A multivariate family-based association test using generalized estimating equations : FBAT-GEE
Lange, C; Silverman, SK; Xu, [No Value; Weiss, ST; Laird, NM
In this paper we propose a multivariate extension of family-based association tests based on generalized estimating equations. The test can be applied to multiple phenotypes and to phenotypic data obtained in longitudinal studies without making any distributional assumptions for the phenotypic
Small sample GEE estimation of regression parameters for longitudinal data.
Paul, Sudhir; Zhang, Xuemao
2014-09-28
Longitudinal (clustered) response data arise in many bio-statistical applications which, in general, cannot be assumed to be independent. Generalized estimating equation (GEE) is a widely used method to estimate marginal regression parameters for correlated responses. The advantage of the GEE is that the estimates of the regression parameters are asymptotically unbiased even if the correlation structure is misspecified, although their small sample properties are not known. In this paper, two bias adjusted GEE estimators of the regression parameters in longitudinal data are obtained when the number of subjects is small. One is based on a bias correction, and the other is based on a bias reduction. Simulations show that the performances of both the bias-corrected methods are similar in terms of bias, efficiency, coverage probability, average coverage length, impact of misspecification of correlation structure, and impact of cluster size on bias correction. Both these methods show superior properties over the GEE estimates for small samples. Further, analysis of data involving a small number of subjects also shows improvement in bias, MSE, standard error, and length of the confidence interval of the estimates by the two bias adjusted methods over the GEE estimates. For small to moderate sample sizes (N ≤50), either of the bias-corrected methods GEEBc and GEEBr can be used. However, the method GEEBc should be preferred over GEEBr, as the former is computationally easier. For large sample sizes, the GEE method can be used. Copyright © 2014 John Wiley & Sons, Ltd.
An, Shengli; Zhang, Yanhong; Chen, Zheng
2012-12-01
To analyze binary classification repeated measurement data with generalized estimating equations (GEE) and generalized linear mixed models (GLMMs) using SPSS19.0. GEE and GLMMs models were tested using binary classification repeated measurement data sample using SPSS19.0. Compared with SAS, SPSS19.0 allowed convenient analysis of categorical repeated measurement data using GEE and GLMMs.
Transport equation solving methods
International Nuclear Information System (INIS)
Granjean, P.M.
1984-06-01
This work is mainly devoted to Csub(N) and Fsub(N) methods. CN method: starting from a lemma stated by Placzek, an equivalence is established between two problems: the first one is defined in a finite medium bounded by a surface S, the second one is defined in the whole space. In the first problem the angular flux on the surface S is shown to be the solution of an integral equation. This equation is solved by Galerkin's method. The Csub(N) method is applied here to one-velocity problems: in plane geometry, slab albedo and transmission with Rayleigh scattering, calculation of the extrapolation length; in cylindrical geometry, albedo and extrapolation length calculation with linear scattering. Fsub(N) method: the basic integral transport equation of the Csub(N) method is integrated on Case's elementary distributions; another integral transport equation is obtained: this equation is solved by a collocation method. The plane problems solved by the Csub(N) method are also solved by the Fsub(N) method. The Fsub(N) method is extended to any polynomial scattering law. Some simple spherical problems are also studied. Chandrasekhar's method, collision probability method, Case's method are presented for comparison with Csub(N) and Fsub(N) methods. This comparison shows the respective advantages of the two methods: a) fast convergence and possible extension to various geometries for Csub(N) method; b) easy calculations and easy extension to polynomial scattering for Fsub(N) method [fr
Methods for Equating Mental Tests.
1984-11-01
1983) compared conventional and IRT methods for equating the Test of English as a Foreign Language ( TOEFL ) after chaining. Three conventional and...three IRT equating methods were examined in this study; two sections of TOEFL were each (separately) equated. The IRT methods included the following: (a...group. A separate base form was established for each of the six equating methods. Instead of equating the base-form TOEFL to itself, the last (eighth
Test equating methods and practices
Kolen, Michael J
1995-01-01
In recent years, many researchers in the psychology and statistical communities have paid increasing attention to test equating as issues of using multiple test forms have arisen and in response to criticisms of traditional testing techniques This book provides a practically oriented introduction to test equating which both discusses the most frequently used equating methodologies and covers many of the practical issues involved The main themes are - the purpose of equating - distinguishing between equating and related methodologies - the importance of test equating to test development and quality control - the differences between equating properties, equating designs, and equating methods - equating error, and the underlying statistical assumptions for equating The authors are acknowledged experts in the field, and the book is based on numerous courses and seminars they have presented As a result, educators, psychometricians, professionals in measurement, statisticians, and students coming to the subject for...
International Nuclear Information System (INIS)
Kotikov, A.V.
1993-01-01
A new method of massive Feynman diagrams calculation is presented. It provides a fairly simple procedure to obtain the result without the D-space integral calculation (for the dimensional regularization). Some diagrams are calculated as an illustration of this method capacities. (author). 7 refs
Differential equations methods and applications
Said-Houari, Belkacem
2015-01-01
This book presents a variety of techniques for solving ordinary differential equations analytically and features a wealth of examples. Focusing on the modeling of real-world phenomena, it begins with a basic introduction to differential equations, followed by linear and nonlinear first order equations and a detailed treatment of the second order linear equations. After presenting solution methods for the Laplace transform and power series, it lastly presents systems of equations and offers an introduction to the stability theory. To help readers practice the theory covered, two types of exercises are provided: those that illustrate the general theory, and others designed to expand on the text material. Detailed solutions to all the exercises are included. The book is excellently suited for use as a textbook for an undergraduate class (of all disciplines) in ordinary differential equations. .
Integral equation methods for electromagnetics
Volakis, John
2012-01-01
This text/reference is a detailed look at the development and use of integral equation methods for electromagnetic analysis, specifically for antennas and radar scattering. Developers and practitioners will appreciate the broad-based approach to understanding and utilizing integral equation methods and the unique coverage of historical developments that led to the current state-of-the-art. In contrast to existing books, Integral Equation Methods for Electromagnetics lays the groundwork in the initial chapters so students and basic users can solve simple problems and work their way up to the mo
Comparison of Kernel Equating and Item Response Theory Equating Methods
Meng, Yu
2012-01-01
The kernel method of test equating is a unified approach to test equating with some advantages over traditional equating methods. Therefore, it is important to evaluate in a comprehensive way the usefulness and appropriateness of the Kernel equating (KE) method, as well as its advantages and disadvantages compared with several popular item…
Variational linear algebraic equations method
International Nuclear Information System (INIS)
Moiseiwitsch, B.L.
1982-01-01
A modification of the linear algebraic equations method is described which ensures a variational bound on the phaseshifts for potentials having a definite sign at all points. The method is illustrated by the elastic scattering of s-wave electrons by the static field of atomic hydrogen. (author)
Generalized estimating equations
Hardin, James W
2002-01-01
Although powerful and flexible, the method of generalized linear models (GLM) is limited in its ability to accurately deal with longitudinal and clustered data. Developed specifically to accommodate these data types, the method of Generalized Estimating Equations (GEE) extends the GLM algorithm to accommodate the correlated data encountered in health research, social science, biology, and other related fields.Generalized Estimating Equations provides the first complete treatment of GEE methodology in all of its variations. After introducing the subject and reviewing GLM, the authors examine th
Iterative Splitting Methods for Differential Equations
Geiser, Juergen
2011-01-01
Iterative Splitting Methods for Differential Equations explains how to solve evolution equations via novel iterative-based splitting methods that efficiently use computational and memory resources. It focuses on systems of parabolic and hyperbolic equations, including convection-diffusion-reaction equations, heat equations, and wave equations. In the theoretical part of the book, the author discusses the main theorems and results of the stability and consistency analysis for ordinary differential equations. He then presents extensions of the iterative splitting methods to partial differential
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
Interdisciplinary Invitations: Exploring Gee's Bend Quilts
Mitchell, Rebecca; Whitin, Phyllis; Whitin, David
2012-01-01
Engaging with the quilts of Gee's Bend offers a rich opportunity for students in grades four through eight to develop appreciation for pattern, rhythm, and innovation while learning about history, entrepreneurship, and political activism. By easily accessing print, film, and Internet resources teachers can include these vibrant quilts and…
Abstract methods in partial differential equations
Carroll, Robert W
2012-01-01
Detailed, self-contained treatment examines modern abstract methods in partial differential equations, especially abstract evolution equations. Suitable for graduate students with some previous exposure to classical partial differential equations. 1969 edition.
International Nuclear Information System (INIS)
Green, P.F.
1981-01-01
It is stated that the common use of Poissonian errors to assign uncertainties in fission-track dating studies has led Johnson, McGee and Naeser (1979) to the mistaken assumption that such errors could be used to measure the spatial variation of track densities. The analysis proposed by JMN 79, employing this assumption, therefore leads to erroneous assessment of the error in an age determination. The basis for the statement is discussed. (U.K.)
