science Foundation of China (Grant Nos. 11332007 and 11402167).

Laser radiation in active amplifying media treated as a transport problem - Transfer equation derived and exactly solved

Science.gov (United States)

Gupta, S. R. D.; Gupta, Santanu D.

1991-10-01

The flow of laser radiation in a plane-parallel cylindrical slab of active amplifying medium with axial symmetry is treated as a problem in radiative transfer. The appropriate one-dimensional transfer equation describing the transfer of laser radiation has been derived by an appeal to Einstein's A, B coefficients (describing the processes of stimulated line absorption, spontaneous line emission, and stimulated line emission sustained by population inversion in the medium) and considering the 'rate equations' to completely establish the rational of the transfer equation obtained. The equation is then exactly solved and the angular distribution of the emergent laser beam intensity is obtained; its numerically computed values are given in tables and plotted in graphs showing the nature of peaks of the emerging laser beam intensity about the axis of the laser cylinder.

T-Stability of the Heun Method and Balanced Method for Solving Stochastic Differential Delay Equations

Directory of Open Access Journals (Sweden)

Xiaolin Zhu

2014-01-01

Full Text Available This paper studies the T-stability of the Heun method and balanced method for solving stochastic differential delay equations (SDDEs. Two T-stable conditions of the Heun method are obtained for two kinds of linear SDDEs. Moreover, two conditions under which the balanced method is T-stable are obtained for two kinds of linear SDDEs. Some numerical examples verify the theoretical results proposed.

A semi-analytical approach for solving of nonlinear systems of functional differential equations with delay

Science.gov (United States)

Rebenda, Josef; Šmarda, Zdeněk

2017-07-01

In the paper, we propose a correct and efficient semi-analytical approach to solve initial value problem for systems of functional differential equations with delay. The idea is to combine the method of steps and differential transformation method (DTM). In the latter, formulas for proportional arguments and nonlinear terms are used. An example of using this technique for a system with constant and proportional delays is presented.

On a numerical method for solving integro-differential equations with variable coefficients with applications in finance

Science.gov (United States)

Kudryavtsev, O.; Rodochenko, V.

2018-03-01

We propose a new general numerical method aimed to solve integro-differential equations with variable coefficients. The problem under consideration arises in finance where in the context of pricing barrier options in a wide class of stochastic volatility models with jumps. To handle the effect of the correlation between the price and the variance, we use a suitable substitution for processes. Then we construct a Markov-chain approximation for the variation process on small time intervals and apply a maturity randomization technique. The result is a system of boundary problems for integro-differential equations with constant coefficients on the line in each vertex of the chain. We solve the arising problems using a numerical Wiener-Hopf factorization method. The approximate formulae for the factors are efficiently implemented by means of the Fast Fourier Transform. Finally, we use a recurrent procedure that moves backwards in time on the variance tree. We demonstrate the convergence of the method using Monte-Carlo simulations and compare our results with the results obtained by the Wiener-Hopf method with closed-form expressions of the factors.

Solving QCD evolution equations in rapidity space with Markovian Monte Carlo

CERN Document Server

Golec-Biernat, K; Placzek, W; Skrzypek, M

2009-01-01

This work covers methodology of solving QCD evolution equation of the parton distribution using Markovian Monte Carlo (MMC) algorithms in a class of models ranging from DGLAP to CCFM. One of the purposes of the above MMCs is to test the other more sophisticated Monte Carlo programs, the so-called Constrained Monte Carlo (CMC) programs, which will be used as a building block in the parton shower MC. This is why the mapping of the evolution variables (eikonal variable and evolution time) into four-momenta is also defined and tested. The evolution time is identified with the rapidity variable of the emitted parton. The presented MMCs are tested independently, with ~0.1% precision, against the non-MC program APCheb especially devised for this purpose.

A fast, parallel algorithm to solve the basic fluvial erosion/transport equations

Science.gov (United States)

Braun, J.

2012-04-01

Quantitative models of landform evolution are commonly based on the solution of a set of equations representing the processes of fluvial erosion, transport and deposition, which leads to predict the geometry of a river channel network and its evolution through time. The river network is often regarded as the backbone of any surface processes model (SPM) that might include other physical processes acting at a range of spatial and temporal scales along hill slopes. The basic laws of fluvial erosion requires the computation of local (slope) and non-local (drainage area) quantities at every point of a given landscape, a computationally expensive operation which limits the resolution of most SPMs. I present here an algorithm to compute the various components required in the parameterization of fluvial erosion (and transport) and thus solve the basic fluvial geomorphic equation, that is very efficient because it is O(n) (the number of required arithmetic operations is linearly proportional to the number of nodes defining the landscape), and is fully parallelizable (the computation cost decreases in a direct inverse proportion to the number of processors used to solve the problem). The algorithm is ideally suited for use on latest multi-core processors. Using this new technique, geomorphic problems can be solved at an unprecedented resolution (typically of the order of 10,000 X 10,000 nodes) while keeping the computational cost reasonable (order 1 sec per time step). Furthermore, I will show that the algorithm is applicable to any regular or irregular representation of the landform, and is such that the temporal evolution of the landform can be discretized by a fully implicit time-marching algorithm, making it unconditionally stable. I will demonstrate that such an efficient algorithm is ideally suited to produce a fully predictive SPM that links observationally based parameterizations of small-scale processes to the evolution of large-scale features of the landscapes on

Constructing Frozen Jacobian Iterative Methods for Solving Systems of Nonlinear Equations, Associated with ODEs and PDEs Using the Homotopy Method

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Uswah Qasim

2016-03-01

Full Text Available A homotopy method is presented for the construction of frozen Jacobian iterative methods. The frozen Jacobian iterative methods are attractive because the inversion of the Jacobian is performed in terms of LUfactorization only once, for a single instance of the iterative method. We embedded parameters in the iterative methods with the help of the homotopy method: the values of the parameters are determined in such a way that a better convergence rate is achieved. The proposed homotopy technique is general and has the ability to construct different families of iterative methods, for solving weakly nonlinear systems of equations. Further iterative methods are also proposed for solving general systems of nonlinear equations.

Instructional Supports for Representational Fluency in Solving Linear Equations with Computer Algebra Systems and Paper-and-Pencil

Science.gov (United States)

Fonger, Nicole L.; Davis, Jon D.; Rohwer, Mary Lou

2018-01-01

This research addresses the issue of how to support students' representational fluency--the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper-and-pencil classroom environment is…

Parallel assembling and equation solving via graph algorithms with an application to the FE simulation of metal extrusion processes

CERN Document Server

Unterkircher, A

2005-01-01

We propose methods for parallel assembling and iterative equation solving based on graph algorithms. The assembling technique is independent of dimension, element type and model shape. As a parallel solving technique we construct a multiplicative symmetric Schwarz preconditioner for the conjugate gradient method. Both methods have been incorporated into a non-linear FE code to simulate 3D metal extrusion processes. We illustrate the efficiency of these methods on shared memory computers by realistic examples.

The Adomian decomposition method for solving partial differential equations of fractal order in finite domains

Energy Technology Data Exchange (ETDEWEB)

El-Sayed, A.M.A. [Faculty of Science University of Alexandria (Egypt)]. E-mail: amasyed@hotmail.com; Gaber, M. [Faculty of Education Al-Arish, Suez Canal University (Egypt)]. E-mail: mghf408@hotmail.com

2006-11-20

The Adomian decomposition method has been successively used to find the explicit and numerical solutions of the time fractional partial differential equations. A different examples of special interest with fractional time and space derivatives of order {alpha}, 0<{alpha}=<1 are considered and solved by means of Adomian decomposition method. The behaviour of Adomian solutions and the effects of different values of {alpha} are shown graphically for some examples.

On the Evaluation of Computational Results Obtained from Solving System of linear Equations With matlab The Dual affine Scalling interior Point

International Nuclear Information System (INIS)

Murfi, Hendri; Basaruddin, T.

2001-01-01

The interior point method for linear programming has gained extraordinary interest as an alternative to simplex method since Karmarkar presented a polynomial-time algorithm for linear programming based on interior point method. In implementation of the algorithm of this method, there are two important things that have impact heavily to performance of the algorithm; they are data structure and used method to solve linear equation system in the algorithm. This paper describes about solving linear equation system in variants of the algorithm called dual-affine scaling algorithm. Next, we evaluate experimentally results of some used methods, either direct method or iterative method. The experimental evaluation used Matlab

An efficient hybrid pseudospectral/finite-difference scheme for solving the TTI pure P-wave equation

KAUST Repository

Zhan, Ge

2013-02-19

The pure P-wave equation for modelling and migration in tilted transversely isotropic (TTI) media has attracted more and more attention in imaging seismic data with anisotropy. The desirable feature is that it is absolutely free of shear-wave artefacts and the consequent alleviation of numerical instabilities generally suffered by some systems of coupled equations. However, due to several forward-backward Fourier transforms in wavefield updating at each time step, the computational cost is significant, and thereby hampers its prevalence. We propose to use a hybrid pseudospectral (PS) and finite-difference (FD) scheme to solve the pure P-wave equation. In the hybrid solution, most of the cost-consuming wavenumber terms in the equation are replaced by inexpensive FD operators, which in turn accelerates the computation and reduces the computational cost. To demonstrate the benefit in cost saving of the new scheme, 2D and 3D reverse-time migration (RTM) examples using the hybrid solution to the pure P-wave equation are carried out, and respective runtimes are listed and compared. Numerical results show that the hybrid strategy demands less computation time and is faster than using the PS method alone. Furthermore, this new TTI RTM algorithm with the hybrid method is computationally less expensive than that with the FD solution to conventional TTI coupled equations. © 2013 Sinopec Geophysical Research Institute.

An efficient hybrid pseudospectral/finite-difference scheme for solving the TTI pure P-wave equation

International Nuclear Information System (INIS)

Zhan, Ge; Pestana, Reynam C; Stoffa, Paul L

2013-01-01

The pure P-wave equation for modelling and migration in tilted transversely isotropic (TTI) media has attracted more and more attention in imaging seismic data with anisotropy. The desirable feature is that it is absolutely free of shear-wave artefacts and the consequent alleviation of numerical instabilities generally suffered by some systems of coupled equations. However, due to several forward–backward Fourier transforms in wavefield updating at each time step, the computational cost is significant, and thereby hampers its prevalence. We propose to use a hybrid pseudospectral (PS) and finite-difference (FD) scheme to solve the pure P-wave equation. In the hybrid solution, most of the cost-consuming wavenumber terms in the equation are replaced by inexpensive FD operators, which in turn accelerates the computation and reduces the computational cost. To demonstrate the benefit in cost saving of the new scheme, 2D and 3D reverse-time migration (RTM) examples using the hybrid solution to the pure P-wave equation are carried out, and respective runtimes are listed and compared. Numerical results show that the hybrid strategy demands less computation time and is faster than using the PS method alone. Furthermore, this new TTI RTM algorithm with the hybrid method is computationally less expensive than that with the FD solution to conventional TTI coupled equations. (paper)

An extended step characteristic method for solving the transport equation in general geometries

International Nuclear Information System (INIS)

DeHart, M.D.; Pevey, R.E.; Parish, T.A.

1994-01-01

A method for applying the discrete ordinates method to solve the Boltzmann transport equation on arbitrary two-dimensional meshes has been developed. The finite difference approach normally used to approximate spatial derivatives in extrapolating angular fluxes across a cell is replaced by direct solution of the characteristic form of the transport equation for each discrete direction. Thus, computational cells are not restricted to the geometrical shape of a mesh element characteristic of a given coordinate system. However, in terms of the treatment of energy and angular dependencies, this method resembles traditional discrete ordinates techniques. By using the method developed here, a general two-dimensional space can be approximated by an irregular mesh comprised of arbitrary polygons. Results for a number of test problems have been compared with solutions obtained from traditional methods, with good agreement. Comparisons include benchmarks against analytical results for problems with simple geometry, as well as numerical results obtained from traditional discrete ordinates methods by applying the ANISN and TWOTRAN-II computer programs

Iterative Algorithm for Solving a Class of Quaternion Matrix Equation over the Generalized (P,Q-Reflexive Matrices

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Ning Li

2013-01-01

Full Text Available The matrix equation ∑l=1uAlXBl+∑s=1vCsXTDs=F, which includes some frequently investigated matrix equations as its special cases, plays important roles in the system theory. In this paper, we propose an iterative algorithm for solving the quaternion matrix equation ∑l=1uAlXBl+∑s=1vCsXTDs=F over generalized (P,Q-reflexive matrices. The proposed iterative algorithm automatically determines the solvability of the quaternion matrix equation over generalized (P,Q-reflexive matrices. When the matrix equation is consistent over generalized (P,Q-reflexive matrices, the sequence {X(k} generated by the introduced algorithm converges to a generalized (P,Q-reflexive solution of the quaternion matrix equation. And the sequence {X(k} converges to the least Frobenius norm generalized (P,Q-reflexive solution of the quaternion matrix equation when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate generalized (P,Q-reflexive solution for a given generalized (P,Q-reflexive matrix X0 can be derived. The numerical results indicate that the iterative algorithm is quite efficient.

Convergence properties of iterative algorithms for solving the nodal diffusion equations

International Nuclear Information System (INIS)

Azmy, Y.Y.; Kirk, B.L.

1990-01-01

We drive the five point form of the nodal diffusion equations in two-dimensional Cartesian geometry and develop three iterative schemes to solve the discrete-variable equations: the unaccelerated, partial Successive Over Relaxation (SOR), and the full SOR methods. By decomposing the iteration error into its Fourier modes, we determine the spectral radius of each method for infinite medium, uniform model problems, and for the unaccelerated and partial SOR methods for finite medium, uniform model problems. Also for the two variants of the SOR method we determine the optimal relaxation factor that results in the smallest number of iterations required for convergence. Our results indicate that the number of iterations for the unaccelerated and partial SOR methods is second order in the number of nodes per dimension, while, for the full SOR this behavior is first order, resulting in much faster convergence for very large problems. We successfully verify the results of the spectral analysis against those of numerical experiments, and we show that for the full SOR method the linear dependence of the number of iterations on the number of nodes per dimension is relatively insensitive to the value of the relaxation parameter, and that it remains linear even for heterogenous problems. 14 refs., 1 fig

Solving the Telegraph and Oscillatory Differential Equations by a Block Hybrid Trigonometrically Fitted Algorithm

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F. F. Ngwane

2015-01-01

Full Text Available We propose a block hybrid trigonometrically fitted (BHT method, whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs, including systems arising from the semidiscretization of hyperbolic Partial Differential Equations (PDEs, such as the Telegraph equation. The BHT is formulated from eight discrete hybrid formulas which are provided by a continuous two-step hybrid trigonometrically fitted method with two off-grid points. The BHT is implemented in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHT is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.