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.
Numerical methods for differential equations and applications
International Nuclear Information System (INIS)
Ixaru, L.G.
1984-01-01
This book is addressed to persons who, without being professionals in applied mathematics, are often faced with the problem of numerically solving differential equations. In each of the first three chapters a definite class of methods is discussed for the solution of the initial value problem for ordinary differential equations: multistep methods; one-step methods; and piecewise perturbation methods. The fourth chapter is mainly focussed on the boundary value problems for linear second-order equations, with a section devoted to the Schroedinger equation. In the fifth chapter the eigenvalue problem for the radial Schroedinger equation is solved in several ways, with computer programs included. (Auth.)
Mac Con Iomaire, Máirtín
2016-01-01
Harold McGee was born on the East coast of America but grew up in Chicago. He writes about the chemistry of food and cooking. He took up this odd vocation after studies at the California Institute of Technology and at Yale University, where he wrote a doctoral thesis with the prophetic title "Keats and the Progress of Taste." After several years as a literature and writing instructor at Yale, he decided to practice what he'd been teaching, and write a book about the science of everyday life. ...
Numerical Methods for Partial Differential Equations
Guo, Ben-yu
1987-01-01
These Proceedings of the first Chinese Conference on Numerical Methods for Partial Differential Equations covers topics such as difference methods, finite element methods, spectral methods, splitting methods, parallel algorithm etc., their theoretical foundation and applications to engineering. Numerical methods both for boundary value problems of elliptic equations and for initial-boundary value problems of evolution equations, such as hyperbolic systems and parabolic equations, are involved. The 16 papers of this volume present recent or new unpublished results and provide a good overview of current research being done in this field in China.
Application of generalized estimating equations to a study in vitro of radiation sensitivity
International Nuclear Information System (INIS)
Cologne, J.B.; Carter, R.L.; Fujita, Shoichiro; Ban, Sadayuki.
1993-08-01
We describes an application of the generalized estimating equation (GEE) method (Liang K-Y, Zeger SL: Longitudinal data analysis using generalized linear models. Biometrika 73:13-22, 1986) for regression analyses of correlated Poisson data. As an alternative to the use of an arbitrarily chosen working correlation matrix, we demonstrate the use of GEE with a reasonable model for the true covariance structure among repeated observations within individuals. We show that, under such a split-plot design with large clusters, the asymptotic relative efficiency of GEE with simple (independence or exchangeable) working correlation matrices is rather low. We also illustrate the use of GEE with an empirically estimated model for overdispersion in a large study of radiation sensitivity where cluster size is small and a simple working correlation structure is sufficient. We conclude by summarizing issues and needs for further work concerning efficiency of the GEE parameter estimates in practice. (author)
Drilling and testing specifications for the McGee well
International Nuclear Information System (INIS)
Patterson, J.K.
1982-01-01
The McGee Well is a part of the Basalt Waste Isolation Project's subsurface site selection and characterization activities. Information from the McGee Well support site hydrologic characterization and repository design. These test specifications include details for the drilling and testing of the McGee. It includes the predicted stratigraphy, the drilling requirements, description of tests to be conducted, intervals selected for hydrologic testing, and a schedule of the drilling and testing activities. 19 refs., 10 figs., 7 tabs
Statistical Methods for Stochastic Differential Equations
Kessler, Mathieu; Sorensen, Michael
2012-01-01
The seventh volume in the SemStat series, Statistical Methods for Stochastic Differential Equations presents current research trends and recent developments in statistical methods for stochastic differential equations. Written to be accessible to both new students and seasoned researchers, each self-contained chapter starts with introductions to the topic at hand and builds gradually towards discussing recent research. The book covers Wiener-driven equations as well as stochastic differential equations with jumps, including continuous-time ARMA processes and COGARCH processes. It presents a sp
Method of controlling chaos in laser equations
International Nuclear Information System (INIS)
Duong-van, M.
1993-01-01
A method of controlling chaotic to laminar flows in the Lorenz equations using fixed points dictated by minimizing the Lyapunov functional was proposed by Singer, Wang, and Bau [Phys. Rev. Lett. 66, 1123 (1991)]. Using different fixed points, we find that the solutions in a chaotic regime can also be periodic. Since the laser equations are isomorphic to the Lorenz equations we use this method to control chaos when the laser is operated over the pump threshold. Furthermore, by solving the laser equations with an occasional proportional feedback mechanism, we recover the essential laser controlling features experimentally discovered by Roy, Murphy, Jr., Maier, Gills, and Hunt [Phys. Rev. Lett. 68, 1259 (1992)
Method of controlling chaos in laser equations
Duong-van, Minh
1993-01-01
A method of controlling chaotic to laminar flows in the Lorenz equations using fixed points dictated by minimizing the Lyapunov functional was proposed by Singer, Wang, and Bau [Phys. Rev. Lett. 66, 1123 (1991)]. Using different fixed points, we find that the solutions in a chaotic regime can also be periodic. Since the laser equations are isomorphic to the Lorenz equations we use this method to control chaos when the laser is operated over the pump threshold. Furthermore, by solving the laser equations with an occasional proportional feedback mechanism, we recover the essential laser controlling features experimentally discovered by Roy, Murphy, Jr., Maier, Gills, and Hunt [Phys. Rev. Lett. 68, 1259 (1992)].
Kahan, Brennan C; Harhay, Michael O
2015-12-01
Adjustment for center in multicenter trials is recommended when there are between-center differences or when randomization has been stratified by center. However, common methods of analysis (such as fixed-effects, Mantel-Haenszel, or stratified Cox models) often require a large number of patients or events per center to perform well. We reviewed 206 multicenter randomized trials published in four general medical journals to assess the average number of patients and events per center and determine whether appropriate methods of analysis were used in trials with few patients or events per center. The median number of events per center/treatment arm combination for trials using a binary or survival outcome was 3 (interquartile range, 1-10). Sixteen percent of trials had less than 1 event per center/treatment combination, 50% fewer than 3, and 63% fewer than 5. Of the trials which adjusted for center using a method of analysis which requires a large number of events per center, 6% had less than 1 event per center-treatment combination, 25% fewer than 3, and 50% fewer than 5. Methods of analysis that allow for few events per center, such as random-effects models or generalized estimating equations (GEEs), were rarely used. Many multicenter trials contain few events per center. Adjustment for center using random-effects models or GEE with model-based (non-robust) standard errors may be beneficial in these scenarios. Copyright © 2015 Elsevier Inc. All rights reserved.
Entropy viscosity method applied to Euler equations
International Nuclear Information System (INIS)
Delchini, M. O.; Ragusa, J. C.; Berry, R. A.
2013-01-01
The entropy viscosity method [4] has been successfully applied to hyperbolic systems of equations such as Burgers equation and Euler equations. The method consists in adding dissipative terms to the governing equations, where a viscosity coefficient modulates the amount of dissipation. The entropy viscosity method has been applied to the 1-D Euler equations with variable area using a continuous finite element discretization in the MOOSE framework and our results show that it has the ability to efficiently smooth out oscillations and accurately resolve shocks. Two equations of state are considered: Ideal Gas and Stiffened Gas Equations Of State. Results are provided for a second-order time implicit schemes (BDF2). Some typical Riemann problems are run with the entropy viscosity method to demonstrate some of its features. Then, a 1-D convergent-divergent nozzle is considered with open boundary conditions. The correct steady-state is reached for the liquid and gas phases with a time implicit scheme. The entropy viscosity method correctly behaves in every problem run. For each test problem, results are shown for both equations of state considered here. (authors)
Advances in iterative methods for nonlinear equations
Busquier, Sonia
2016-01-01
This book focuses on the approximation of nonlinear equations using iterative methods. Nine contributions are presented on the construction and analysis of these methods, the coverage encompassing convergence, efficiency, robustness, dynamics, and applications. Many problems are stated in the form of nonlinear equations, using mathematical modeling. In particular, a wide range of problems in Applied Mathematics and in Engineering can be solved by finding the solutions to these equations. The book reveals the importance of studying convergence aspects in iterative methods and shows that selection of the most efficient and robust iterative method for a given problem is crucial to guaranteeing a good approximation. A number of sample criteria for selecting the optimal method are presented, including those regarding the order of convergence, the computational cost, and the stability, including the dynamics. This book will appeal to researchers whose field of interest is related to nonlinear problems and equations...
Spline methods for conversation equations
International Nuclear Information System (INIS)
Bottcher, C.; Strayer, M.R.
1991-01-01
The consider the numerical solution of physical theories, in particular hydrodynamics, which can be formulated as systems of conservation laws. To this end we briefly describe the Basis Spline and collocation methods, paying particular attention to representation theory, which provides discrete analogues of the continuum conservation and dispersion relations, and hence a rigorous understanding of errors and instabilities. On this foundation we propose an algorithm for hydrodynamic problems in which most linear and nonlinear instabilities are brought under control. Numerical examples are presented from one-dimensional relativistic hydrodynamics. 9 refs., 10 figs
Ozdemir, Burhanettin
2017-01-01
The purpose of this study is to equate Trends in International Mathematics and Science Study (TIMSS) mathematics subtest scores obtained from TIMSS 2011 to scores obtained from TIMSS 2007 form with different nonlinear observed score equating methods under Non-Equivalent Anchor Test (NEAT) design where common items are used to link two or more test…
Partial differential equations methods, applications and theories
Hattori, Harumi
2013-01-01
This volume is an introductory level textbook for partial differential equations (PDE's) and suitable for a one-semester undergraduate level or two-semester graduate level course in PDE's or applied mathematics. Chapters One to Five are organized according to the equations and the basic PDE's are introduced in an easy to understand manner. They include the first-order equations and the three fundamental second-order equations, i.e. the heat, wave and Laplace equations. Through these equations we learn the types of problems, how we pose the problems, and the methods of solutions such as the separation of variables and the method of characteristics. The modeling aspects are explained as well. The methods introduced in earlier chapters are developed further in Chapters Six to Twelve. They include the Fourier series, the Fourier and the Laplace transforms, and the Green's functions. The equations in higher dimensions are also discussed in detail. This volume is application-oriented and rich in examples. Going thr...
Kinetic equation solution by inverse kinetic method
International Nuclear Information System (INIS)
Salas, G.
1983-01-01
We propose a computer program (CAMU) which permits to solve the inverse kinetic equation. The CAMU code is written in HPL language for a HP 982 A microcomputer with a peripheral interface HP 9876 A ''thermal graphic printer''. The CAMU code solves the inverse kinetic equation by taking as data entry the output of the ionization chambers and integrating the equation with the help of the Simpson method. With this program we calculate the evolution of the reactivity in time for a given disturbance
Waveform relaxation methods for implicit differential equations
P.J. van der Houwen; W.A. van der Veen
1996-01-01
textabstractWe apply a Runge-Kutta-based waveform relaxation method to initial-value problems for implicit differential equations. In the implementation of such methods, a sequence of nonlinear systems has to be solved iteratively in each step of the integration process. The size of these systems
Savoye, Philippe
2009-01-01
In recent years, I started covering difference equations and z transform methods in my introductory differential equations course. This allowed my students to extend the "classical" methods for (ordinary differential equation) ODE's to discrete time problems arising in many applications.
Structural equation modeling methods and applications
Wang, Jichuan
2012-01-01
A reference guide for applications of SEM using Mplus Structural Equation Modeling: Applications Using Mplus is intended as both a teaching resource and a reference guide. Written in non-mathematical terms, this book focuses on the conceptual and practical aspects of Structural Equation Modeling (SEM). Basic concepts and examples of various SEM models are demonstrated along with recently developed advanced methods, such as mixture modeling and model-based power analysis and sample size estimate for SEM. The statistical modeling program, Mplus, is also featured and provides researchers with a
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.
Method of ATMS operators in the formalism of Faddeev equations
International Nuclear Information System (INIS)
Zubarev, D.A.
1991-01-01
The method of ATMS operators is generalized for the case of Faddeev equations. The method to construct effective equations for both elastic scattering and scattering with rearrangement is presented. Properties to obtained equations are considered
Partial differential equations with numerical methods
Larsson, Stig
2003-01-01
The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering. The main theme is the integration of the theory of linear PDEs and the numerical solution of such equations. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. As preparation, the two-point boundary value problem and the initial-value problem for ODEs are discussed in separate chapters. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. Some background on linear functional analysis and Sobolev spaces, and also on numerical linear algebra, is reviewed in two appendices.
Efimova, Olga Yu.
2010-01-01
The modification of simplest equation method to look for exact solutions of nonlinear partial differential equations is presented. Using this method we obtain exact solutions of generalized Korteweg-de Vries equation with cubic source and exact solutions of third-order Kudryashov-Sinelshchikov equation describing nonlinear waves in liquids with gas bubbles.
Numerov iteration method for second order integral-differential equation
International Nuclear Information System (INIS)
Zeng Fanan; Zhang Jiaju; Zhao Xuan
1987-01-01
In this paper, Numerov iterative method for second order integral-differential equation and system of equations are constructed. Numerical examples show that this method is better than direct method (Gauss elimination method) in CPU time and memoy requireing. Therefore, this method is an efficient method for solving integral-differential equation in nuclear physics
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
Adaptive finite element methods for differential equations
Bangerth, Wolfgang
2003-01-01
These Lecture Notes discuss concepts of `self-adaptivity' in the numerical solution of differential equations, with emphasis on Galerkin finite element methods. The key issues are a posteriori error estimation and it automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method for goal-oriented error estimation, is discussed in detail. This method aims at economical computation of arbitrary quantities of physical interest by properly adapting the computational mesh. This is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters, and finally the stability of the resulting flow is investigated by solving an eigenvalue problem. `Goal-oriented' adaptivity is designed to achieve these tasks with minimal cost. At the end of each chapter some exercises are posed in order ...
Telescopic projective methods for parabolic differential equations
Gear, C W
2003-01-01
Projective methods were introduced in an earlier paper [C.W. Gear, I.G. Kevrekidis, Projective Methods for Stiff Differential Equations: problems with gaps in their eigenvalue spectrum, NEC Research Institute Report 2001-029, available from http://www.neci.nj.nec.com/homepages/cwg/projective.pdf Abbreviated version to appear in SISC] as having potential for the efficient integration of problems with a large gap between two clusters in their eigenvalue spectrum, one cluster containing eigenvalues corresponding to components that have already been damped in the numerical solution and one corresponding to components that are still active. In this paper we introduce iterated projective methods that allow for explicit integration of stiff problems that have a large spread of eigenvalues with no gaps in their spectrum as arise in the semi-discretization of PDEs with parabolic components.
Telescopic projective methods for parabolic differential equations
International Nuclear Information System (INIS)
Gear, C.W.; Kevrekidis, Ioannis G.
2003-01-01
Projective methods were introduced in an earlier paper [C.W. Gear, I.G. Kevrekidis, Projective Methods for Stiff Differential Equations: problems with gaps in their eigenvalue spectrum, NEC Research Institute Report 2001-029, available from http://www.neci.nj.nec.com/homepages/cwg/projective.pdf Abbreviated version to appear in SISC] as having potential for the efficient integration of problems with a large gap between two clusters in their eigenvalue spectrum, one cluster containing eigenvalues corresponding to components that have already been damped in the numerical solution and one corresponding to components that are still active. In this paper we introduce iterated projective methods that allow for explicit integration of stiff problems that have a large spread of eigenvalues with no gaps in their spectrum as arise in the semi-discretization of PDEs with parabolic components
McGee Creek Drainage and Levee District, Illinois.
1973-04-01
J COUNTY MC GEE REEK DRAINAGE BLEVEE DISTRICT N~ i, 0. ~ I I LEGEND Co ,. 3 ~ . -~------~----EXISTING LEVEE _____________PROPOSED LEVEE...Development, Conservation, Environmental Protection Agency, Health, Institute for Environmental Quality, Mines and Minerals. Pollution Control Board...Protection Agency, Health, Institute for Environmental Quality, Mines and Minerals, Pollution Control Board, Registration and Education, Transportation
An irrational trial equation method and its applications
Indian Academy of Sciences (India)
equation method which is different from those direct methods. Liu's key idea is that exact solution to a differential equation can be given by solving an integration. For example, consider a differential equation of u. We always assume that its exact solution satisfies a solvable equation u = F(u). Therefore, our task is just to find.
Deriving average soliton equations with a perturbative method
International Nuclear Information System (INIS)
Ballantyne, G.J.; Gough, P.T.; Taylor, D.P.
1995-01-01
The method of multiple scales is applied to periodically amplified, lossy media described by either the nonlinear Schroedinger (NLS) equation or the Korteweg--de Vries (KdV) equation. An existing result for the NLS equation, derived in the context of nonlinear optical communications, is confirmed. The method is then applied to the KdV equation and the result is confirmed numerically
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.