COMPARISON OF IMPLICIT SCHEMES TO SOLVE EQUATIONS OF RADIATION HYDRODYNAMICS WITH A FLUX-LIMITED DIFFUSION APPROXIMATION: NEWTON–RAPHSON, OPERATOR SPLITTING, AND LINEARIZATION

Energy Technology Data Exchange (ETDEWEB)

Tetsu, Hiroyuki; Nakamoto, Taishi, E-mail: h.tetsu@geo.titech.ac.jp [Earth and Planetary Sciences, Tokyo Institute of Technology, Tokyo 152-8551 (Japan)

2016-03-15

Radiation is an important process of energy transport, a force, and a basis for synthetic observations, so radiation hydrodynamics (RHD) calculations have occupied an important place in astrophysics. However, although the progress in computational technology is remarkable, their high numerical cost is still a persistent problem. In this work, we compare the following schemes used to solve the nonlinear simultaneous equations of an RHD algorithm with the flux-limited diffusion approximation: the Newton–Raphson (NR) method, operator splitting, and linearization (LIN), from the perspective of the computational cost involved. For operator splitting, in addition to the traditional simple operator splitting (SOS) scheme, we examined the scheme developed by Douglas and Rachford (DROS). We solve three test problems (the thermal relaxation mode, the relaxation and the propagation of linear waves, and radiating shock) using these schemes and then compare their dependence on the time step size. As a result, we find the conditions of the time step size necessary for adopting each scheme. The LIN scheme is superior to other schemes if the ratio of radiation pressure to gas pressure is sufficiently low. On the other hand, DROS can be the most efficient scheme if the ratio is high. Although the NR scheme can be adopted independently of the regime, especially in a problem that involves optically thin regions, the convergence tends to be worse. In all cases, SOS is not practical.

Numerical Solution of Heun Equation Via Linear Stochastic Differential Equation

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Hamidreza Rezazadeh

2014-05-01

Full Text Available In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation(S.D.E.. So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding differential equations.

A theory of solving TAP equations for Ising models with general invariant random matrices

DEFF Research Database (Denmark)

Opper, Manfred; Çakmak, Burak; Winther, Ole

2016-01-01

We consider the problem of solving TAP mean field equations by iteration for Ising models with coupling matrices that are drawn at random from general invariant ensembles. We develop an analysis of iterative algorithms using a dynamical functional approach that in the thermodynamic limit yields...... the iteration dependent on a Gaussian distributed field only. The TAP magnetizations are stable fixed points if a de Almeida–Thouless stability criterion is fulfilled. We illustrate our method explicitly for coupling matrices drawn from the random orthogonal ensemble....

A numerical technique for solving fractional optimal control problems and fractional Riccati differential equations

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F. Ghomanjani

2016-10-01

Full Text Available In the present paper, we apply the Bezier curves method for solving fractional optimal control problems (OCPs and fractional Riccati differential equations. The main advantage of this method is that it can reduce the error of the approximate solutions. Hence, the solutions obtained using the Bezier curve method give good approximations. Some numerical examples are provided to confirm the accuracy of the proposed method. All of the numerical computations have been performed on a PC using several programs written in MAPLE 13.

A highly efficient parallel algorithm for solving the neutron diffusion nodal equations on shared-memory computers

International Nuclear Information System (INIS)

Azmy, Y.Y.; Kirk, B.L.

1990-01-01

Modern parallel computer architectures offer an enormous potential for reducing CPU and wall-clock execution times of large-scale computations commonly performed in various applications in science and engineering. Recently, several authors have reported their efforts in developing and implementing parallel algorithms for solving the neutron diffusion equation on a variety of shared- and distributed-memory parallel computers. Testing of these algorithms for a variety of two- and three-dimensional meshes showed significant speedup of the computation. Even for very large problems (i.e., three-dimensional fine meshes) executed concurrently on a few nodes in serial (nonvector) mode, however, the measured computational efficiency is very low (40 to 86%). In this paper, the authors present a highly efficient (∼85 to 99.9%) algorithm for solving the two-dimensional nodal diffusion equations on the Sequent Balance 8000 parallel computer. Also presented is a model for the performance, represented by the efficiency, as a function of problem size and the number of participating processors. The model is validated through several tests and then extrapolated to larger problems and more processors to predict the performance of the algorithm in more computationally demanding situations

Numerical solution of the Schroedinger equation with a polynomial potential

International Nuclear Information System (INIS)

Campoy, G.; Palma, A.

1986-01-01

A numerical method for solving the Schroedinger equation for a potential expressed as a polynomial is proposed. The basic assumption relies on the asymptotic properties of the solution of this equation. It is possible to obtain the energies and the stationary state functions simultaneously. They analyze, in particular, the cases of the quartic anharmonic oscillator and a hydrogen atom perturbed by a quadratic term, obtaining its energy eigenvalues for some values of the perturbation parameter. Together with the Hellmann-Feynman theorem, they use their algorithm to calculate expectation values of x'' for arbitrary positive values of n. 4 tables

Development of the hierarchical domain decomposition boundary element method for solving the three-dimensional multiregion neutron diffusion equations

International Nuclear Information System (INIS)

Chiba, Gou; Tsuji, Masashi; Shimazu, Yoichiro

2001-01-01

A hierarchical domain decomposition boundary element method (HDD-BEM) that was developed to solve a two-dimensional neutron diffusion equation has been modified to deal with three-dimensional problems. In the HDD-BEM, the domain is decomposed into homogeneous regions. The boundary conditions on the common inner boundaries between decomposed regions and the neutron multiplication factor are initially assumed. With these assumptions, the neutron diffusion equations defined in decomposed homogeneous regions can be solved respectively by applying the boundary element method. This part corresponds to the 'lower level' calculations. At the 'higher level' calculations, the assumed values, the inner boundary conditions and the neutron multiplication factor, are modified so as to satisfy the continuity conditions for the neutron flux and the neutron currents on the inner boundaries. These procedures of the lower and higher levels are executed alternately and iteratively until the continuity conditions are satisfied within a convergence tolerance. With the hierarchical domain decomposition, it is possible to deal with problems composing a large number of regions, something that has been difficult with the conventional BEM. In this paper, it is showed that a three-dimensional problem even with 722 regions can be solved with a fine accuracy and an acceptable computation time. (author)

TENSOLVE: A software package for solving systems of nonlinear equations and nonlinear least squares problems using tensor methods

Energy Technology Data Exchange (ETDEWEB)

Bouaricha, A. [Argonne National Lab., IL (United States). Mathematics and Computer Science Div.; Schnabel, R.B. [Colorado Univ., Boulder, CO (United States). Dept. of Computer Science

1996-12-31

This paper describes a modular software package for solving systems of nonlinear equations and nonlinear least squares problems, using a new class of methods called tensor methods. It is intended for small to medium-sized problems, say with up to 100 equations and unknowns, in cases where it is reasonable to calculate the Jacobian matrix or approximate it by finite differences at each iteration. The software allows the user to select between a tensor method and a standard method based upon a linear model. The tensor method models F({ital x}) by a quadratic model, where the second-order term is chosen so that the model is hardly more expensive to form, store, or solve than the standard linear model. Moreover, the software provides two different global strategies, a line search and a two- dimensional trust region approach. Test results indicate that, in general, tensor methods are significantly more efficient and robust than standard methods on small and medium-sized problems in iterations and function evaluations.

Modified homotopy perturbation method for solving hypersingular integral equations of the first kind.

Science.gov (United States)

Eshkuvatov, Z K; Zulkarnain, F S; Nik Long, N M A; Muminov, Z

2016-01-01

Modified homotopy perturbation method (HPM) was used to solve the hypersingular integral equations (HSIEs) of the first kind on the interval [-1,1] with the assumption that the kernel of the hypersingular integral is constant on the diagonal of the domain. Existence of inverse of hypersingular integral operator leads to the convergence of HPM in certain cases. Modified HPM and its norm convergence are obtained in Hilbert space. Comparisons between modified HPM, standard HPM, Bernstein polynomials approach Mandal and Bhattacharya (Appl Math Comput 190:1707-1716, 2007), Chebyshev expansion method Mahiub et al. (Int J Pure Appl Math 69(3):265-274, 2011) and reproducing kernel Chen and Zhou (Appl Math Lett 24:636-641, 2011) are made by solving five examples. Theoretical and practical examples revealed that the modified HPM dominates the standard HPM and others. Finally, it is found that the modified HPM is exact, if the solution of the problem is a product of weights and polynomial functions. For rational solution the absolute error decreases very fast by increasing the number of collocation points.

Computer programs for solving systems of nonlinear equations

International Nuclear Information System (INIS)

Asaoka, Takumi

1978-03-01

Computer programs to find a solution, usually the one closest to some guess, of a system of simultaneous nonlinear equations are provided for real functions of the real arguments. These are based on quasi-Newton methods or projection methods, which are briefly reviewed in the present report. Benchmark tests were performed on these subroutines to grasp their characteristics. As the program not requiring analytical forms of the derivatives of the Jacobian matrix, we have dealt with NS01A of Powell, NS03A of Reid for a system with the sparse Jacobian and NONLIN of Brown. Of these three subroutines of quasi-Newton methods, NONLIN is shown to be the most useful because of its stable algorithm and short computation time. On the other hand, as the subroutine for which the derivatives of the Jacobian are to be supplied analytically, we have tested INTECH of a quasi-Newton method based on the Boggs' algorithm, PROJA of Georg and Keller based on the projection method and an option of NS03A. The results have shown that INTECH, treating variables which appear only linearly in the functions separately, takes the shortest computation time, on the whole, while the projection method requires further research to find an optimal algorithm. (auth.)

Modified Taylor series method for solving nonlinear differential equations with mixed boundary conditions defined on finite intervals.

Science.gov (United States)

Vazquez-Leal, Hector; Benhammouda, Brahim; Filobello-Nino, Uriel Antonio; Sarmiento-Reyes, Arturo; Jimenez-Fernandez, Victor Manuel; Marin-Hernandez, Antonio; Herrera-May, Agustin Leobardo; Diaz-Sanchez, Alejandro; Huerta-Chua, Jesus

2014-01-01

In this article, we propose the application of a modified Taylor series method (MTSM) for the approximation of nonlinear problems described on finite intervals. The issue of Taylor series method with mixed boundary conditions is circumvented using shooting constants and extra derivatives of the problem. In order to show the benefits of this proposal, three different kinds of problems are solved: three-point boundary valued problem (BVP) of third-order with a hyperbolic sine nonlinearity, two-point BVP for a second-order nonlinear differential equation with an exponential nonlinearity, and a two-point BVP for a third-order nonlinear differential equation with a radical nonlinearity. The result shows that the MTSM method is capable to generate easily computable and highly accurate approximations for nonlinear equations. 34L30.

Fibonacci collocation method with a residual error Function to solve linear Volterra integro differential equations

Directory of Open Access Journals (Sweden)

Salih Yalcinbas

2016-01-01

Full Text Available In this paper, a new collocation method based on the Fibonacci polynomials is introduced to solve the high-order linear Volterra integro-differential equations under the conditions. Numerical examples are included to demonstrate the applicability and validity of the proposed method and comparisons are made with the existing results. In addition, an error estimation based on the residual functions is presented for this method. The approximate solutions are improved by using this error estimation.

Solving Differential Equations in R

Science.gov (United States)

Although R is still predominantly applied for statistical analysis and graphical representation, it is rapidly becoming more suitable for mathematical computing. One of the fields where considerable progress has been made recently is the solution of differential equations. Here w...

Solutions of system of P1 equations without use of auxiliary differential equations coupled

International Nuclear Information System (INIS)

Martinez, Aquilino Senra; Silva, Fernando Carvalho da; Cardoso, Carlos Eduardo Santos

2000-01-01

The system of P1 equations is composed by two equations coupled itself one for the neutron flux and other for the current. Usually this system is solved by definitions of two integrals parameters, which are named slowing down densities of the flux and the current. Hence, the system P1 can be change from integral to only two differential equations. However, there are two new differentials equations that may be solved with the initial system. The present work analyzes this procedure and studies a method, which solve the P1 equations directly, without definitions of slowing down densities. (author)

Finite difference method and algebraic polynomial interpolation for numerically solving Poisson's equation over arbitrary domains

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Tsugio Fukuchi

2014-06-01

Full Text Available The finite difference method (FDM based on Cartesian coordinate systems can be applied to numerical analyses over any complex domain. A complex domain is usually taken to mean that the geometry of an immersed body in a fluid is complex; here, it means simply an analytical domain of arbitrary configuration. In such an approach, we do not need to treat the outer and inner boundaries differently in numerical calculations; both are treated in the same way. Using a method that adopts algebraic polynomial interpolations in the calculation around near-wall elements, all the calculations over irregular domains reduce to those over regular domains. Discretization of the space differential in the FDM is usually derived using the Taylor series expansion; however, if we use the polynomial interpolation systematically, exceptional advantages are gained in deriving high-order differences. In using the polynomial interpolations, we can numerically solve the Poisson equation freely over any complex domain. Only a particular type of partial differential equation, Poisson's equations, is treated; however, the arguments put forward have wider generality in numerical calculations using the FDM.