Entropy methods for diffusive partial differential equations
Jüngel, Ansgar
2016-01-01
This book presents a range of entropy methods for diffusive PDEs devised by many researchers in the course of the past few decades, which allow us to understand the qualitative behavior of solutions to diffusive equations (and Markov diffusion processes). Applications include the large-time asymptotics of solutions, the derivation of convex Sobolev inequalities, the existence and uniqueness of weak solutions, and the analysis of discrete and geometric structures of the PDEs. The purpose of the book is to provide readers an introduction to selected entropy methods that can be found in the research literature. In order to highlight the core concepts, the results are not stated in the widest generality and most of the arguments are only formal (in the sense that the functional setting is not specified or sufficient regularity is supposed). The text is also suitable for advanced master and PhD students and could serve as a textbook for special courses and seminars.
Method of lines solution of Richards` equation
Energy Technology Data Exchange (ETDEWEB)
Kelley, C.T.; Miller, C.T.; Tocci, M.D.
1996-12-31
We consider the method of lines solution of Richard`s equation, which models flow through porous media, as an example of a situation in which the method can give incorrect results because of premature termination of the nonlinear corrector iteration. This premature termination arises when the solution has a sharp moving front and the Jacobian is ill-conditioned. While this problem can be solved by tightening the tolerances provided to the ODE or DAE solver used for the temporal integration, it is more efficient to modify the termination criteria of the nonlinear solver and/or recompute the Jacobian more frequently. In this paper we continue previous work on this topic by analyzing the modifications in more detail and giving a strategy on how the modifications can be turned on and off in response to changes in the character of the solution.
PARALLEL SOLUTION METHODS OF PARTIAL DIFFERENTIAL EQUATIONS
Directory of Open Access Journals (Sweden)
Korhan KARABULUT
1998-03-01
Full Text Available Partial differential equations arise in almost all fields of science and engineering. Computer time spent in solving partial differential equations is much more than that of in any other problem class. For this reason, partial differential equations are suitable to be solved on parallel computers that offer great computation power. In this study, parallel solution to partial differential equations with Jacobi, Gauss-Siedel, SOR (Succesive OverRelaxation and SSOR (Symmetric SOR algorithms is studied.
Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation
Directory of Open Access Journals (Sweden)
Vitanov Nikolay K.
2018-03-01
Full Text Available We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations for deep water waves: the nonlinear Schrödinger equation. It is shown that the methodology works also for other equations of the nonlinear Schrödinger kind.
Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation
Vitanov, Nikolay K.; Dimitrova, Zlatinka I.
2018-03-01
We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations for deep water waves: the nonlinear Schrödinger equation. It is shown that the methodology works also for other equations of the nonlinear Schrödinger kind.
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)
International Nuclear Information System (INIS)
Zhao, Zhonglong; Zhang, Yufeng; Han, Zhong; Rui, Wenjuan
2014-01-01
In this paper, the simplest equation method is used to construct exact traveling solutions of the (3+1)-dimensional KP equation and generalized Fisher equation. We summarize the main steps of the simplest equation method. The Bernoulli and Riccati equation are used as simplest equations. This method is straightforward and concise, and it can be applied to other nonlinear partial differential equations
Relaxation methods for gauge field equilibrium equations
International Nuclear Information System (INIS)
Adler, S.L.; Piran, T.
1984-01-01
This article gives a pedagogical introduction to relaxation methods for the numerical solution of elliptic partial differential equations, with particular emphasis on treating nonlinear problems with delta-function source terms and axial symmetry, which arise in the context of effective Lagrangian approximations to the dynamics of quantized gauge fields. The authors present a detailed theoretical analysis of three models which are used as numerical examples: the classical Abelian Higgs model (illustrating charge screening), the semiclassical leading logarithm model (illustrating flux confinement within a free boundary or ''bag''), and the axially symmetric Bogomol'nyi-Prasad-Sommerfield monopoles (illustrating the occurrence of p topological quantum numbers in non-Abelian gauge fields). They then proceed to a self-contained introduction to the theory of relaxation methods and allied iterative numerical methods and to the practical aspects of their implementation, with attention to general issues which arise in the three examples. The authors conclude with a brief discussion of details of the numerical solution of the models, presenting sample numerical results
Analytic method for solitary solutions of some partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Ugurlu, Yavuz [Firat University, Department of Mathematics, 23119 Elazig (Turkey); Kaya, Dogan [Firat University, Department of Mathematics, 23119 Elazig (Turkey)], E-mail: dkaya@firat.edu.tr
2007-10-22
In this Letter by considering an improved tanh function method, we found some exact solutions of the clannish random walker's parabolic equation, the modified Korteweg-de Vries (KdV) equation, and the Sharma-Tasso-Olver (STO) equation with its fission and fusion, the Jaulent-Miodek equation.
Analytic method for solitary solutions of some partial differential equations
International Nuclear Information System (INIS)
Ugurlu, Yavuz; Kaya, Dogan
2007-01-01
In this Letter by considering an improved tanh function method, we found some exact solutions of the clannish random walker's parabolic equation, the modified Korteweg-de Vries (KdV) equation, and the Sharma-Tasso-Olver (STO) equation with its fission and fusion, the Jaulent-Miodek equation
A Line-Tau Collocation Method for Partial Differential Equations ...
African Journals Online (AJOL)
This paper deals with the numerical solution of second order linear partial differential equations with the use of the method of lines coupled with the tau collocation method. The method of lines is used to convert the partial differential equation (PDE) to a sequence of ordinary differential equations (ODEs) which is then ...
The Extended Fractional Subequation Method for Nonlinear Fractional Differential Equations
Zhao, Jianping; Tang, Bo; Kumar, Sunil; Hou, Yanren
2012-01-01
An extended fractional subequation method is proposed for solving fractional differential equations by introducing a new general ansätz and Bäcklund transformation of the fractional Riccati equation with known solutions. Being concise and straightforward, this method is applied to the space-time fractional coupled Burgers’ equations and coupled MKdV equations. As a result, many exact solutions are obtained. It is shown that the considered method provides a very effective, convenient, and powe...
Semigroup methods for evolution equations on networks
Mugnolo, Delio
2014-01-01
This concise text is based on a series of lectures held only a few years ago and originally intended as an introduction to known results on linear hyperbolic and parabolic equations. Yet the topic of differential equations on graphs, ramified spaces, and more general network-like objects has recently gained significant momentum and, well beyond the confines of mathematics, there is a lively interdisciplinary discourse on all aspects of so-called complex networks. Such network-like structures can be found in virtually all branches of science, engineering and the humanities, and future research thus calls for solid theoretical foundations. This book is specifically devoted to the study of evolution equations – i.e., of time-dependent differential equations such as the heat equation, the wave equation, or the Schrödinger equation (quantum graphs) – bearing in mind that the majority of the literature in the last ten years on the subject of differential equations of graphs has been devoted to ellip...
Introduction to numerical methods for time dependent differential equations
Kreiss, Heinz-Otto
2014-01-01
Introduces both the fundamentals of time dependent differential equations and their numerical solutions Introduction to Numerical Methods for Time Dependent Differential Equations delves into the underlying mathematical theory needed to solve time dependent differential equations numerically. Written as a self-contained introduction, the book is divided into two parts to emphasize both ordinary differential equations (ODEs) and partial differential equations (PDEs). Beginning with ODEs and their approximations, the authors provide a crucial presentation of fundamental notions, such as the t
A generalized simplest equation method and its application to the Boussinesq-Burgers equation.
Sudao, Bilige; Wang, Xiaomin
2015-01-01
In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method.
Applying homotopy analysis method for solving differential-difference equation
International Nuclear Information System (INIS)
Wang Zhen; Zou Li; Zhang Hongqing
2007-01-01
In this Letter, we apply the homotopy analysis method to solving the differential-difference equations. A simple but typical example is applied to illustrate the validity and the great potential of the generalized homotopy analysis method in solving differential-difference equation. Comparisons are made between the results of the proposed method and exact solutions. The results show that the homotopy analysis method is an attractive method in solving the differential-difference equations
Hilbert space methods in partial differential equations
Showalter, Ralph E
1994-01-01
This graduate-level text opens with an elementary presentation of Hilbert space theory sufficient for understanding the rest of the book. Additional topics include boundary value problems, evolution equations, optimization, and approximation.1979 edition.
Adaptive integral equation methods in transport theory
International Nuclear Information System (INIS)
Kelley, C.T.
1992-01-01
In this paper, an adaptive multilevel algorithm for integral equations is described that has been developed with the Chandrasekhar H equation and its generalizations in mind. The algorithm maintains good performance when the Frechet derivative of the nonlinear map is singular at the solution, as happens in radiative transfer with conservative scattering and in critical neutron transport. Numerical examples that demonstrate the algorithm's effectiveness are presented
Symmetrized neutron transport equation and the fast Fourier transform method
International Nuclear Information System (INIS)
Sinh, N.Q.; Kisynski, J.; Mika, J.
1978-01-01
The differential equation obtained from the neutron transport equation by the application of the source iteration method in two-dimensional rectangular geometry is transformed into a symmetrized form with respect to one of the angular variables. The discretization of the symmetrized equation leads to finite difference equations based on the five-point scheme and solved by use of the fast Fourier transform method. Possible advantages of the approach are shown on test calculations
Choi, Sae Il
2009-01-01
This study used simulation (a) to compare the kernel equating method to traditional equipercentile equating methods under the equivalent-groups (EG) design and the nonequivalent-groups with anchor test (NEAT) design and (b) to apply the parametric bootstrap method for estimating standard errors of equating. A two-parameter logistic item response…
The parabolic equation method for outdoor sound propagation
DEFF Research Database (Denmark)
Arranz, Marta Galindo
The parabolic equation method is a versatile tool for outdoor sound propagation. The present study has focused on the Cranck-Nicolson type Parabolic Equation method (CNPE). Three different applications of the CNPE method have been investigated. The first two applications study variations of the g......The parabolic equation method is a versatile tool for outdoor sound propagation. The present study has focused on the Cranck-Nicolson type Parabolic Equation method (CNPE). Three different applications of the CNPE method have been investigated. The first two applications study variations...
Integral Equation Methods for Electromagnetic and Elastic Waves
Chew, Weng; Hu, Bin
2008-01-01
Integral Equation Methods for Electromagnetic and Elastic Waves is an outgrowth of several years of work. There have been no recent books on integral equation methods. There are books written on integral equations, but either they have been around for a while, or they were written by mathematicians. Much of the knowledge in integral equation methods still resides in journal papers. With this book, important relevant knowledge for integral equations are consolidated in one place and researchers need only read the pertinent chapters in this book to gain important knowledge needed for integral eq
Solutions of hyperbolic equations with the CIP-BS method
International Nuclear Information System (INIS)
Utsumi, Takayuki; Koga, James; Yamagiwa, Mitsuru; Yabe, Takashi; Aoki, Takayuki
2004-01-01
In this paper, we show that a new numerical method, the Constrained Interpolation Profile - Basis Set (CIP-BS) method, can solve general hyperbolic equations efficiently. This method uses a simple polynomial basis set that is easily extendable to any desired higher-order accuracy. The interpolating profile is chosen so that the subgrid scale solution approaches the local real solution owing to the constraints from the spatial derivatives of the master equations. Then, introducing scalar products, the linear and nonlinear partial differential equations are uniquely reduced to the ordinary differential equations for values and spatial derivatives at the grid points. The method gives stable, less diffusive, and accurate results. It is successfully applied to the continuity equation, the Burgers equation, the Korteweg-de Vries equation, and one-dimensional shock tube problems. (author)
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 ...
A General Linear Method for Equating with Small Samples
Albano, Anthony D.
2015-01-01
Research on equating with small samples has shown that methods with stronger assumptions and fewer statistical estimates can lead to decreased error in the estimated equating function. This article introduces a new approach to linear observed-score equating, one which provides flexible control over how form difficulty is assumed versus estimated…
Differential and difference equations a comparison of methods of solution
Maximon, Leonard C
2016-01-01
This book, intended for researchers and graduate students in physics, applied mathematics and engineering, presents a detailed comparison of the important methods of solution for linear differential and difference equations - variation of constants, reduction of order, Laplace transforms and generating functions - bringing out the similarities as well as the significant differences in the respective analyses. Equations of arbitrary order are studied, followed by a detailed analysis for equations of first and second order. Equations with polynomial coefficients are considered and explicit solutions for equations with linear coefficients are given, showing significant differences in the functional form of solutions of differential equations from those of difference equations. An alternative method of solution involving transformation of both the dependent and independent variables is given for both differential and difference equations. A comprehensive, detailed treatment of Green’s functions and the associat...
A Comparison of Methods of Vertical Equating.
Loyd, Brenda H.; Hoover, H. D.
Rasch model vertical equating procedures were applied to three mathematics computation tests for grades six, seven, and eight. Each level of the test was composed of 45 items in three sets of 15 items, arranged in such a way that tests for adjacent grades had two sets (30 items) in common, and the sixth and eighth grades had 15 items in common. In…
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)
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.
Green's function method for perturbed Korteweg-de Vries equation
International Nuclear Information System (INIS)
Cai Hao; Huang Nianning
2003-01-01
The x-derivatives of squared Jost solution are the eigenfunctions with the zero eigenvalue of the linearized equation derived from the perturbed Korteweg-de Vries equation. A method similar to Green's function formalism is introduced to show the completeness of the squared Jost solutions in multi-soliton cases. It is not related to Lax equations directly, and thus it is beneficial to deal with the nonlinear equations with complicated Lax pair
Exact solution of some linear matrix equations using algebraic methods
Djaferis, T. E.; Mitter, S. K.
1977-01-01
A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.
International Nuclear Information System (INIS)
Wang Qi; Chen Yong; Zhang Hongqing
2005-01-01
In this paper, we present a new Riccati equation rational expansion method to uniformly construct a series of exact solutions for nonlinear evolution equations. Compared with most existing tanh methods and other sophisticated methods, the proposed method not only recover some known solutions, but also find some new and general solutions. The solutions obtained in this paper include rational triangular periodic wave solutions, rational solitary wave solutions and rational wave solutions. The efficiency of the method can be demonstrated on (2 + 1)-dimensional Burgers equation
International Nuclear Information System (INIS)
Chen Yong; Wang Qi; Li Biao
2005-01-01
Based on a new general ansatz and a general subepuation, a new general algebraic method named elliptic equation rational expansion method is devised for constructing multiple travelling wave solutions in terms of rational special function for nonlinear evolution equations (NEEs). We apply the proposed method to solve Whitham-Broer-Kaup equation and explicitly construct a series of exact solutions which include rational form solitary wave solution, rational form triangular periodic wave solutions and rational wave solutions as special cases. In addition, the links among our proposed method with the method by Fan [Chaos, Solitons and Fractals 2004;20:609], are also clarified generally
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
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 fractional partial differential equations.
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.
Generalized differential transform method to differential-difference equation
International Nuclear Information System (INIS)
Zou Li; Wang Zhen; Zong Zhi
2009-01-01
In this Letter, we generalize the differential transform method to solve differential-difference equation for the first time. Two simple but typical examples are applied to illustrate the validity and the great potential of the generalized differential transform method in solving differential-difference equation. A Pade technique is also introduced and combined with GDTM in aim of extending the convergence area of presented series solutions. Comparisons are made between the results of the proposed method and exact solutions. Then we apply the differential transform method to the discrete KdV equation and the discrete mKdV equation, and successfully obtain solitary wave solutions. The results reveal that the proposed method is very effective and simple. We should point out that generalized differential transform method is also easy to be applied to other nonlinear differential-difference equation.
Convergence of a random walk method for the Burgers equation
International Nuclear Information System (INIS)
Roberts, S.
1985-10-01
In this paper we consider a random walk algorithm for the solution of Burgers' equation. The algorithm uses the method of fractional steps. The non-linear advection term of the equation is solved by advecting ''fluid'' particles in a velocity field induced by the particles. The diffusion term of the equation is approximated by adding an appropriate random perturbation to the positions of the particles. Though the algorithm is inefficient as a method for solving Burgers' equation, it does model a similar method, the random vortex method, which has been used extensively to solve the incompressible Navier-Stokes equations. The purpose of this paper is to demonstrate the strong convergence of our random walk method and so provide a model for the proof of convergence for more complex random walk algorithms; for instance, the random vortex method without boundaries
The Integral Equation Method and the Neumann Problem for the Poisson Equation on NTA Domains
Czech Academy of Sciences Publication Activity Database
Medková, Dagmar
2009-01-01
Roč. 63, č. 21 (2009), s. 227-247 ISSN 0378-620X Institutional research plan: CEZ:AV0Z10190503 Keywords : Poisson equation * Neumann problem * integral equation method Subject RIV: BA - General Mathematics Impact factor: 0.477, year: 2009
Receptor binding kinetics equations: Derivation using the Laplace transform method.