Augmented Lagrangian methods to solve Navier-Stokes equations for a Bingham fluid flow

International Nuclear Information System (INIS)

Boscardin, Laetitia

1999-01-01

The objective of this research thesis is to develop one or more methods for the numerical resolution of equations of movement obtained for a Bingham fluid. The resolution of Navier-Stokes equations is processed by splitting elliptic and hyperbolic operators (Galerkin transport). In this purpose, the author first studied the Stokes problem, and then addressed issues of stability and consistency of the global scheme. The variational formulation of the Stokes problem can be expressed under the form of a minimisation problem under the constraint of non linear and non differentiable functions. Then, the author proposes a discretization of the Stokes problem based on a hybrid finite element method. Then he extends the demonstrations of stability and consistency of the Galerkin-transport scheme which have been established for a Newtonian fluid, to the case of a Bingham fluid. A relaxation algorithm and a Newton-GMRES algorithm are developed to solve the problem, and their convergence is studied. To ensure this convergence, some constraints must be verified. In order to do so, a specific speed element has been developed [fr

Expansion methods for solving integral equations with multiple time lags using Bernstein polynomial of the second kind

Directory of Open Access Journals (Sweden)

Mahmoud Paripour

2014-08-01

Full Text Available In this paper, the Bernstein polynomials are used to approximatethe solutions of linear integral equations with multiple time lags (IEMTL through expansion methods (collocation method, partition method, Galerkin method. The method is discussed in detail and illustrated by solving some numerical examples. Comparison between the exact and approximated results obtained from these methods is carried out

On the comparison of perturbation-iteration algorithm and residual power series method to solve fractional Zakharov-Kuznetsov equation

Science.gov (United States)

Şenol, Mehmet; Alquran, Marwan; Kasmaei, Hamed Daei

2018-06-01

In this paper, we present analytic-approximate solution of time-fractional Zakharov-Kuznetsov equation. This model demonstrates the behavior of weakly nonlinear ion acoustic waves in a plasma bearing cold ions and hot isothermal electrons in the presence of a uniform magnetic field. Basic definitions of fractional derivatives are described in the Caputo sense. Perturbation-iteration algorithm (PIA) and residual power series method (RPSM) are applied to solve this equation with success. The convergence analysis is also presented for both methods. Numerical results are given and then they are compared with the exact solutions. Comparison of the results reveal that both methods are competitive, powerful, reliable, simple to use and ready to apply to wide range of fractional partial differential equations.

An energy-stable method for solving the incompressible Navier-Stokes equations with non-slip boundary condition

Science.gov (United States)

Lee, Byungjoon; Min, Chohong

2018-05-01

We introduce a stable method for solving the incompressible Navier-Stokes equations with variable density and viscosity. Our method is stable in the sense that it does not increase the total energy of dynamics that is the sum of kinetic energy and potential energy. Instead of velocity, a new state variable is taken so that the kinetic energy is formulated by the L2 norm of the new variable. Navier-Stokes equations are rephrased with respect to the new variable, and a stable time discretization for the rephrased equations is presented. Taking into consideration the incompressibility in the Marker-And-Cell (MAC) grid, we present a modified Lax-Friedrich method that is L2 stable. Utilizing the discrete integration-by-parts in MAC grid and the modified Lax-Friedrich method, the time discretization is fully discretized. An explicit CFL condition for the stability of the full discretization is given and mathematically proved.

A generalized fractional sub-equation method for fractional differential equations with variable coefficients

International Nuclear Information System (INIS)

Tang, Bo; He, Yinnian; Wei, Leilei; Zhang, Xindong

2012-01-01

In this Letter, a generalized fractional sub-equation method is proposed for solving fractional differential equations with variable coefficients. Being concise and straightforward, this method is applied to the space–time fractional Gardner equation with variable coefficients. As a result, many exact solutions are obtained including hyperbolic function solutions, trigonometric function solutions and rational solutions. It is shown that the considered method provides a very effective, convenient and powerful mathematical tool for solving many other fractional differential equations in mathematical physics. -- Highlights: ► Study of fractional differential equations with variable coefficients plays a role in applied physical sciences. ► It is shown that the proposed algorithm is effective for solving fractional differential equations with variable coefficients. ► The obtained solutions may give insight into many considerable physical processes.

Analytic solutions of hydrodynamics equations

International Nuclear Information System (INIS)

Coggeshall, S.V.

1991-01-01

Many similarity solutions have been found for the equations of one-dimensional (1-D) hydrodynamics. These special combinations of variables allow the partial differential equations to be reduced to ordinary differential equations, which must then be solved to determine the physical solutions. Usually, these reduced ordinary differential equations are solved numerically. In some cases it is possible to solve these reduced equations analytically to obtain explicit solutions. In this work a collection of analytic solutions of the 1-D hydrodynamics equations is presented. These can be used for a variety of purposes, including (i) numerical benchmark problems, (ii) as a basis for analytic models, and (iii) to provide insight into more complicated solutions

PRECONDITIONED CONJUGATE-GRADIENT 2 (PCG2), a computer program for solving ground-water flow equations

Science.gov (United States)

Hill, Mary C.

1990-01-01

This report documents PCG2 : a numerical code to be used with the U.S. Geological Survey modular three-dimensional, finite-difference, ground-water flow model . PCG2 uses the preconditioned conjugate-gradient method to solve the equations produced by the model for hydraulic head. Linear or nonlinear flow conditions may be simulated. PCG2 includes two reconditioning options : modified incomplete Cholesky preconditioning, which is efficient on scalar computers; and polynomial preconditioning, which requires less computer storage and, with modifications that depend on the computer used, is most efficient on vector computers . Convergence of the solver is determined using both head-change and residual criteria. Nonlinear problems are solved using Picard iterations. This documentation provides a description of the preconditioned conjugate gradient method and the two preconditioners, detailed instructions for linking PCG2 to the modular model, sample data inputs, a brief description of PCG2, and a FORTRAN listing.

Accelerated procedure to solve kinetic equation for neutral atoms in a hot plasma

Science.gov (United States)

Tokar, Mikhail Z.

2017-12-01

The recombination of plasma charged components, electrons and ions of hydrogen isotopes, on the wall of a fusion reactor is a source of neutral molecules and atoms, recycling back into the plasma volume. Here neutral species participate, in particular, in charge-exchange (c-x) collisions with the plasma ions and, as a result, atoms of high energies with chaotically directed velocities are generated. Some fraction of these hot atoms hit the wall. Statistical Monte Carlo methods normally used to model c-x atoms are too time consuming for reasonably small level of accident errors and extensive parameter studies are problematic. By applying pass method to evaluate integrals from functions, including the ion velocity distribution, an iteration approach to solve one-dimensional kinetic equation [1], being alternative to Monte Carlo procedure, has been tremendously accelerated, at least by a factor of 30-50 [2]. Here this approach is developed further to solve the 2-D kinetic equation, applied to model the transport of c-x atoms in the vicinity of an opening in the wall, e.g., the entrance of the duct guiding to a diagnostic installation. This is necessary to determine firmly the energy spectrum of c-x atoms penetrating into the duct and to assess the erosion of the installation there. The results of kinetic modeling are compared with those obtained with the diffusion description for c-x atoms, being strictly relevant under plasma conditions of low temperature and high density, where the mean free path length between c-x collisions is much smaller than that till the atom ionization by electrons. It is demonstrated that the previous calculations [3], done with the diffusion approximation for c-x atoms, overestimate the erosion rate of Mo mirrors in a reactor by a factor of 3 compared to the result of the present kinetic study.

Expert Strategies in Solving Algebraic Structure Sense Problems: The Case of Quadratic Equations

Science.gov (United States)

Jupri, Al; Sispiyati, R.

2017-02-01

Structure sense, an intuitive ability towards symbolic expressions, including skills to interpret, to manipulate, and to perceive symbols in different roles, is considered as a key success in learning algebra. In this article, we report results of three phases of a case study on solving algebraic structure sense problems aiming at testing the appropriateness of algebraic structure sense tasks and at investigating expert strategies dealing with the tasks. First, we developed three tasks on quadratic equations based on the characteristics of structure sense for high school algebra. Next, we validated the tasks to seven experts. In the validation process, we requested these experts to solve each task using two different strategies. Finally, we analyzing expert solution strategies in the light of structure sense characteristics. We found that even if eventual expert strategies are in line with the characteristics of structure sense; some of their initial solution strategies used standard procedures which might pay less attention to algebraic structures. This finding suggests that experts have reconsidered their procedural work and have provided more efficient solution strategies. For further investigation, we consider to test the tasks to high school algebra students and to see whether they produce similar results as experts.

Variable-mesh method of solving differential equations

Science.gov (United States)

Van Wyk, R.

1969-01-01

Multistep predictor-corrector method for numerical solution of ordinary differential equations retains high local accuracy and convergence properties. In addition, the method was developed in a form conducive to the generation of effective criteria for the selection of subsequent step sizes in step-by-step solution of differential equations.

Solving Differential Equations Using Modified Picard Iteration

Science.gov (United States)

Robin, W. A.

2010-01-01

Many classes of differential equations are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential equations considered include typical initial value, boundary value and eigenvalue problems arising in physics and…

Solving the radiation diffusion and energy balance equations using pseudo-transient continuation

International Nuclear Information System (INIS)

Shestakov, A.I.; Greenough, J.A.; Howell, L.H.

2005-01-01

We develop a scheme for the system coupling the radiation diffusion and matter energy balance equations. The method is based on fully implicit, first-order, backward Euler differencing; Picard-Newton iterations solve the nonlinear system. We show that iterating on the radiation energy density and the emission source is more robust. Since the Picard-Newton scheme may not converge for all initial conditions and time steps, pseudo-transient continuation (Ψtc) is introduced. The combined Ψtc-Picard-Newton scheme is analyzed. We derive conditions on the Ψtc parameter that guarantee physically meaningful iterates, e.g., positive energies. Successive Ψtc iterates are bounded and the radiation energy density and emission source tend to equilibrate. The scheme is incorporated into a multiply dimensioned, massively parallel, Eulerian, radiation-hydrodynamic computer program with automatic mesh refinement (AMR). Three examples are presented that exemplify the scheme's performance. (1) The Pomraning test problem that models radiation flow into cold matter. (2) A similar, but more realistic problem simulating the propagation of an ionization front into tenuous hydrogen gas with a Saha model for the equation-of-state. (3) A 2D axisymmetric (R,Z) simulation with real materials featuring jetting, radiatively driven, interacting shocks

Aumann Fuzzy Improper Integral and Its Application to Solve Fuzzy Integro-Differential Equations by Laplace Transform Method

Directory of Open Access Journals (Sweden)

Elhassan Eljaoui

2018-01-01

Full Text Available We introduce the Aumann fuzzy improper integral to define the convolution product of a fuzzy mapping and a crisp function in this paper. The Laplace convolution formula is proved in this case and used to solve fuzzy integro-differential equations with kernel of convolution type. Then, we report and correct an error in the article by Salahshour et al. dealing with the same topic.

Application of Trotter approximation for solving time dependent neutron transport equation; Primena Trotterove aproksimacije za resavanje vremenski zavisne transportne jednacine neutrona

Energy Technology Data Exchange (ETDEWEB)

Stancic, V [Institut za nuklearne nauke Boris Kidric, Vinca, Beograd (Yugoslavia)

1987-07-01

A method is proposed to solve multigroup time dependent neutron transport equation with arbitrary scattering anisotropy. The recurrence relation thus obtained is simple, numerically stable and especially suitable for treatment of complicated geometries. (author)

Solving applied mathematical problems with Matlab

CERN Document Server

Xue, Dingyu

2008-01-01

Computer Mathematics Language-An Overview. Fundamentals of MATLAB Programming. Calculus Problems. MATLAB Computations of Linear Algebra Problems. Integral Transforms and Complex Variable Functions. Solutions to Nonlinear Equations and Optimization Problems. MATLAB Solutions to Differential Equation Problems. Solving Interpolations and Approximations Problems. Solving Probability and Mathematical Statistics Problems. Nontraditional Solution Methods for Mathematical Problems.

A regression approach for Zircaloy-2 in-reactor creep constitutive equations

International Nuclear Information System (INIS)

Yung Liu, Y.; Bement, A.L.

1977-01-01

In this paper the methodology of multiple regressions as applied to Zircaloy-2 in-reactor creep data analysis and construction of constitutive equation are illustrated. While the resulting constitutive equation can be used in creep analysis of in-reactor Zircaloy structural components, the methodology itself is entirely general and can be applied to any creep data analysis. The promising aspects of multiple regression creep data analysis are briefly outlined as follows: (1) When there are more than one variable involved, there is no need to make the assumption that each variable affects the response independently. No separate normalizations are required either and the estimation of parameters is obtained by solving many simultaneous equations. The number of simultaneous equations is equal to the number of data sets. (2) Regression statistics such as R 2 - and F-statistics provide measures of the significance of regression creep equation in correlating the overall data. The relative weights of each variable on the response can also be obtained. (3) Special regression techniques such as step-wise, ridge, and robust regressions and residual plots, etc., provide diagnostic tools for model selections. Multiple regression analysis performed on a set of carefully selected Zircaloy-2 in-reactor creep data leads to a model which provides excellent correlations for the data. (Auth.)

Transport methods: general. 7. Formulation of a Fourier-Boltzmann Transformation to Solve the Three-Dimensional Transport Equation

International Nuclear Information System (INIS)

Stancic, V.

2001-01-01

This paper presents some elements of a new approach to solve analytically the linearized three-dimensional (3-D) transport equation of neutral particles. Since this task is of such special importance, we present some results of a paper that is still in progress. The most important is that using this transformation, an integro-differential equation with an analytical solution is obtained. For this purpose, a simplest 3-D equation is being considered which describes the transport process in an infinite medium. Until now, this equation has been analytically considered either using the Laplace transform with respect to time parameter t or applying the Fourier transform over the space coordinate. Both of them reduce the number of differential terms in the equation; however, evaluation of the inverse transformation is complicated. In this paper, we introduce for the first time a Fourier transform induced by the Boltzmann operator. For this, we use a complete set of 3-D eigenfunctions of the Boltzmann transport operator defined in a similar way as those that have been already used in 3-D transport theory as a basic set to transform the transport equation. This set consists of a continuous part and a discrete one with spectral measure. The density distribution equation shows the known form asymptotic behavior. Several applications are to be performed using this equation and compared to the benchmark one. Such an analysis certainly would be out of the available space

Parallel algorithms for solving the diffusion equation by finite elements methods and by nodal methods

International Nuclear Information System (INIS)

Coulomb, F.