Hoare, Sam R J
Measuring unlabeled ligand receptor binding kinetics is valuable in optimizing and understanding drug action. Unfortunately, deriving equations for estimating kinetic parameters is challenging because it involves calculus; integration can be a frustrating barrier to the pharmacologist seeking to measure simple rate parameters. Here, a well-known tool for simplifying the derivation, the Laplace transform, is applied to models of receptor-ligand interaction. The method transforms differential equations to a form in which simple algebra can be applied to solve for the variable of interest, for example the concentration of ligand-bound receptor. The goal is to provide instruction using familiar examples, to enable investigators familiar with handling equilibrium binding equations to derive kinetic equations for receptor-ligand interaction. First, the Laplace transform is used to derive the equations for association and dissociation of labeled ligand binding. Next, its use for unlabeled ligand kinetic equations is exemplified by a full derivation of the kinetics of competitive binding equation. Finally, new unlabeled ligand equations are derived using the Laplace transform. These equations incorporate a pre-incubation step with unlabeled or labeled ligand. Four equations for measuring unlabeled ligand kinetics were compared and the two new equations verified by comparison with numerical solution. Importantly, the equations have not been verified with experimental data because no such experiments are evident in the literature. Equations were formatted for use in the curve-fitting program GraphPad Prism 6.0 and fitted to simulated data. This description of the Laplace transform method will enable pharmacologists to derive kinetic equations for their model or experimental paradigm under study. Application of the transform will expand the set of equations available for the pharmacologist to measure unlabeled ligand binding kinetics, and for other time
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
Variable-mesh method of solving differential equations
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.
Discontinuous Galerkin finite element methods for hyperbolic differential equations
van der Vegt, Jacobus J.W.; van der Ven, H.; Boelens, O.J.; Boelens, O.J.; Toro, E.F.
2002-01-01
In this paper a suryey is given of the important steps in the development of discontinuous Galerkin finite element methods for hyperbolic partial differential equations. Special attention is paid to the application of the discontinuous Galerkin method to the solution of the Euler equations of gas
Insights: A New Method to Balance Chemical Equations.
Garcia, Arcesio
1987-01-01
Describes a method designed to balance oxidation-reduction chemical equations. Outlines a method which is based on changes in the oxidation number that can be applied to both molecular reactions and ionic reactions. Provides examples and delineates the steps to follow for each type of equation balancing. (TW)
Field Method for Integrating the First Order Differential Equation
Institute of Scientific and Technical Information of China (English)
JIA Li-qun; ZHENG Shi-wang; ZHANG Yao-yu
2007-01-01
An important modern method in analytical mechanics for finding the integral, which is called the field-method, is used to research the solution of a differential equation of the first order. First, by introducing an intermediate variable, a more complicated differential equation of the first order can be expressed by two simple differential equations of the first order, then the field-method in analytical mechanics is introduced for solving the two differential equations of the first order. The conclusion shows that the field-method in analytical mechanics can be fully used to find the solutions of a differential equation of the first order, thus a new method for finding the solutions of the first order is provided.
Introduction to partial differential equations and Hilbert space methods
Gustafson, Karl E
1997-01-01
Easy-to-use text examines principal method of solving partial differential equations, 1st-order systems, computation methods, and much more. Over 600 exercises, with answers for many. Ideal for a 1-semester or full-year course.
students' preference of method of solving simultaneous equations
African Journals Online (AJOL)
Ugboduma,Samuel.O.
substitution method irrespective of their gender for solving simultaneous equations. A recommendation ... advantage given to one method over others. Students' interest .... from two (2) single girls' schools, two (2) single boys schools and ten.
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.
Five-equation and robust three-equation methods for solution verification of large eddy simulation
Dutta, Rabijit; Xing, Tao
2018-02-01
This study evaluates the recently developed general framework for solution verification methods for large eddy simulation (LES) using implicitly filtered LES of periodic channel flows at friction Reynolds number of 395 on eight systematically refined grids. The seven-equation method shows that the coupling error based on Hypothesis I is much smaller as compared with the numerical and modeling errors and therefore can be neglected. The authors recommend five-equation method based on Hypothesis II, which shows a monotonic convergence behavior of the predicted numerical benchmark ( S C ), and provides realistic error estimates without the need of fixing the orders of accuracy for either numerical or modeling errors. Based on the results from seven-equation and five-equation methods, less expensive three and four-equation methods for practical LES applications were derived. It was found that the new three-equation method is robust as it can be applied to any convergence types and reasonably predict the error trends. It was also observed that the numerical and modeling errors usually have opposite signs, which suggests error cancellation play an essential role in LES. When Reynolds averaged Navier-Stokes (RANS) based error estimation method is applied, it shows significant error in the prediction of S C on coarse meshes. However, it predicts reasonable S C when the grids resolve at least 80% of the total turbulent kinetic energy.
Fast sweeping method for the factored eikonal equation
Fomel, Sergey; Luo, Songting; Zhao, Hongkai
2009-09-01
We develop a fast sweeping method for the factored eikonal equation. By decomposing the solution of a general eikonal equation as the product of two factors: the first factor is the solution to a simple eikonal equation (such as distance) or a previously computed solution to an approximate eikonal equation. The second factor is a necessary modification/correction. Appropriate discretization and a fast sweeping strategy are designed for the equation of the correction part. The key idea is to enforce the causality of the original eikonal equation during the Gauss-Seidel iterations. Using extensive numerical examples we demonstrate that (1) the convergence behavior of the fast sweeping method for the factored eikonal equation is the same as for the original eikonal equation, i.e., the number of iterations for the Gauss-Seidel iterations is independent of the mesh size, (2) the numerical solution from the factored eikonal equation is more accurate than the numerical solution directly computed from the original eikonal equation, especially for point sources.
International Nuclear Information System (INIS)
Zhang Huiqun
2009-01-01
By using a new coupled Riccati equations, a direct algebraic method, which was applied to obtain exact travelling wave solutions of some complex nonlinear equations, is improved. And the exact travelling wave solutions of the complex KdV equation, Boussinesq equation and Klein-Gordon equation are investigated using the improved method. The method presented in this paper can also be applied to construct exact travelling wave solutions for other nonlinear complex equations.
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.
Physics of the mechanical toy Gee-Haw Whammy Diddle.
Marek, Martin; Badin, Matej; Plesch, Martin
2018-02-27
Gee-Haw Whammy Diddle is a seemingly simple mechanical toy consisting of a wooden stick and a second stick that is made up of a series of notches with a propeller at its end. When the wooden stick is pulled over the notches, the propeller starts to rotate. Despite its simplicity, physical principles governing the motion of the stick and the propeller are rather complicated and interesting. Here we provide a thorough analysis of the system and parameters influencing the motion. We show that contrary to the results published on this topic so far, neither elliptic motion of the stick nor frequency synchronization is needed for starting a stable motion of the propeller.
Solving hyperbolic equations with finite volume methods
Vázquez-Cendón, M Elena
2015-01-01
Finite volume methods are used in numerous applications and by a broad multidisciplinary scientific community. The book communicates this important tool to students, researchers in training and academics involved in the training of students in different science and technology fields. The selection of content is based on the author’s experience giving PhD and master courses in different universities. In the book the introduction of new concepts and numerical methods go together with simple exercises, examples and applications that contribute to reinforce them. In addition, some of them involve the execution of MATLAB codes. The author promotes an understanding of common terminology with a balance between mathematical rigor and physical intuition that characterizes the origin of the methods. This book aims to be a first contact with finite volume methods. Once readers have studied it, they will be able to follow more specific bibliographical references and use commercial programs or open source software withi...
Quadratic algebras in the noncommutative integration method of wave equation
International Nuclear Information System (INIS)
Varaksin, O.L.
1995-01-01
The paper deals with the investigation of applications of the method of noncommutative integration of linear differential equations by partial derivatives. Nontrivial example was taken for integration of three-dimensions wave equation with the use of non-Abelian quadratic algebras
Application of the trial equation method for solving some nonlinear ...
Indian Academy of Sciences (India)
Therefore, our aim is just to find the function F. Liu has obtained a number of exact solutions to many nonlinear differential equations when F(u) is a polynomial or a rational function. ... In this study, we apply the trial equation method to seek exact solutions of the ... twice and setting the integration constant to zero, we have.
An auxiliary differential equation FDTD method for anisotropic magnetized plasmas
International Nuclear Information System (INIS)
Liu Shaobin; Mo Jinjun; Yuan Naichang
2004-01-01
An auxiliary differential equation finite-difference time-domain (ADE-FDTD) methodology for anisotropic magnetized plasmas is derived. The method is based on a difference approximation of the auxiliary differential equation. A comparison with the JEC method is included. The CPU time saving by several times and accuracy of the method are confirmed by computing the reflection and transmission through a magnetized plasma layer with the direction of propagation parallel to the direction of the biasing field
Systems of evolution equations and the singular perturbation method
International Nuclear Information System (INIS)
Mika, J.
Several fundamental theorems are presented important for the solution of linear evolution equations in the Banach space. The algorithm is deduced extending the solution of the system of singularly perturbed evolution equations into an asymptotic series with respect to a small positive parameter. The asymptotic convergence is shown of an approximate solution to the accurate solution. Singularly perturbed evolution equations of the resonance type were analysed. The special role is considered of the asymptotic equivalence of P1 equations obtained as the first order approximation if the spherical harmonics method is applied to the linear Boltzmann equation, and the diffusion equations of the linear transport theory where the small parameter approaches zero. (J.B.)