1989-06-01

The aim of this work is to study methods for solving the diffusion equation, based on a primal or mixed-dual finite elements discretization and well suited for use on multiprocessors computers; domain decomposition methods are the subject of the main part of this study, the linear systems being solved by the block-Jacobi method. The origin of the diffusion equation is explained in short, and various variational formulations are reminded. A survey of iterative methods is given. The elemination of the flux or current is treated in the case of a mixed method. Numerical tests are performed on two examples of reactors, in order to compare mixed elements and Lagrange elements. A theoretical study of domain decomposition is led in the case of Lagrange finite elements, and convergence conditions for the block-Jacobi method are derived; the dissection decomposition is previously the purpose of a particular numerical analysis. In the case of mixed-dual finite elements, a study is led on examples and is confirmed by numerical tests performed for the dissection decomposition; furthermore, after being justified, decompositions along axes of symmetry are numerically tested. In the case of a decomposition into two subdomains, the dissection decomposition and the decomposition with an integrated interface are compared. Alternative directions methods are defined; the convergence of those relative to Lagrange elements is shown; in the case of mixed elements, convergence conditions are found [fr

Differential equation analysis in biomedical science and engineering ordinary differential equation applications with R

CERN Document Server

Schiesser, William E

2014-01-01

Features a solid foundation of mathematical and computational tools to formulate and solve real-world ODE problems across various fields With a step-by-step approach to solving ordinary differential equations (ODEs), Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R successfully applies computational techniques for solving real-worldODE problems that are found in a variety of fields, including chemistry, physics, biology,and physiology. The book provides readers with the necessary knowledge to reproduce andextend the comp

OpenMP GNU and Intel Fortran programs for solving the time-dependent Gross-Pitaevskii equation

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Young-S., Luis E.; Muruganandam, Paulsamy; Adhikari, Sadhan K.; Lončar, Vladimir; Vudragović, Dušan; Balaž, Antun

2017-11-01

reduce the execution time cannot be overemphasized. To address this issue, we provide here such OpenMP Fortran programs, optimized for both Intel and GNU Fortran compilers and capable of using all available CPU cores, which can significantly reduce the execution time. Summary of revisions: Previous Fortran programs [1] for solving the time-dependent GP equation in 1d, 2d, and 3d with different trap symmetries have been parallelized using the OpenMP interface to reduce the execution time on multi-core processors. There are six different trap symmetries considered, resulting in six programs for imaginary-time propagation and six for real-time propagation, totaling to 12 programs included in BEC-GP-OMP-FOR software package. All input data (number of atoms, scattering length, harmonic oscillator trap length, trap anisotropy, etc.) are conveniently placed at the beginning of each program, as before [2]. Present programs introduce a new input parameter, which is designated by Number_of_Threads and defines the number of CPU cores of the processor to be used in the calculation. If one sets the value 0 for this parameter, all available CPU cores will be used. For the most efficient calculation it is advisable to leave one CPU core unused for the background system's jobs. For example, on a machine with 20 CPU cores such that we used for testing, it is advisable to use up to 19 CPU cores. However, the total number of used CPU cores can be divided into more than one job. For instance, one can run three simulations simultaneously using 10, 4, and 5 CPU cores, respectively, thus totaling to 19 used CPU cores on a 20-core computer. The Fortran source programs are located in the directory src, and can be compiled by the make command using the makefile in the root directory BEC-GP-OMP-FOR of the software package. The examples of produced output files can be found in the directory output, although some large density files are omitted, to save space. The programs calculate the values of

PSsolver: A Maple implementation to solve first order ordinary differential equations with Liouvillian solutions

Science.gov (United States)

Avellar, J.; Duarte, L. G. S.; da Mota, L. A. C. P.

2012-10-01

We present a set of software routines in Maple 14 for solving first order ordinary differential equations (FOODEs). The package implements the Prelle-Singer method in its original form together with its extension to include integrating factors in terms of elementary functions. The package also presents a theoretical extension to deal with all FOODEs presenting Liouvillian solutions. Applications to ODEs taken from standard references show that it solves ODEs which remain unsolved using Maple's standard ODE solution routines. New version program summary Program title: PSsolver Catalogue identifier: ADPR_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADPR_v2_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 2302 No. of bytes in distributed program, including test data, etc.: 31962 Distribution format: tar.gz Programming language: Maple 14 (also tested using Maple 15 and 16). Computer: Intel Pentium Processor P6000, 1.86 GHz. Operating system: Windows 7. RAM: 4 GB DDR3 Memory Classification: 4.3. Catalogue identifier of previous version: ADPR_v1_0 Journal reference of previous version: Comput. Phys. Comm. 144 (2002) 46 Does the new version supersede the previous version?: Yes Nature of problem: Symbolic solution of first order differential equations via the Prelle-Singer method. Solution method: The method of solution is based on the standard Prelle-Singer method, with extensions for the cases when the FOODE contains elementary functions. Additionally, an extension of our own which solves FOODEs with Liouvillian solutions is included. Reasons for new version: The program was not running anymore due to changes in the latest versions of Maple. Additionally, we corrected/changed some bugs/details that were hampering the smoother functioning of the routines. Summary

Solving the master equation without kinetic Monte Carlo: Tensor train approximations for a CO oxidation model

International Nuclear Information System (INIS)

Gelß, Patrick; Matera, Sebastian; Schütte, Christof

2016-01-01

In multiscale modeling of heterogeneous catalytic processes, one crucial point is the solution of a Markovian master equation describing the stochastic reaction kinetics. Usually, this is too high-dimensional to be solved with standard numerical techniques and one has to rely on sampling approaches based on the kinetic Monte Carlo method. In this study we break the curse of dimensionality for the direct solution of the Markovian master equation by exploiting the Tensor Train Format for this purpose. The performance of the approach is demonstrated on a first principles based, reduced model for the CO oxidation on the RuO 2 (110) surface. We investigate the complexity for increasing system size and for various reaction conditions. The advantage over the stochastic simulation approach is illustrated by a problem with increased stiffness.

Solving the master equation without kinetic Monte Carlo: Tensor train approximations for a CO oxidation model

Science.gov (United States)

Gelß, Patrick; Matera, Sebastian; Schütte, Christof

2016-06-01

In multiscale modeling of heterogeneous catalytic processes, one crucial point is the solution of a Markovian master equation describing the stochastic reaction kinetics. Usually, this is too high-dimensional to be solved with standard numerical techniques and one has to rely on sampling approaches based on the kinetic Monte Carlo method. In this study we break the curse of dimensionality for the direct solution of the Markovian master equation by exploiting the Tensor Train Format for this purpose. The performance of the approach is demonstrated on a first principles based, reduced model for the CO oxidation on the RuO2(110) surface. We investigate the complexity for increasing system size and for various reaction conditions. The advantage over the stochastic simulation approach is illustrated by a problem with increased stiffness.

Solving the master equation without kinetic Monte Carlo: Tensor train approximations for a CO oxidation model

Energy Technology Data Exchange (ETDEWEB)

Gelß, Patrick, E-mail: p.gelss@fu-berlin.de; Matera, Sebastian, E-mail: matera@math.fu-berlin.de; Schütte, Christof, E-mail: schuette@mi.fu-berlin.de

2016-06-01

In multiscale modeling of heterogeneous catalytic processes, one crucial point is the solution of a Markovian master equation describing the stochastic reaction kinetics. Usually, this is too high-dimensional to be solved with standard numerical techniques and one has to rely on sampling approaches based on the kinetic Monte Carlo method. In this study we break the curse of dimensionality for the direct solution of the Markovian master equation by exploiting the Tensor Train Format for this purpose. The performance of the approach is demonstrated on a first principles based, reduced model for the CO oxidation on the RuO{sub 2}(110) surface. We investigate the complexity for increasing system size and for various reaction conditions. The advantage over the stochastic simulation approach is illustrated by a problem with increased stiffness.

Some operational tools for solving fractional and higher integer order differential equations: A survey on their mutual relations

Science.gov (United States)

Kiryakova, Virginia S.

2012-11-01

The Laplace Transform (LT) serves as a basis of the Operational Calculus (OC), widely explored by engineers and applied scientists in solving mathematical models for their practical needs. This transform is closely related to the exponential and trigonometric functions (exp, cos, sin) and to the classical differentiation and integration operators, reducing them to simple algebraic operations. Thus, the classical LT and the OC give useful tool to handle differential equations and systems with constant coefficients. Several generalizations of the LT have been introduced to allow solving, in a similar way, of differential equations with variable coefficients and of higher integer orders, as well as of fractional (arbitrary non-integer) orders. Note that fractional order mathematical models are recently widely used to describe better various systems and phenomena of the real world. This paper surveys briefly some of our results on classes of such integral transforms, that can be obtained from the LT by means of "transmutations" which are operators of the generalized fractional calculus (GFC). On the list of these Laplace-type integral transforms, we consider the Borel-Dzrbashjan, Meijer, Krätzel, Obrechkoff, generalized Obrechkoff (multi-index Borel-Dzrbashjan) transforms, etc. All of them are G- and H-integral transforms of convolutional type, having as kernels Meijer's G- or Fox's H-functions. Besides, some special functions (also being G- and H-functions), among them - the generalized Bessel-type and Mittag-Leffler (M-L) type functions, are generating Gel'fond-Leontiev (G-L) operators of generalized differentiation and integration, which happen to be also operators of GFC. Our integral transforms have operational properties analogous to those of the LT - they do algebrize the G-L generalized integrations and differentiations, and thus can serve for solving wide classes of differential equations with variable coefficients of arbitrary, including non-integer order

Electronic representation of wave equation

Energy Technology Data Exchange (ETDEWEB)

Veigend, Petr; Kunovský, Jiří, E-mail: kunovsky@fit.vutbr.cz; Kocina, Filip; Nečasová, Gabriela; Valenta, Václav [University of Technology, Faculty of Information Technology, Božetěchova 2, 612 66 Brno (Czech Republic); Šátek, Václav [IT4Innovations, VŠB Technical University of Ostrava, 17. listopadu 15/2172, 708 33 Ostrava-Poruba (Czech Republic); University of Technology, Faculty of Information Technology, Božetěchova 2, 612 66 Brno (Czech Republic)

2016-06-08

The Taylor series method for solving differential equations represents a non-traditional way of a numerical solution. Even though this method is not much preferred in the literature, experimental calculations done at the Department of Intelligent Systems of the Faculty of Information Technology of TU Brno have verified that the accuracy and stability of the Taylor series method exceeds the currently used algorithms for numerically solving differential equations. This paper deals with solution of Telegraph equation using modelling of a series small pieces of the wire. Corresponding differential equations are solved by the Modern Taylor Series Method.

Differential equation analysis in biomedical science and engineering partial differential equation applications with R

CERN Document Server

Schiesser, William E

2014-01-01

Features a solid foundation of mathematical and computational tools to formulate and solve real-world PDE problems across various fields With a step-by-step approach to solving partial differential equations (PDEs), Differential Equation Analysis in Biomedical Science and Engineering: Partial Differential Equation Applications with R successfully applies computational techniques for solving real-world PDE problems that are found in a variety of fields, including chemistry, physics, biology, and physiology. The book provides readers with the necessary knowledge to reproduce and extend the com

Different seeds to solve the equations of stochastic point kinetics using the Euler-Maruyama method

International Nuclear Information System (INIS)

Suescun D, D.; Oviedo T, M.

2017-09-01

In this paper, a numerical study of stochastic differential equations that describe the kinetics in a nuclear reactor is presented. These equations, known as the stochastic equations of punctual kinetics they model temporal variations in neutron population density and concentrations of deferred neutron precursors. Because these equations are probabilistic in nature (since random oscillations in the neutrons and population of precursors were considered to be approximately normally distributed, and these equations also possess strong coupling and stiffness properties) the proposed method for the numerical simulations is the Euler-Maruyama scheme that provides very good approximations for calculating the neutron population and concentrations of deferred neutron precursors. The method proposed for this work was computationally tested for different seeds, initial conditions, experimental data and forms of reactivity for a group of precursors and then for six groups of deferred neutron precursors at each time step with 5000 Brownian movements per seed. In a paper reported in the literature, the Euler-Maruyama method was proposed, but there are many doubts about the reported values, in addition to not reporting the seed used, so in this work is expected to rectify the reported values. After taking the average of the different seeds used to generate the pseudo-random numbers the results provided by the Euler-Maruyama scheme will be compared in mean and standard deviation with other methods reported in the literature and results of the deterministic model of the equations of the punctual kinetics. This comparison confirms in particular that the Euler-Maruyama scheme is an efficient method to solve the equations of stochastic point kinetics but different from the values found and reported by another author. The Euler-Maruyama method is simple and easy to implement, provides acceptable results for neutron population density and concentration of deferred neutron precursors and

Nodal approximations of varying order by energy group for solving the diffusion equation

International Nuclear Information System (INIS)

Broda, J.T.

1992-02-01

The neutron flux across the nuclear reactor core is of interest to reactor designers and others. The diffusion equation, an integro-differential equation in space and energy, is commonly used to determine the flux level. However, the solution of a simplified version of this equation when automated is very time consuming. Since the flux level changes with time, in general, this calculation must be made repeatedly. Therefore solution techniques that speed the calculation while maintaining accuracy are desirable. One factor that contributes to the solution time is the spatial flux shape approximation used. It is common practice to use the same order flux shape approximation in each energy group even though this method may not be the most efficient. The one-dimensional, two-energy group diffusion equation was solved, for the node average flux and core k-effective, using two sets of spatial shape approximations for each of three reactor types. A fourth-order approximation in both energy groups forms the first set of approximations used. The second set used combines a second-order approximation with a fourth-order approximation in energy group two. Comparison of the results from the two approximation sets show that the use of a different order spatial flux shape approximation results in considerable loss in accuracy for the pressurized water reactor modeled. However, the loss in accuracy is small for the heavy water and graphite reactors modeled. The use of different order approximations in each energy group produces mixed results. Further investigation into the accuracy and computing time is required before any quantitative advantage of the use of the second-order approximation in energy group one and the fourth-order approximation in energy group two can be determined

A Pseudo-Temporal Multi-Grid Relaxation Scheme for Solving the Parabolized Navier-Stokes Equations

Science.gov (United States)

White, J. A.; Morrison, J. H.

1999-01-01

A multi-grid, flux-difference-split, finite-volume code, VULCAN, is presented for solving the elliptic and parabolized form of the equations governing three-dimensional, turbulent, calorically perfect and non-equilibrium chemically reacting flows. The space marching algorithms developed to improve convergence rate and or reduce computational cost are emphasized. The algorithms presented are extensions to the class of implicit pseudo-time iterative, upwind space-marching schemes. A full approximate storage, full multi-grid scheme is also described which is used to accelerate the convergence of a Gauss-Seidel relaxation method. The multi-grid algorithm is shown to significantly improve convergence on high aspect ratio grids.

An accurate anisotropic adaptation method for solving the level set advection equation

International Nuclear Information System (INIS)

Bui, C.; Dapogny, C.; Frey, P.

2012-01-01

In the present paper, a mesh adaptation process for solving the advection equation on a fully unstructured computational mesh is introduced, with a particular interest in the case it implicitly describes an evolving surface. This process mainly relies on a numerical scheme based on the method of characteristics. However, low order, this scheme lends itself to a thorough analysis on the theoretical side. It gives rise to an anisotropic error estimate which enjoys a very natural interpretation in terms of the Hausdorff distance between the exact and approximated surfaces. The computational mesh is then adapted according to the metric supplied by this estimate. The whole process enjoys a good accuracy as far as the interface resolution is concerned. Some numerical features are discussed and several classical examples are presented and commented in two or three dimensions. (authors)

A method for solving the spherical harmonics equations applied for space-energy transport of fast and resonance neutrons

International Nuclear Information System (INIS)

Matausek, M.

1972-01-01

A new proposed method for solving the space-energy dependent spherical harmonics equations represents a methodological contribution to neutron transport theory. The proposed method was applied for solving the problem of spec-energy transport of fast and resonance neutrons in multi-zone, cylindrical y symmetric infinite reactor cell and is related to previously developed procedure for treating the thermal energy region. The advantages of this method are as follows: a unique algorithm was obtained for detailed determination of spatial and energy distribution of neutrons (from thermal to fast) in the reactor cell; these detailed distributions enable more precise calculations of criticality conditions, obtaining adequate multigroup data and better interpretation of experimental data; computing time is rather short

Fourth order Douglas implicit scheme for solving three dimension reaction diffusion equation with non-linear source term

Science.gov (United States)

Hasnain, Shahid; Saqib, Muhammad; Mashat, Daoud Suleiman

2017-07-01

This research paper represents a numerical approximation to non-linear three dimension reaction diffusion equation with non-linear source term from population genetics. Since various initial and boundary value problems exist in three dimension reaction diffusion phenomena, which are studied numerically by different numerical methods, here we use finite difference schemes (Alternating Direction Implicit and Fourth Order Douglas Implicit) to approximate the solution. Accuracy is studied in term of L2, L∞ and relative error norms by random selected grids along time levels for comparison with analytical results. The test example demonstrates the accuracy, efficiency and versatility of the proposed schemes. Numerical results showed that Fourth Order Douglas Implicit scheme is very efficient and reliable for solving 3-D non-linear reaction diffusion equation.

Fourth order Douglas implicit scheme for solving three dimension reaction diffusion equation with non-linear source term

Directory of Open Access Journals (Sweden)

Shahid Hasnain

2017-07-01

Full Text Available This research paper represents a numerical approximation to non-linear three dimension reaction diffusion equation with non-linear source term from population genetics. Since various initial and boundary value problems exist in three dimension reaction diffusion phenomena, which are studied numerically by different numerical methods, here we use finite difference schemes (Alternating Direction Implicit and Fourth Order Douglas Implicit to approximate the solution. Accuracy is studied in term of L2, L∞ and relative error norms by random selected grids along time levels for comparison with analytical results. The test example demonstrates the accuracy, efficiency and versatility of the proposed schemes. Numerical results showed that Fourth Order Douglas Implicit scheme is very efficient and reliable for solving 3-D non-linear reaction diffusion equation.

CTD: a computer program to solve the three dimensional multi-group diffusion equation in X, Y, Z, and triangular Z geometries

Energy Technology Data Exchange (ETDEWEB)

Fletcher, J K

1973-05-01

CTD is a computer program written in Fortran 4 to solve the multi-group diffusion theory equations in X, Y, Z and triangular Z geometries. A power print- out neutron balance and breeding gain are also produced. 4 references. (auth)

An Explicit Upwind Algorithm for Solving the Parabolized Navier-Stokes Equations

Science.gov (United States)

Korte, John J.

1991-01-01

An explicit, upwind algorithm was developed for the direct (noniterative) integration of the 3-D Parabolized Navier-Stokes (PNS) equations in a generalized coordinate system. The new algorithm uses upwind approximations of the numerical fluxes for the pressure and convection terms obtained by combining flux difference splittings (FDS) formed from the solution of an approximate Riemann (RP). The approximate RP is solved using an extension of the method developed by Roe for steady supersonic flow of an ideal gas. Roe's method is extended for use with the 3-D PNS equations expressed in generalized coordinates and to include Vigneron's technique of splitting the streamwise pressure gradient. The difficulty associated with applying Roe's scheme in the subsonic region is overcome. The second-order upwind differencing of the flux derivatives are obtained by adding FDS to either an original forward or backward differencing of the flux derivative. This approach is used to modify an explicit MacCormack differencing scheme into an upwind differencing scheme. The second order upwind flux approximations, applied with flux limiters, provide a method for numerically capturing shocks without the need for additional artificial damping terms which require adjustment by the user. In addition, a cubic equation is derived for determining Vegneron's pressure splitting coefficient using the updated streamwise flux vector. Decoding the streamwise flux vector with the updated value of Vigneron's pressure splitting improves the stability of the scheme. The new algorithm is applied to 2-D and 3-D supersonic and hypersonic laminar flow test cases. Results are presented for the experimental studies of Holden and of Tracy. In addition, a flow field solution is presented for a generic hypersonic aircraft at a Mach number of 24.5 and angle of attack of 1 degree. The computed results compare well to both experimental data and numerical results from other algorithms. Computational times required

LOCFES-B: A program for solving the one-dimensional particle transport equation with user-selected CLOF methods

International Nuclear Information System (INIS)

Jarvis, R.D.; Nelson, P.

1995-01-01

LOCFES-B solves the steady-state, monoenergetic and azimuthally symmetric neutral-particle transport equation in one-dimensional plane-parallel geometry. LOCFES-B is designed to facilitate testing and comparison of different spatial approximations in neutron transport. Accordingly, it permits performance of user-provided CLOF spatial approximations to be compared directly on successively refined mesh sizes and user-input physical problems with automatic comparison of results. if desired, to user-supplied benchmark results

Analysis and development of spatial hp-refinement methods for solving the neutron transport equation

International Nuclear Information System (INIS)

Fournier, D.

2011-01-01

The different neutronic parameters have to be calculated with a higher accuracy in order to design the 4. generation reactor cores. As memory storage and computation time are limited, adaptive methods are a solution to solve the neutron transport equation. The neutronic flux, solution of this equation, depends on the energy, angle and space. The different variables are successively discretized. The energy with a multigroup approach, considering the different quantities to be constant on each group, the angle by a collocation method called SN approximation. Once the energy and angle variable are discretized, a system of spatially-dependent hyperbolic equations has to be solved. Discontinuous finite elements are used to make possible the development of hp-refinement methods. Thus, the accuracy of the solution can be improved by spatial refinement (h-refinement), consisting into subdividing a cell into sub-cells, or by order refinement (p-refinement), by increasing the order of the polynomial basis. In this thesis, the properties of this methods are analyzed showing the importance of the regularity of the solution to choose the type of refinement. Thus, two error estimators are used to lead the refinement process. Whereas the first one requires high regularity hypothesis (analytical solution), the second one supposes only the minimal hypothesis required for the solution to exist. The comparison of both estimators is done on benchmarks where the analytic solution is known by the method of manufactured solutions. Thus, the behaviour of the solution as a regard of the regularity can be studied. It leads to a hp-refinement method using the two estimators. Then, a comparison is done with other existing methods on simplified but also realistic benchmarks coming from nuclear cores. These adaptive methods considerably reduces the computational cost and memory footprint. To further improve these two points, an approach with energy-dependent meshes is proposed. Actually, as the

The shallow water equations in Lagrangian coordinates

International Nuclear Information System (INIS)

Mead, J.L.

2004-01-01

Recent advances in the collection of Lagrangian data from the ocean and results about the well-posedness of the primitive equations have led to a renewed interest in solving flow equations in Lagrangian coordinates. We do not take the view that solving in Lagrangian coordinates equates to solving on a moving grid that can become twisted or distorted. Rather, the grid in Lagrangian coordinates represents the initial position of particles, and it does not change with time. We apply numerical methods traditionally used to solve differential equations in Eulerian coordinates, to solve the shallow water equations in Lagrangian coordinates. The difficulty with solving in Lagrangian coordinates is that the transformation from Eulerian coordinates results in solving a highly nonlinear partial differential equation. The non-linearity is mainly due to the Jacobian of the coordinate transformation, which is a precise record of how the particles are rotated and stretched. The inverse Jacobian must be calculated, thus Lagrangian coordinates cannot be used in instances where the Jacobian vanishes. For linear (spatial) flows we give an explicit formula for the Jacobian and describe the two situations where the Lagrangian shallow water equations cannot be used because either the Jacobian vanishes or the shallow water assumption is violated. We also prove that linear (in space) steady state solutions of the Lagrangian shallow water equations have Jacobian equal to one. In the situations where the shallow water equations can be solved in Lagrangian coordinates, accurate numerical solutions are found with finite differences, the Chebyshev pseudospectral method, and the fourth order Runge-Kutta method. The numerical results shown here emphasize the need for high order temporal approximations for long time integrations

The orthogonal gradients method: A radial basis functions method for solving partial differential equations on arbitrary surfaces

KAUST Repository

Piret, Cécile

2012-05-01

Much work has been done on reconstructing arbitrary surfaces using the radial basis function (RBF) method, but one can hardly find any work done on the use of RBFs to solve partial differential equations (PDEs) on arbitrary surfaces. In this paper, we investigate methods to solve PDEs on arbitrary stationary surfaces embedded in . R3 using the RBF method. We present three RBF-based methods that easily discretize surface differential operators. We take advantage of the meshfree character of RBFs, which give us a high accuracy and the flexibility to represent the most complex geometries in any dimension. Two out of the three methods, which we call the orthogonal gradients (OGr) methods are the result of our work and are hereby presented for the first time. © 2012 Elsevier Inc.

Simultaneous Localization and Mapping with Iterative Sparse Extended Information Filter for Autonomous Vehicles

Directory of Open Access Journals (Sweden)

Bo He

2015-08-01

Full Text Available In this paper, a novel iterative sparse extended information filter (ISEIF was proposed to solve the simultaneous localization and mapping problem (SLAM, which is very crucial for autonomous vehicles. The proposed algorithm solves the measurement update equations with iterative methods adaptively to reduce linearization errors. With the scalability advantage being kept, the consistency and accuracy of SEIF is improved. Simulations and practical experiments were carried out with both a land car benchmark and an autonomous underwater vehicle. Comparisons between iterative SEIF (ISEIF, standard EKF and SEIF are presented. All of the results convincingly show that ISEIF yields more consistent and accurate estimates compared to SEIF and preserves the scalability advantage over EKF, as well.

Simultaneous Localization and Mapping with Iterative Sparse Extended Information Filter for Autonomous Vehicles.

Science.gov (United States)

He, Bo; Liu, Yang; Dong, Diya; Shen, Yue; Yan, Tianhong; Nian, Rui

2015-08-13

In this paper, a novel iterative sparse extended information filter (ISEIF) was proposed to solve the simultaneous localization and mapping problem (SLAM), which is very crucial for autonomous vehicles. The proposed algorithm solves the measurement update equations with iterative methods adaptively to reduce linearization errors. With the scalability advantage being kept, the consistency and accuracy of SEIF is improved. Simulations and practical experiments were carried out with both a land car benchmark and an autonomous underwater vehicle. Comparisons between iterative SEIF (ISEIF), standard EKF and SEIF are presented. All of the results convincingly show that ISEIF yields more consistent and accurate estimates compared to SEIF and preserves the scalability advantage over EKF, as well.

Bayesian simultaneous equation models for the analysis of energy intake and partitioning in growing pigs

DEFF Research Database (Denmark)

Strathe, Anders Bjerring; Jørgensen, Henry; Kebreab, E

2012-01-01

ABSTRACT SUMMARY The objective of the current study was to develop Bayesian simultaneous equation models for modelling energy intake and partitioning in growing pigs. A key feature of the Bayesian approach is that parameters are assigned prior distributions, which may reflect the current state...... of nature. In the models, rates of metabolizable energy (ME) intake, protein deposition (PD) and lipid deposition (LD) were treated as dependent variables accounting for residuals being correlated. Two complementary equation systems were used to model ME intake (MEI), PD and LD. Informative priors were...... developed, reflecting current knowledge about metabolic scaling and partial efficiencies of PD and LD rates, whereas flat non-informative priors were used for the reminder of the parameters. The experimental data analysed originate from a balance and respiration trial with 17 cross-bred pigs of three...

A numerical method to solve the 1D and the 2D reaction diffusion equation based on Bessel functions and Jacobian free Newton-Krylov subspace methods

Science.gov (United States)

Parand, K.; Nikarya, M.