Directory of Open Access Journals (Sweden)
Browne Dillon
2008-05-01
Full Text Available Abstract Background The generalized estimating equations (GEE technique is often used in longitudinal data modeling, where investigators are interested in population-averaged effects of covariates on responses of interest. GEE involves specifying a model relating covariates to outcomes and a plausible correlation structure between responses at different time periods. While GEE parameter estimates are consistent irrespective of the true underlying correlation structure, the method has some limitations that include challenges with model selection due to lack of absolute goodness-of-fit tests to aid comparisons among several plausible models. The quadratic inference functions (QIF method extends the capabilities of GEE, while also addressing some GEE limitations. Methods We conducted a comparative study between GEE and QIF via an illustrative example, using data from the "National Longitudinal Survey of Children and Youth (NLSCY" database. The NLSCY dataset consists of long-term, population based survey data collected since 1994, and is designed to evaluate the determinants of developmental outcomes in Canadian children. We modeled the relationship between hyperactivity-inattention and gender, age, family functioning, maternal depression symptoms, household income adequacy, maternal immigration status and maternal educational level using GEE and QIF. Basis for comparison include: (1 ease of model selection; (2 sensitivity of results to different working correlation matrices; and (3 efficiency of parameter estimates. Results The sample included 795, 858 respondents (50.3% male; 12% immigrant; 6% from dysfunctional families. QIF analysis reveals that gender (male (odds ratio [OR] = 1.73; 95% confidence interval [CI] = 1.10 to 2.71, family dysfunctional (OR = 2.84, 95% CI of 1.58 to 5.11, and maternal depression (OR = 2.49, 95% CI of 1.60 to 2.60 are significantly associated with higher odds of hyperactivity-inattention. The results remained robust
Numerical methods for stochastic partial differential equations with white noise
Zhang, Zhongqiang
2017-01-01
This book covers numerical methods for stochastic partial differential equations with white noise using the framework of Wong-Zakai approximation. The book begins with some motivational and background material in the introductory chapters and is divided into three parts. Part I covers numerical stochastic ordinary differential equations. Here the authors start with numerical methods for SDEs with delay using the Wong-Zakai approximation and finite difference in time. Part II covers temporal white noise. Here the authors consider SPDEs as PDEs driven by white noise, where discretization of white noise (Brownian motion) leads to PDEs with smooth noise, which can then be treated by numerical methods for PDEs. In this part, recursive algorithms based on Wiener chaos expansion and stochastic collocation methods are presented for linear stochastic advection-diffusion-reaction equations. In addition, stochastic Euler equations are exploited as an application of stochastic collocation methods, where a numerical compa...
Numerical Methods for Partial Differential Equations.
1984-01-09
iteration or the conjugate gradient method. The smoothing sweeps are used to annihilate the highly oscillatory (compared to the grid spacing) components of...53 52 "- 33 41 *32 * . 31 * 21 - 11 O- carrius plane rotacions o I ~~arr: ’.trix vrS2-0 Cf A Figure 4. QM fiitorization of a BLTE (1,2) mnitrix
Galerkin method for solving diffusion equations
International Nuclear Information System (INIS)
Tsapelkin, E.S.
1975-01-01
A programme for the solution of the three-dimensional two-group multizone neutron diffusion problem in (x, y, z)-geometry is described. The programme XYZ-5 gives the currents of both groups, the effective neutron multiplication coefficient and several integral properties of the reactor. The solution was found with the Galerkin method using speciallly constructed and chosen coordinate functions. The programme is written in ALGOL-60 and consists of 5 parts. Its text is given
Incompressible spectral-element method: Derivation of equations
Deanna, Russell G.
1993-01-01
A fractional-step splitting scheme breaks the full Navier-Stokes equations into explicit and implicit portions amenable to the calculus of variations. Beginning with the functional forms of the Poisson and Helmholtz equations, we substitute finite expansion series for the dependent variables and derive the matrix equations for the unknown expansion coefficients. This method employs a new splitting scheme which differs from conventional three-step (nonlinear, pressure, viscous) schemes. The nonlinear step appears in the conventional, explicit manner, the difference occurs in the pressure step. Instead of solving for the pressure gradient using the nonlinear velocity, we add the viscous portion of the Navier-Stokes equation from the previous time step to the velocity before solving for the pressure gradient. By combining this 'predicted' pressure gradient with the nonlinear velocity in an explicit term, and the Crank-Nicholson method for the viscous terms, we develop a Helmholtz equation for the final velocity.
Knezevic, David; Patera, Anthony T.; Huynh, Dinh Bao Phuong
2010-01-01
We present a certified reduced basis (RB) method for the heat equation and wave equation. The critical ingredients are certified RB approximation of the Laplace transform; the inverse Laplace transform to develop the time-domain RB output approximation and rigorous error bound; a (Butterworth) filter in time to effect the necessary “modal” truncation; RB eigenfunction decomposition and contour integration for Offline–Online decomposition. We present numerical results to demonstrate the accura...
Various Newton-type iterative methods for solving nonlinear equations
Directory of Open Access Journals (Sweden)
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.
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.
APPLICATION OF BOUNDARY INTEGRAL EQUATION METHOD FOR THERMOELASTICITY PROBLEMS
Directory of Open Access Journals (Sweden)
Vorona Yu.V.
2015-12-01
Full Text Available Boundary Integral Equation Method is used for solving analytically the problems of coupled thermoelastic spherical wave propagation. The resulting mathematical expressions coincide with the solutions obtained in a conventional manner.
Approximate solution fuzzy pantograph equation by using homotopy perturbation method
Jameel, A. F.; Saaban, A.; Ahadkulov, H.; Alipiah, F. M.
2017-09-01
In this paper, Homotopy Perturbation Method (HPM) is modified and formulated to find the approximate solution for its employment to solve (FDDEs) involving a fuzzy pantograph equation. The solution that can be obtained by using HPM is in the form of infinite series that converge to the actual solution of the FDDE and this is one of the benefits of this method In addition, it can be used for solving high order fuzzy delay differential equations directly without reduction to a first order system. Moreover, the accuracy of HPM can be detected without needing the exact solution. The HPM is studied for fuzzy initial value problems involving pantograph equation. Using the properties of fuzzy set theory, we reformulate the standard approximate method of HPM and obtain the approximate solutions. The effectiveness of the proposed method is demonstrated for third order fuzzy pantograph equation.
A Numerical Method for Lane-Emden Equations Using Hybrid Functions and the Collocation Method
Directory of Open Access Journals (Sweden)
Changqing Yang
2012-01-01
Full Text Available A numerical method to solve Lane-Emden equations as singular initial value problems is presented in this work. This method is based on the replacement of unknown functions through a truncated series of hybrid of block-pulse functions and Chebyshev polynomials. The collocation method transforms the differential equation into a system of algebraic equations. It also has application in a wide area of differential equations. Corresponding numerical examples are presented to demonstrate the accuracy of the proposed method.
Workshop on Numerical Methods for Ordinary Differential Equations
Gear, Charles; Russo, Elvira
1989-01-01
Developments in numerical initial value ode methods were the focal topic of the meeting at L'Aquila which explord the connections between the classical background and new research areas such as differental-algebraic equations, delay integral and integro-differential equations, stability properties, continuous extensions (interpolants for Runge-Kutta methods and their applications, effective stepsize control, parallel algorithms for small- and large-scale parallel architectures). The resulting proceedings address many of these topics in both research and survey papers.
Solution methods for large systems of linear equations in BACCHUS
International Nuclear Information System (INIS)
Homann, C.; Dorr, B.
1993-05-01
The computer programme BACCHUS is used to describe steady state and transient thermal-hydraulic behaviour of a coolant in a fuel element with intact geometry in a fast breeder reactor. In such computer programmes generally large systems of linear equations with sparse matrices of coefficients, resulting from discretization of coolant conservation equations, must be solved thousands of times giving rise to large demands of main storage and CPU time. Direct and iterative solution methods of the systems of linear equations, available in BACCHUS, are described, giving theoretical details and experience with their use in the programme. Besides use of a method of lines, a Runge-Kutta-method, for solution of the partial differential equation is outlined. (orig.) [de
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
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.
Perturbation method for periodic solutions of nonlinear jerk equations
International Nuclear Information System (INIS)
Hu, H.
2008-01-01
A Lindstedt-Poincare type perturbation method with bookkeeping parameters is presented for determining accurate analytical approximate periodic solutions of some third-order (jerk) differential equations with cubic nonlinearities. In the process of the solution, higher-order approximate angular frequencies are obtained by Newton's method. A typical example is given to illustrate the effectiveness and simplicity of the proposed method
An improved front tracking method for the Euler equations
Witteveen, J.A.S.; Koren, B.; Bakker, P.G.