2017-11-01

In this paper a novel method will be introduced to solve a nonlinear partial differential equation (PDE). In the proposed method, we use the spectral collocation method based on Bessel functions of the first kind and the Jacobian free Newton-generalized minimum residual (JFNGMRes) method with adaptive preconditioner. In this work a nonlinear PDE has been converted to a nonlinear system of algebraic equations using the collocation method based on Bessel functions without any linearization, discretization or getting the help of any other methods. Finally, by using JFNGMRes, the solution of the nonlinear algebraic system is achieved. To illustrate the reliability and efficiency of the proposed method, we solve some examples of the famous Fisher equation. We compare our results with other methods.

An introduction to the history of algebra solving equations from Mesopotamian times to the Renaissance

CERN Document Server

Sesiano, Jacques

2009-01-01

This text should not be viewed as a comprehensive history of algebra before 1600, but as a basic introduction to the types of problems that illustrate the earliest forms of algebra. It would be particularly useful for an instructor who is looking for examples to help enliven a course on elementary algebra with problems drawn from actual historical texts. -Warren Van Egmond about the French edition for MathSciNet This book does not aim to give an exhaustive survey of the history of algebra up to early modern times but merely to present some significant steps in solving equations and, wherever

Spectral finite element methods for solving fractional differential equations with applications in anomalous transport

Energy Technology Data Exchange (ETDEWEB)

Carella, Alfredo Raul

2012-09-15

Quantifying species transport rates is a main concern in chemical and petrochemical industries. In particular, the design and operation of many large-scale industrial chemical processes is as much dependent on diffusion as it is on reaction rates. However, the existing diffusion models sometimes fail to predict experimentally observed behaviors and their accuracy is usually insufficient for process optimization purposes. Fractional diffusion models offer multiple possibilities for generalizing Flick's law in a consistent manner in order to account for history dependence and nonlocal effects. These models have not been extensively applied to the study of real systems, mainly due to their computational cost and mathematical complexity. A least squares spectral formulation was developed for solving fractional differential equations. The proposed method was proven particularly well-suited for dealing with the numerical difficulties inherent to fractional differential operators. The practical implementation was explained in detail in order to enhance reproducibility, and directions were specified for extending it to multiple dimensions and arbitrarily shaped domains. A numerical framework based on the least-squares spectral element method was developed for studying and comparing anomalous diffusion models in pellets. This simulation tool is capable of solving arbitrary integro-differential equations and can be effortlessly adapted to various problems in any number of dimensions. Simulations of the flow around a cylindrical particle were achieved by extending the functionality of the developed framework. A test case was analyzed by coupling the boundary condition yielded by the fluid model with two families of anomalous diffusion models: hyperbolic diffusion and fractional diffusion. Qualitative guidelines for determining the suitability of diffusion models can be formulated by complementing experimental data with the results obtained from this approach.(Author)

Differential equations for dummies

CERN Document Server

Holzner, Steven

2008-01-01

The fun and easy way to understand and solve complex equations Many of the fundamental laws of physics, chemistry, biology, and economics can be formulated as differential equations. This plain-English guide explores the many applications of this mathematical tool and shows how differential equations can help us understand the world around us. Differential Equations For Dummies is the perfect companion for a college differential equations course and is an ideal supplemental resource for other calculus classes as well as science and engineering courses. It offers step-by-step techniques, practical tips, numerous exercises, and clear, concise examples to help readers improve their differential equation-solving skills and boost their test scores.

A simple method for solving the Bussian equation for electrical conduction in rocks

Directory of Open Access Journals (Sweden)

P. W. J. Glover

2010-09-01

Full Text Available One of the most general and effective models for calculating the complex electrical conductivity and relative dielectric permittivity of rocks saturated with pore fluids is that of Bussian. Unlike most models, it is non-linear and cannot be solved algebraically. Consequently, researchers use reiterating numerical routines to obtain a solution of the equation, and then only for the real part of the solution. Here we present a different approach to the solution that uses conformal mapping in the complex plane, and implements it within Maple^TM. The method is simple and elegant in that it requires, for example, only 3 lines of code in Maple^TM 11 and little programming experience. The approach has been shown to be as precise as using the classical reiterating bisection method for real data implemented in C++ on an ordinary desktop computer to within a probability over 1 in 10⁹. However, the conformal mapping approach is 52 times as fast. We show once more that the Bussian equation breaks down for low fluid conductivities, but recommend it (with the modified Archie's law for use with rocks saturated with high salinity fluids when the matrix is conductive.

Transmission probability method for solving neutron transport equation in three-dimensional triangular-z geometry

Energy Technology Data Exchange (ETDEWEB)

Liu Guoming [Department of Nuclear Engineering, Xi' an Jiaotong University, Xi' an, Shaanxi 710049 (China)], E-mail: gmliusy@gmail.com; Wu Hongchun; Cao Liangzhi [Department of Nuclear Engineering, Xi' an Jiaotong University, Xi' an, Shaanxi 710049 (China)

2008-09-15

This paper presents a transmission probability method (TPM) to solve the neutron transport equation in three-dimensional triangular-z geometry. The source within the mesh is assumed to be spatially uniform and isotropic. At the mesh surface, the constant and the simplified P{sub 1} approximation are invoked for the anisotropic angular flux distribution. Based on this model, a code TPMTDT is encoded. It was verified by three 3D Takeda benchmark problems, in which the first two problems are in XYZ geometry and the last one is in hexagonal-z geometry, and an unstructured geometry problem. The results of the present method agree well with those of Monte-Carlo calculation method and Spherical Harmonics (P{sub N}) method.

Program to solve the multigroup discrete ordinates transport equation in (x,y,z) geometry

International Nuclear Information System (INIS)

Lathrop, K.D.

1976-04-01

Numerical formulations and programming algorithms are given for the THREETRAN computer program which solves the discrete ordinates, multigroup transport equation in (x,y,z) geometry. An efficient, flexible, and general data-handling strategy is derived to make use of three hierarchies of storage: small core memory, large core memory, and disk file. Data management, input instructions, and sample problem output are described. A six-group, S 4 , 18 502 mesh point, 2 800 zone, k/sub eff/ calculation of the ZPPR-4 critical assembly required 144 min of CDC-7600 time to execute to a convergence tolerance of 5 x 10 -4 and gave results in good qualitative agreement with experiment and other calculations. 6 references

On matrix fractional differential equations

Directory of Open Access Journals (Sweden)

Adem Kılıçman

2017-01-01

Full Text Available The aim of this article is to study the matrix fractional differential equations and to find the exact solution for system of matrix fractional differential equations in terms of Riemann–Liouville using Laplace transform method and convolution product to the Riemann–Liouville fractional of matrices. Also, we show the theorem of non-homogeneous matrix fractional partial differential equation with some illustrative examples to demonstrate the effectiveness of the new methodology. The main objective of this article is to discuss the Laplace transform method based on operational matrices of fractional derivatives for solving several kinds of linear fractional differential equations. Moreover, we present the operational matrices of fractional derivatives with Laplace transform in many applications of various engineering systems as control system. We present the analytical technique for solving fractional-order, multi-term fractional differential equation. In other words, we propose an efficient algorithm for solving fractional matrix equation.

On the solution of the Schroedinger equation through continued fractions

International Nuclear Information System (INIS)

Mignaco, J.A.

1979-05-01

The domain of interest for the applications of a method to solve the Schroedinger equation through continued fractions is studied. It is argued that the method applies almost equally well to quantum mechanical regimes (lower energy levels, low energy scattering) as well as to semiclassical ones simultaneously; this is illustrated by the example of the central power law potentials r sup(ν)(ν>o). The explanation of this behaviour is given in terms of the mathematical approximations involved and its relationship to physically interesting quantities. (Author) [pt

Block Hybrid Collocation Method with Application to Fourth Order Differential Equations

Directory of Open Access Journals (Sweden)

Lee Ken Yap

2015-01-01

Full Text Available The block hybrid collocation method with three off-step points is proposed for the direct solution of fourth order ordinary differential equations. The interpolation and collocation techniques are applied on basic polynomial to generate the main and additional methods. These methods are implemented in block form to obtain the approximation at seven points simultaneously. Numerical experiments are conducted to illustrate the efficiency of the method. The method is also applied to solve the fourth order problem from ship dynamics.

Solving variational problems and partial differential equations that map between manifolds via the closest point method

Science.gov (United States)

King, Nathan D.; Ruuth, Steven J.

2017-05-01

Maps from a source manifold M to a target manifold N appear in liquid crystals, color image enhancement, texture mapping, brain mapping, and many other areas. A numerical framework to solve variational problems and partial differential equations (PDEs) that map between manifolds is introduced within this paper. Our approach, the closest point method for manifold mapping, reduces the problem of solving a constrained PDE between manifolds M and N to the simpler problems of solving a PDE on M and projecting to the closest points on N. In our approach, an embedding PDE is formulated in the embedding space using closest point representations of M and N. This enables the use of standard Cartesian numerics for general manifolds that are open or closed, with or without orientation, and of any codimension. An algorithm is presented for the important example of harmonic maps and generalized to a broader class of PDEs, which includes p-harmonic maps. Improved efficiency and robustness are observed in convergence studies relative to the level set embedding methods. Harmonic and p-harmonic maps are computed for a variety of numerical examples. In these examples, we denoise texture maps, diffuse random maps between general manifolds, and enhance color images.

CO2 emissions, energy consumption and economic growth nexus in MENA countries: Evidence from simultaneous equations models

International Nuclear Information System (INIS)

Omri, Anis

2013-01-01

This paper examines the nexus between CO 2 emissions, energy consumption and economic growth using simultaneous-equations models with panel data of 14 MENA countries over the period 1990–2011. Our empirical results show that there exists a bidirectional causal relationship between energy consumption and economic growth. However, the results support the occurrence of unidirectional causality from energy consumption to CO 2 emissions without any feedback effects, and there exists a bidirectional causal relationship between economic growth and CO 2 emissions for the region as a whole. The study suggests that environmental and energy policies should recognize the differences in the nexus between energy consumption and economic growth in order to maintain sustainable economic growth in the MENA region. - Graphical abstract: Interaction between CO 2 , energy and GDP for MENA countries. - Highlights: • We investigate the energy–environment–GDP nexus for 14 MENA countries. • We have used simultaneous equations models estimated by the GMM-estimator. • Results show bi-directional causal relationship between energy consumption and economic growth. • There is uni-directional causality from energy consumption to CO 2 . • There exists bi-directional causal relationship between economic growth and pollutant emissions

The modified simple equation method for solving some fractional ...

Indian Academy of Sciences (India)

... and processes in various areas of natural science. Thus, many effective and powerful methods have been established and improved. In this study, we establish exact solutions of the time fractional biological population model equation and nonlinearfractional Klein–Gordon equation by using the modified simple equation ...

Elliptic and solitary wave solutions for Bogoyavlenskii equations system, couple Boiti-Leon-Pempinelli equations system and Time-fractional Cahn-Allen equation

Directory of Open Access Journals (Sweden)

Mostafa M.A. Khater

Full Text Available In this article and for the first time, we introduce and describe Khater method which is a new technique for solving nonlinear partial differential equations (PDEs.. We apply this method for each of the following models Bogoyavlenskii equation, couple Boiti-Leon-Pempinelli system and Time-fractional Cahn-Allen equation. Khater method is very powerful, Effective, felicitous and fabulous method to get exact and solitary wave solution of (PDEs.. Not only just like that but it considers too one of the general methods for solving that kind of equations since it involves some methods as we will see in our discuss of the results. We make a comparison between the results of this new method and another method. Keywords: Bogoyavlenskii equations system, Couple Boiti-Leon-Pempinelli equations system, Time-fractional Cahn-Allen equation, Khater method, Traveling wave solutions, Solitary wave solutions

Hybrid B-Spline Collocation Method for Solving the Generalized Burgers-Fisher and Burgers-Huxley Equations

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Imtiaz Wasim

2018-01-01

Full Text Available In this study, we introduce a new numerical technique for solving nonlinear generalized Burgers-Fisher and Burgers-Huxley equations using hybrid B-spline collocation method. This technique is based on usual finite difference scheme and Crank-Nicolson method which are used to discretize the time derivative and spatial derivatives, respectively. Furthermore, hybrid B-spline function is utilized as interpolating functions in spatial dimension. The scheme is verified unconditionally stable using the Von Neumann (Fourier method. Several test problems are considered to check the accuracy of the proposed scheme. The numerical results are in good agreement with known exact solutions and the existing schemes in literature.

A Stable Marching on-in-time Scheme for Solving the Time Domain Electric Field Volume Integral Equation on High-contrast Scatterers

KAUST Repository

Sayed, Sadeed Bin

2015-05-05

A time domain electric field volume integral equation (TD-EFVIE) solver is proposed for characterizing transient electromagnetic wave interactions on high-contrast dielectric scatterers. The TD-EFVIE is discretized using the Schaubert- Wilton-Glisson (SWG) and approximate prolate spherical wave (APSW) functions in space and time, respectively. The resulting system of equations can not be solved by a straightforward application of the marching on-in-time (MOT) scheme since the two-sided APSW interpolation functions require the knowledge of unknown “future” field samples during time marching. Causality of the MOT scheme is restored using an extrapolation technique that predicts the future samples from known “past” ones. Unlike the extrapolation techniques developed for MOT schemes that are used in solving time domain surface integral equations, this scheme trains the extrapolation coefficients using samples of exponentials with exponents on the complex frequency plane. This increases the stability of the MOT-TD-EFVIE solver significantly, since the temporal behavior of decaying and oscillating electromagnetic modes induced inside the scatterers is very accurately taken into account by this new extrapolation scheme. Numerical results demonstrate that the proposed MOT solver maintains its stability even when applied to analyzing wave interactions on high-contrast scatterers.

A Stable Marching on-in-time Scheme for Solving the Time Domain Electric Field Volume Integral Equation on High-contrast Scatterers

KAUST Repository

Sayed, Sadeed Bin; Ulku, Huseyin; Bagci, Hakan

2015-01-01

A time domain electric field volume integral equation (TD-EFVIE) solver is proposed for characterizing transient electromagnetic wave interactions on high-contrast dielectric scatterers. The TD-EFVIE is discretized using the Schaubert- Wilton-Glisson (SWG) and approximate prolate spherical wave (APSW) functions in space and time, respectively. The resulting system of equations can not be solved by a straightforward application of the marching on-in-time (MOT) scheme since the two-sided APSW interpolation functions require the knowledge of unknown “future” field samples during time marching. Causality of the MOT scheme is restored using an extrapolation technique that predicts the future samples from known “past” ones. Unlike the extrapolation techniques developed for MOT schemes that are used in solving time domain surface integral equations, this scheme trains the extrapolation coefficients using samples of exponentials with exponents on the complex frequency plane. This increases the stability of the MOT-TD-EFVIE solver significantly, since the temporal behavior of decaying and oscillating electromagnetic modes induced inside the scatterers is very accurately taken into account by this new extrapolation scheme. Numerical results demonstrate that the proposed MOT solver maintains its stability even when applied to analyzing wave interactions on high-contrast scatterers.

Constructing and solving equations - inverse operations

Czech Academy of Sciences Publication Activity Database

Neuman, František

2005-01-01

Roč. 70, č. 1 (2005), s. 77-87 ISSN 0001-9054 R&D Projects: GA AV ČR(CZ) IAA1163401 Institutional research plan: CEZ:AV0Z10190503 Keywords : functional and differential equations * representation of solution spaces * discretization of smooth relations Subject RIV: BA - General Mathematics

The improved fractional sub-equation method and its applications to the space–time fractional differential equations in fluid mechanics

International Nuclear Information System (INIS)

Guo, Shimin; Mei, Liquan; Li, Ying; Sun, Youfa

2012-01-01

By introducing a new general ansätz, the improved fractional sub-equation method is proposed to construct analytical solutions of nonlinear evolution equations involving Jumarie's modified Riemann–Liouville derivative. By means of this method, the space–time fractional Whitham–Broer–Kaup and generalized Hirota–Satsuma coupled KdV equations are successfully solved. The obtained results show that the proposed method is quite effective, promising and convenient for solving nonlinear fractional differential equations. -- Highlights: ► We propose a novel method for nonlinear fractional differential equations. ► Two important fractional differential equations in fluid mechanics are solved successfully. ► Some new exact solutions of the fractional differential equations are obtained. ► These solutions will advance the understanding of nonlinear physical phenomena.

FDTD for Hydrodynamic Electron Fluid Maxwell Equations

Directory of Open Access Journals (Sweden)

Yingxue Zhao

2015-05-01

Full Text Available In this work, we develop a numerical method for solving the three dimensional hydrodynamic electron fluid Maxwell equations that describe the electron gas dynamics driven by an external electromagnetic wave excitation. Our numerical approach is based on the Finite-Difference Time-Domain (FDTD method for solving the Maxwell’s equations and an explicit central finite difference method for solving the hydrodynamic electron fluid equations containing both electron density and current equations. Numerical results show good agreement with the experiment of studying the second-harmonic generation (SHG from metallic split-ring resonator (SRR.

Hartree--Fock density matrix equation

International Nuclear Information System (INIS)

Cohen, L.; Frishberg, C.

1976-01-01

An equation for the Hartree--Fock density matrix is discussed and the possibility of solving this equation directly for the density matrix instead of solving the Hartree--Fock equation for orbitals is considered. Toward that end the density matrix is expanded in a finite basis to obtain the matrix representative equation. The closed shell case is considered. Two numerical schemes are developed and applied to a number of examples. One example is given where the standard orbital method does not converge while the method presented here does

A dynamical regularization algorithm for solving inverse source problems of elliptic partial differential equations

Science.gov (United States)

Zhang, Ye; Gong, Rongfang; Cheng, Xiaoliang; Gulliksson, Mårten

2018-06-01

This study considers the inverse source problem for elliptic partial differential equations with both Dirichlet and Neumann boundary data. The unknown source term is to be determined by additional boundary conditions. Unlike the existing methods found in the literature, which usually employ the first-order in time gradient-like system (such as the steepest descent methods) for numerically solving the regularized optimization problem with a fixed regularization parameter, we propose a novel method with a second-order in time dissipative gradient-like system and a dynamical selected regularization parameter. A damped symplectic scheme is proposed for the numerical solution. Theoretical analysis is given for both the continuous model and the numerical algorithm. Several numerical examples are provided to show the robustness of the proposed algorithm.

Analysis of rice policy based on presidential instruction on household food security: simultaneous equation model

Science.gov (United States)

Sembiring, S. A.

2018-02-01

The objective of this research is to analyze the impacts of rice policy on the household food security. The research used cross section data, were collected from 74 respondent as determined by purposive sampling in Sei Rejo villages, the Sub District of Sei Rampah, Serdang Bedagai District in the Province of North Sumatera. Rice policy model specification uses the simultaneous equations consisting of 6 structural equations and 6 identity equations which was estimated using Two Stages Least Squares (2SLS) method. The results show that the effectiveness of government purchase price of dried harvest paddy gave a positive impact on paddy planted area and lead to an increase paddy production and an increase of the rice production gave a positive impact on household rice availability and household rice surplus, and the increase of household rice surplus gave the quantity of Raskin decrease, whereas the increase of fertilizers gave a negative impact on paddy planted area and lead to decrease paddy production and to decrease in rice production was followed by an decrease in household rice availability and household rice surplus, and the decrease of household rice surplus gave the quantity of Raskin increase.

Bäcklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations

International Nuclear Information System (INIS)

Lu, Bin

2012-01-01

In this Letter, the fractional derivatives in the sense of modified Riemann–Liouville derivative and the Bäcklund transformation of fractional Riccati equation are employed for constructing the exact solutions of nonlinear fractional partial differential equations. The power of this manageable method is presented by applying it to several examples. This approach can also be applied to other nonlinear fractional differential equations. -- Highlights: ► Backlund transformation of fractional Riccati equation is presented. ► A new method for solving nonlinear fractional differential equations is proposed. ► Three important fractional differential equations are solved successfully. ► Some new exact solutions of the fractional differential equations are obtained.

Calculation of reinforced-concrete frame strength under a simultaneous static cross section load and a column lateral impact

International Nuclear Information System (INIS)

Belov, Nikolay; Kopanitsa, Dmitry; Yugov, Alexey; Kaparulin, Sergey; Plyaskin, Andrey; Kalichkina, Anna; Ustinov, Artyom; Yugov, Nikolay; Kopanitsa, Georgy

2016-01-01

When designing buildings with reinforced concrete that are planned to resist dynamic loads it is necessary to calculate this structural behavior under operational static and emergency impact and blast loads. Calculations of the structures under shock-wave loads can be performed by solving dynamic equations that do not consider static loads. Due to this fact the calculation of reinforced concrete frame under a simultaneous static and dynamic load in full 3d settings becomes a very non trivial and resource consuming problem. This problem can be split into two tasks. The first one is a shock-wave problem that can be solved using software package RANET-3, which allows solving the problem using finite elements method adapted for dynamic task. This method calculates strain-stress state of the material and its dynamic destruction, which is considered as growth and consolidation of micro defects under loading. On the second step the results of the first step are taken as input parameters for quasi static calculation of simultaneous static and dynamic load using finite elements method in AMP Civil Engineering-11

Calculation of reinforced-concrete frame strength under a simultaneous static cross section load and a column lateral impact

Energy Technology Data Exchange (ETDEWEB)

Belov, Nikolay, E-mail: n.n.belov@mail.ru; Kopanitsa, Dmitry, E-mail: kopanitsa@mail.ru; Yugov, Alexey, E-mail: yugalex@mail.ru; Kaparulin, Sergey, E-mail: kaparulin@mail.ru; Plyaskin, Andrey, E-mail: plyaskinandrei@mail.ru; Kalichkina, Anna, E-mail: aniotka@mail.ru; Ustinov, Artyom, E-mail: artemustinov@bk.ru [Tomsk State University of Architecture and Building, 2, Solyanaya Sq. Tomsk, 634003 (Russian Federation); Yugov, Nikolay, E-mail: n.t.yugov@mail.ru [Tomsk State University for Radio Electronics and Control Systems, 40, Lenin Av. Tomsk, 634050 (Russian Federation); Kopanitsa, Georgy, E-mail: kopanitsa@mail.ru [National Research Tomsk Polytechnic University, 30, Lenin Av. Tomsk, 634050 (Russian Federation)

2016-01-15

When designing buildings with reinforced concrete that are planned to resist dynamic loads it is necessary to calculate this structural behavior under operational static and emergency impact and blast loads. Calculations of the structures under shock-wave loads can be performed by solving dynamic equations that do not consider static loads. Due to this fact the calculation of reinforced concrete frame under a simultaneous static and dynamic load in full 3d settings becomes a very non trivial and resource consuming problem. This problem can be split into two tasks. The first one is a shock-wave problem that can be solved using software package RANET-3, which allows solving the problem using finite elements method adapted for dynamic task. This method calculates strain-stress state of the material and its dynamic destruction, which is considered as growth and consolidation of micro defects under loading. On the second step the results of the first step are taken as input parameters for quasi static calculation of simultaneous static and dynamic load using finite elements method in AMP Civil Engineering-11.

New Numerical Treatment for Solving the KDV Equation

Directory of Open Access Journals (Sweden)

khalid ali

2017-01-01

Full Text Available In the present article, a numerical method is proposed for the numerical solution of the KdV equation by using collocation method with the modified exponential cubic B-spline. In this paper we convert the KdV equation to system of two equations. The method is shown to be unconditionally stable using von-Neumann technique. To test accuracy the error norms2L, ?L are computed. Three invariants of motion are predestined to determine the preservation properties of the problem, and the numerical scheme leads to careful and active results. Furthermore, interaction of two and three solitary waves is shown. These results show that the technique introduced here is easy to apply.

LOCFES-B: Solving the one-dimensional transport equation with user-selected spatial approximations

International Nuclear Information System (INIS)

Jarvis, R.D.; Nelson, P.

1993-01-01

Closed linear one-cell functional (CLOF) methods constitute an abstractly defined class of spatial approximations to the one-dimensional discrete ordinates equations of linear particle transport that encompass, as specific instances, the vast majority of the spatial approximations that have been either used or suggested in the computational solution of these equations. A specific instance of the class of CLOF methods is defined by a (typically small) number of functions of the cell width, total cross section, and direction cosine of particle motion. The LOCFES code takes advantage of the latter observation by permitting the use, within a more-or-less standard source iteration solution process, of an arbitrary CLOF method as defined by a user-supplied subroutine. The design objective of LOCFES was to provide automated determination of the order of accuracy (i.e., order of the discretization error) in the fine-mesh limit for an arbitrary user-selected CLOF method. This asymptotic order of accuracy is one widely used measure of the merit of a spatial approximation. This paper discusses LOCFES-B, which is a code that uses methods developed in LOCFES to solve one-dimensional linear particle transport problems with any user-selected CLOF method. LOCFES-B provides automatic solution of a given problem to within an accuracy specified by user input and provides comparison of the computational results against results from externally provided benchmark results

Exp-function method for solving fractional partial differential equations.

Science.gov (United States)

Zheng, Bin

2013-01-01

We extend the Exp-function method to fractional partial differential equations in the sense of modified Riemann-Liouville derivative based on nonlinear fractional complex transformation. For illustrating the validity of this method, we apply it to the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation. As a result, some new exact solutions for them are successfully established.

Simple equation method for nonlinear partial differential equations and its applications

Directory of Open Access Journals (Sweden)

Taher A. Nofal

2016-04-01

Full Text Available In this article, we focus on the exact solution of the some nonlinear partial differential equations (NLPDEs such as, Kodomtsev–Petviashvili (KP equation, the (2 + 1-dimensional breaking soliton equation and the modified generalized Vakhnenko equation by using the simple equation method. In the simple equation method the trial condition is the Bernoulli equation or the Riccati equation. It has been shown that the method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering problems.

Simultaneous UV Spectrophotometric Determination of Cetrizine and Dextromethorphan in Tablet Dosage Form

Directory of Open Access Journals (Sweden)

R. Vijayalakshmi

2010-01-01

Full Text Available Two accurate, precise, sensitve and economical procedures for simultaneous estimation of cetrizine and dextromethorphan in tablet dosage forms have been developed. First method employs formation and solving of simultaneous equations using 230 nm and 280 nm as two analytical wavelengths for both drugs in methanol. The second method is Q-analysis based on measurement of absorptivity at 224 nm (as isobestic point and 280 nm (λmax of CTZ. Cetrizine and dextromethorphan at their respective λmax 280 nm and 230 nm and at 224 nm (isobestic point shows linearity in a concentration range of 10-30 mcg/mL for both the drugs. The recovery studies confirmed accuracy of the proposed methods and low values of standard deviation confirmed precision of the methods. The methods were validated as per ICH guidelines.

Coupling of discrete ordinates methods by transmission of boundary conditions in solving the neutron transport equation in slab geometry; Couplage de discretisations aux ordonnees discretes d`equations de transport 1D par passage de conditions frontieres

Energy Technology Data Exchange (ETDEWEB)

Bal, G. [Departement MMN, Service IMA, Direction des Etudes et Recherches, Electricite de France (EDF), 92 - Clamart (France)

1995-10-01

Neutron transport in nuclear reactors is quite well modelled by the linear Boltzmann transport equation. Its solution is relatively easy, but unfortunately too expensive to achieve whole core computations. Thus, we have to simplify it, for example by homogenizing some physical characteristics. However, the solution may then be inaccurate. Moreover, in strongly homogeneous areas, the error may be too big. Then we would like to deal with such an inconvenient by solving the equation accurately on this area, but more coarsely away from it, so that the computation is not too expensive. This problem is the subject of a thesis. We present here some results obtained for slab geometry. The couplings between the fine and coarse discretization regions could be conceived in a number of approaches. Here, we only deal with the coupling at crossing the interface between two sub-domains. In the first section, we present the coupling of discrete ordinate methods for solving the homogeneous, isotropic and mono-kinetic equation. Coupling operators are defined and shown to be optimal. The second and the third sections are devoted to an extension of the previous results when the equation is non-homogeneous, anisotropic and multigroup (under some restrictive assumptions). Some numerical results are given in the case of isotropic and mono-kinetic equations. (author) 15 refs.

A nonclassical Radau collocation method for solving the Lane-Emden equations of the polytropic index 4.75 ≤ α < 5

Science.gov (United States)

Tirani, M. D.; Maleki, M.; Kajani, M. T.

2014-11-01

A numerical method for solving the Lane-Emden equations of the polytropic index α when 4.75 ≤ α ≤ 5 is introduced. The method is based upon nonclassical Gauss-Radau collocation points and Freud type weights. Nonclassical orthogonal polynomials, nonclassical Radau points and weighted interpolation are introduced and are utilized in the interval [0,1]. A smooth, strictly monotonic transformation is used to map the infinite domain x ∈ [0,∞) onto a half-open interval t ∈ [0,1). The resulting problem on the finite interval is then transcribed to a system of nonlinear algebraic equations using collocation. The method is easy to implement and yields very accurate results.

Differential Equation over Banach Algebra

OpenAIRE

Kleyn, Aleks

2018-01-01

In the book, I considered differential equations of order $1$ over Banach $D$-algebra: differential equation solved with respect to the derivative; exact differential equation; linear homogeneous equation. In noncommutative Banach algebra, initial value problem for linear homogeneous equation has infinitely many solutions.

Weighted particle method for solving the Boltzmann equation

International Nuclear Information System (INIS)

Tohyama, M.; Suraud, E.

1990-01-01

We propose a new, deterministic, method of solution of the nuclear Boltzmann equation. In this Weighted Particle Method two-body collisions are treated by a Master equation for an occupation probability of each numerical particle. We apply the method to the quadrupole motion of 12 C. A comparison with usual stochastic methods is made. Advantages and disadvantages of the Weighted Particle Method are discussed

Causal relationships between energy consumption, foreign direct investment and economic growth: Fresh evidence from dynamic simultaneous-equations models

International Nuclear Information System (INIS)

Omri, Anis; Kahouli, Bassem

2014-01-01

This paper examines the interrelationships between energy consumption, foreign direct investment and economic growth using dynamic panel data models in simultaneous-equations for a global panel consisting of 65 countries. The time component of our dataset is 1990–2011 inclusive. To make the panel data analysis more homogenous, we also investigate this interrelationship for a number of sub-panels which are constructed based on the income level of countries. In this way, we end up with three income panels; namely, high income, middle income, and low income panels. In the empirical part, we draw on the growth theory and augment the classical growth model, which consists of capital stock, labor force and inflation, with foreign direct investment and energy. Generally, we show mixed results about the interrelationship between energy consumption, FDI and economic growth. - Highlights: • We examine the energy–FDI–growth nexus for a global panel of 65 countries. • Dynamic simultaneous-equation panel data models are used to address this issue. • We also investigate this nexus for three sub-panels which are constructed based on the income level of countries. • We show mixed results about the interrelationship between the three variables

Rigorous solution to Bargmann-Wigner equation for integer spin

CERN Document Server

Huang Shi Zhong; Wu Ning; Zheng Zhi Peng

2002-01-01

A rigorous method is developed to solve the Bargamann-Wigner equation for arbitrary integer spin in coordinate representation in a step by step way. The Bargmann-Wigner equation is first transformed to a form easier to solve, the new equations are then solved rigorously in coordinate representation, and the wave functions in a closed form are thus derived

Simultaneous effects of slip and MHD on peristaltic blood flow of Jeffrey fluid model through a porous medium

Directory of Open Access Journals (Sweden)

M.M. Bhatti

2016-06-01

Full Text Available In this article, the simultaneous effects of slip and Magnetohydrodynamics (MHD on peristaltic blood flow of Jeffrey fluid model have been investigated in a non-uniform porous channel. The governing equation of blood flow for Jeffrey fluid model is solved with the help of long wavelength and creeping flow regime. The solution of the resulting differential equation is solved analytically and a closed form solution is presented. The impact of all the physical parameters is plotted for velocity profile and pressure rise. Nowadays, Magnetohydrodynamics is applicable in various magnetic drug targeting for cancer diseases and also very helpful to control the flow. The present analysis is also described for Newtonian fluid (λ1→0 as a special case of our study. It is observed that magnitude of the velocity is opposite near the walls due to slip effects whereas similar behavior has been observed for magnetic field.

Application of wavelets to singular integral scattering equations

International Nuclear Information System (INIS)

Kessler, B.M.; Payne, G.L.; Polyzou, W.N.

2004-01-01

The use of orthonormal wavelet basis functions for solving singular integral scattering equations is investigated. It is shown that these basis functions lead to sparse matrix equations which can be solved by iterative techniques. The scaling properties of wavelets are used to derive an efficient method for evaluating the singular integrals. The accuracy and efficiency of the wavelet transforms are demonstrated by solving the two-body T-matrix equation without partial wave projection. The resulting matrix equation which is characteristic of multiparticle integral scattering equations is found to provide an efficient method for obtaining accurate approximate solutions to the integral equation. These results indicate that wavelet transforms may provide a useful tool for studying few-body systems

Development and Validation of Spectrophotometric Methods for Simultaneous Estimation of Valsartan and Hydrochlorothiazide in Tablet Dosage Form

Directory of Open Access Journals (Sweden)

Monika L. Jadhav

2014-01-01

Full Text Available Two UV-spectrophotometric methods have been developed and validated for simultaneous estimation of valsartan and hydrochlorothiazide in a tablet dosage form. The first method employed solving of simultaneous equations based on the measurement of absorbance at two wavelengths, 249.4 nm and 272.6 nm, λmax for valsartan and hydrochlorothiazide, respectively. The second method was absorbance ratio method, which involves formation of Q-absorbance equation at 258.4 nm (isoabsorptive point and also at 272.6 nm (λmax of hydrochlorothiazide. The methods were found to be linear between the range of 5–30 µg/mL for valsartan and 4–24 μg/mL for hydrochlorothiazide using 0.1 N NaOH as solvent. The mean percentage recovery was found to be 100.20% and 100.19% for the simultaneous equation method and 98.56% and 97.96% for the absorbance ratio method, for valsartan and hydrochlorothiazide, respectively, at three different levels of standard additions. The precision (intraday, interday of methods was found within limits (RSD<2%. It could be concluded from the results obtained in the present investigation that the two methods for simultaneous estimation of valsartan and hydrochlorothiazide in tablet dosage form are simple, rapid, accurate, precise and economical and can be used, successfully, in the quality control of pharmaceutical formulations and other routine laboratory analysis.

Computation of the stability derivatives via CFD and the sensitivity equations

Science.gov (United States)

Lei, Guo-Dong; Ren, Yu-Xin

2011-04-01

The method to calculate the aerodynamic stability derivates of aircrafts by using the sensitivity equations is extended to flows with shock waves in this paper. Using the newly developed second-order cell-centered finite volume scheme on the unstructured-grid, the unsteady Euler equations and sensitivity equations are solved simultaneously in a non-inertial frame of reference, so that the aerodynamic stability derivatives can be calculated for aircrafts with complex geometries. Based on the numerical results, behavior of the aerodynamic sensitivity parameters near the shock wave is discussed. Furthermore, the stability derivatives are analyzed for supersonic and hypersonic flows. The numerical results of the stability derivatives are found in good agreement with theoretical results for supersonic flows, and variations of the aerodynamic force and moment predicted by the stability derivatives are very close to those obtained by CFD simulation for both supersonic and hypersonic flows.

A novel approach for solving fractional Fisher equation using

Indian Academy of Sciences (India)

Differential transform method; fractional Fisher equation. ... confirmed by applying this method on different forms of functional equations. Author Affiliations. MIRZAZADEH M1. Department of Engineering Sciences, Faculty of Technology and ...

Discovery of a general method of solving the Schrödinger and dirac equations that opens a way to accurately predictive quantum chemistry.

Science.gov (United States)

Nakatsuji, Hiroshi

2012-09-18

Just as Newtonian law governs classical physics, the Schrödinger equation (SE) and the relativistic Dirac equation (DE) rule the world of chemistry. So, if we can solve these equations accurately, we can use computation to predict chemistry precisely. However, for approximately 80 years after the discovery of these equations, chemists believed that they could not solve SE and DE for atoms and molecules that included many electrons. This Account reviews ideas developed over the past decade to further the goal of predictive quantum chemistry. Between 2000 and 2005, I discovered a general method of solving the SE and DE accurately. As a first inspiration, I formulated the structure of the exact wave function of the SE in a compact mathematical form. The explicit inclusion of the exact wave function's structure within the variational space allows for the calculation of the exact wave function as a solution of the variational method. Although this process sounds almost impossible, it is indeed possible, and I have published several formulations and applied them to solve the full configuration interaction (CI) with a very small number of variables. However, when I examined analytical solutions for atoms and molecules, the Hamiltonian integrals in their secular equations diverged. This singularity problem occurred in all atoms and molecules because it originates from the singularity of the Coulomb potential in their Hamiltonians. To overcome this problem, I first introduced the inverse SE and then the scaled SE. The latter simpler idea led to immediate and surprisingly accurate solution for the SEs of the hydrogen atom, helium atom, and hydrogen molecule. The free complement (FC) method, also called the free iterative CI (free ICI) method, was efficient for solving the SEs. In the FC method, the basis functions that span the exact wave function are produced by the Hamiltonian of the system and the zeroth-order wave function. These basis functions are called complement

Comparison between results of solution of Burgers' equation and Laplace's equation by Galerkin and least-square finite element methods

Science.gov (United States)

Adib, Arash; Poorveis, Davood; Mehraban, Farid

2018-03-01

In this research, two equations are considered as examples of hyperbolic and elliptic equations. In addition, two finite element methods are applied for solving of these equations. The purpose of this research is the selection of suitable method for solving each of two equations. Burgers' equation is a hyperbolic equation. This equation is a pure advection (without diffusion) equation. This equation is one-dimensional and unsteady. A sudden shock wave is introduced to the model. This wave moves without deformation. In addition, Laplace's equation is an elliptical equation. This equation is steady and two-dimensional. The solution of Laplace's equation in an earth dam is considered. By solution of Laplace's equation, head pressure and the value of seepage in the directions X and Y are calculated in different points of earth dam. At the end, water table is shown in the earth dam. For Burgers' equation, least-square method can show movement of wave with oscillation but Galerkin method can not show it correctly (the best method for solving of the Burgers' equation is discrete space by least-square finite element method and discrete time by forward difference.). For Laplace's equation, Galerkin and least square methods can show water table correctly in earth dam.

Theoretical study of simultaneous water and VOCs adsorption and desorption in a silica gel rotor

DEFF Research Database (Denmark)

Zhang, G.; Zhang, Y.F.; Fang, Lei

2008-01-01

One-dimensional partial differential equations were used to model the simultaneous water and VOC (Volatile Organic Compound) adsorption and desorption in a silica gel rotor which was recommended for indoor air cleaning. The interaction among VOCs and moisture in the adsorption and desorption...... process was neglected in the model as the concentrations of VOC pollutants in typical indoor environment were much lower than that of moisture and the adsorbed VOCs occupied only a minor portion of adsorption capacity of the rotor. Consequently VOC transfer was coupled with heat and moisture transfer only...... by the temperatures of the rotor and the air stream. The VOC transfer equations were solved by discretizing them into explicit up-wind finite differential equations. The model was validated with experimental data. The calculated results suggested that the regeneration time designed for dehumidification may...

A nonlinear boundary integral equations method for the solving of quasistatic elastic contact problem with Coulomb friction

Directory of Open Access Journals (Sweden)

Yurii M. Streliaiev

2016-06-01

Full Text Available Three-dimensional quasistatic contact problem of two linearly elastic bodies' interaction with Coulomb friction taken into account is considered. The boundary conditions of the problem have been simplified by the modification of the Coulomb's law of friction. This modification is based on the introducing of a delay in normal contact tractions that bound tangent contact tractions in the Coulomb's law of friction expressions. At this statement the problem is reduced to a sequence of similar systems of nonlinear integral equations describing bodies' interaction at each step of loading. A method for an approximate solution of the integral equations system corresponded to each step of loading is applied. This method consists of system regularization, discretization of regularized system and iterative process application for solving the discretized system. A numerical solution of a contact problem of an elastic sphere with an elastic half-space interaction under increasing and subsequently decreasing normal compressive force has been obtained.

B-splines and Faddeev equations

International Nuclear Information System (INIS)

Huizing, A.J.

1990-01-01

Two numerical methods for solving the three-body equations describing relativistic pion deuteron scattering have been investigated. For separable two body interactions these equations form a set of coupled one-dimensional integral equations. They are plagued by singularities which occur in the kernel of the integral equations as well as in the solution. The methods to solve these equations differ in the way they treat the singularities. First the Fuda-Stuivenberg method is discussed. The basic idea of this method is an one time iteration of the set of integral equations to treat the logarithmic singularities. In the second method, the spline method, the unknown solution is approximated by splines. Cubic splines have been used with cubic B-splines as basis. If the solution is approximated by a linear combination of basis functions, an integral equation can be transformed into a set of linear equations for the expansion coefficients. This set of linear equations is solved by standard means. Splines are determined by points called knots. A proper choice of splines to approach the solution stands for a proper choice of the knots. The solution of the three-body scattering equations has a square root behaviour at a certain point. Hence it was investigated how the knots should be chosen to approximate the square root function by cubic B-splines in an optimal way. Before applying this method to solve numerically the three-body equations describing pion-deuteron scattering, an analytically solvable example has been constructed with a singularity structure of both kernel and solution comparable to those of the three-body equations. The accuracy of the numerical solution was determined to a large extent by the accuracy of the approximation of the square root part. The results for a pion laboratory energy of 47.4 MeV agree very well with those from literature. In a complete calculation for 47.7 MeV the spline method turned out to be a factor thousand faster than the Fuda