2007-01-01
An improved front tracking method for hyperbolic conservation laws is presented. The improved method accurately resolves discontinuities as well as continuous phenomena. The method is based on an improved front interaction model for a physically more accurate modeling of the Euler equations, as
Classical and modular methods applied to Diophantine equations
Dahmen, S.R.
2008-01-01
Deep methods from the theory of elliptic curves and modular forms have been used to prove Fermat's last theorem and solve other Diophantine equations. These so-called modular methods can often benefit from information obtained by other, classical, methods from number theory; and vice versa. In our
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.
Psychoanalytic and musical ambiguity: the tritone in gee, officer krupke.
Jaffee Nagel, Julie
2010-02-01
The poignant and timeless Broadway musical West Side Story is viewed from the standpoint of taking musical forms as psychoanalytic data. The musical configuration of notes called the tritone (or diabolus in musica) is taken as a sonic metaphor expressing ambiguity both in musical vocabulary and in mental life. The tritone, which historically and harmonically represents instability, is heard throughout the score and emphasizes the intrapsychic, interpersonal, and social dramas that unfold within and between the two gangs in West Side Story. Particular emphasis is given to the comic but exceedingly sober song Gee, Officer Krupke. Bernstein's sensitivity to the ambiguity and tension inherent in the tritone in West Side Story is conceptualized as an intersection of music theory and theories of mind; this perspective holds implications for clinical practice and transports psychoanalytic concepts from the couch to the Broadway stage and into the community to address the complexities of love, hate, aggression, prejudice, and violence. Ultimately, West Side Story cross-pollinates music and theater, as well as music and psychoanalytic concepts.
Solution of the Schroedinger equation by a spectral method
International Nuclear Information System (INIS)
Feit, M.D.; Fleck, J.A. Jr.; Steiger, A.
1982-01-01
A new computational method for determining the eigenvalues and eigenfunctions of the Schroedinger equation is described. Conventional methods for solving this problem rely on diagonalization of a Hamiltonian matrix or iterative numerical solutions of a time independent wave equation. The new method, in contrast, is based on the spectral properties of solutions to the time-dependent Schroedinger equation. The method requires the computation of a correlation function from a numerical solution psi(r, t). Fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system. Analysis of the location of these peaks reveals the eigenvalues with high accuracy. Additional Fourier transforms of psi(r, t) with respect to time generate the eigenfunctions. The effectiveness of the method is demonstrated for a one-dimensional asymmetric double well potential and for the two-dimensional Henon--Heiles potential
Analysis of spectral methods for the homogeneous Boltzmann equation
Filbet, Francis
2011-04-01
The development of accurate and fast algorithms for the Boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis. Recently, several works were devoted to the derivation of spectrally accurate schemes for the Boltzmann equation, but very few of them were concerned with the stability analysis of the method. In particular there was no result of stability except when the method was modified in order to enforce the positivity preservation, which destroys the spectral accuracy. In this paper we propose a new method to study the stability of homogeneous Boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution. This method takes advantage of the "spreading" property of the collision, together with estimates on regularity and entropy production. As an application we prove stability and convergence of spectral methods for the Boltzmann equation, when the discretization parameter is large enough (with explicit bound). © 2010 American Mathematical Society.
Analysis of spectral methods for the homogeneous Boltzmann equation
Filbet, Francis; Mouhot, Clé ment
2011-01-01
The development of accurate and fast algorithms for the Boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis. Recently, several works were devoted to the derivation of spectrally accurate schemes for the Boltzmann equation, but very few of them were concerned with the stability analysis of the method. In particular there was no result of stability except when the method was modified in order to enforce the positivity preservation, which destroys the spectral accuracy. In this paper we propose a new method to study the stability of homogeneous Boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution. This method takes advantage of the "spreading" property of the collision, together with estimates on regularity and entropy production. As an application we prove stability and convergence of spectral methods for the Boltzmann equation, when the discretization parameter is large enough (with explicit bound). © 2010 American Mathematical Society.
Convergence of method of lines approximations to partial differential equations
International Nuclear Information System (INIS)
Verwer, J.G.; Sanz-Serna, J.M.
1984-01-01
Many existing numerical schemes for evolutionary problems in partial differential equations (PDEs) can be viewed as method of lines (MOL) schemes. This paper treats the convergence of one-step MOL schemes. The main purpose is to set up a general framework for a convergence analysis applicable to nonlinear problems. The stability materials for this framework are taken from the field of nonlinear stiff ODEs. In this connection, important concepts are the logarithmic matrix norm and C-stability. A nonlinear parabolic equation and the cubic Schroedinger equation are used for illustrating the ideas. (Auth.)
On the evolution equations, solvable through the inverse scattering method
International Nuclear Information System (INIS)
Gerdjikov, V.S.; Khristov, E.Kh.
1979-01-01
The nonlinear evolution equations (NLEE), related to the one-parameter family of Dirac operators are considered in a uniform manner. The class of NLEE solvable through the inverse scatterina method and their conservation laws are described. The description of the hierarchy of Hamiltonian structures and the proof of complete integrability of the NLEE is presented. The class of Baecklund transformations for these NLEE is derived. The general formulae are illustrated by two important examples: the nonlinear Schroedinger equation and the sine-Gordon equation
New method for solving three-dimensional Schroedinger equation
International Nuclear Information System (INIS)
Melezhik, V.S.
1992-01-01
A new method is developed for solving the multidimensional Schroedinger equation without the variable separation. To solve the Schroedinger equation in a multidimensional coordinate space X, a difference grid Ω i (i=1,2,...,N) for some of variables, Ω, from X={R,Ω} is introduced and the initial partial-differential equation is reduced to a system of N differential-difference equations in terms of one of the variables R. The arising multi-channel scattering (or eigenvalue) problem is solved by the algorithm based on a continuous analog of the Newton method. The approach has been successfully tested for several two-dimensional problems (scattering on a nonspherical potential well and 'dipole' scatterer, a hydrogen atom in a homogenous magnetic field) and for a three-dimensional problem of the helium-atom bound states. (author)
Instability of the filtering method for Vlasov's equation
International Nuclear Information System (INIS)
Figua, H.; Bouchut, F.; Fijalkow, E.
1999-01-01
Klimas has introduced a smoothed Fourier-Fourier method. This method consists in convolving the original distribution function with a Gaussian distribution function, and, next, in solving the new system with a transformed splitting algorithm. Unfortunately, a second-order term appears in the new equation. In this work, it is studied how this term affects the numerical equation. In particular it is proven that instability occurs in the linear version of the Vlasov equation obtained by considering only free non-interacting particles. It is proved that the use of Fourier-Fourier transform is a fundamental requirement to solve this new equation. An important property is pointed out concerning the filtered distribution function in the transformed space. (K.A.)
A method of solving simple harmonic oscillator Schroedinger equation
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.
An introduction to neural network methods for differential equations
Yadav, Neha; Kumar, Manoj
2015-01-01
This book introduces a variety of neural network methods for solving differential equations arising in science and engineering. The emphasis is placed on a deep understanding of the neural network techniques, which has been presented in a mostly heuristic and intuitive manner. This approach will enable the reader to understand the working, efficiency and shortcomings of each neural network technique for solving differential equations. The objective of this book is to provide the reader with a sound understanding of the foundations of neural networks, and a comprehensive introduction to neural network methods for solving differential equations together with recent developments in the techniques and their applications. The book comprises four major sections. Section I consists of a brief overview of differential equations and the relevant physical problems arising in science and engineering. Section II illustrates the history of neural networks starting from their beginnings in the 1940s through to the renewed...
Methods of mathematical modelling continuous systems and differential equations
Witelski, Thomas
2015-01-01
This book presents mathematical modelling and the integrated process of formulating sets of equations to describe real-world problems. It describes methods for obtaining solutions of challenging differential equations stemming from problems in areas such as chemical reactions, population dynamics, mechanical systems, and fluid mechanics. Chapters 1 to 4 cover essential topics in ordinary differential equations, transport equations and the calculus of variations that are important for formulating models. Chapters 5 to 11 then develop more advanced techniques including similarity solutions, matched asymptotic expansions, multiple scale analysis, long-wave models, and fast/slow dynamical systems. Methods of Mathematical Modelling will be useful for advanced undergraduate or beginning graduate students in applied mathematics, engineering and other applied sciences.
New Efficient Fourth Order Method for Solving Nonlinear Equations
Directory of Open Access Journals (Sweden)
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 Parameter Robust Method for Singularly Perturbed Delay Differential Equations
Directory of Open Access Journals (Sweden)
Erdogan Fevzi
2010-01-01
Full Text Available Uniform finite difference methods are constructed via nonstandard finite difference methods for the numerical solution of singularly perturbed quasilinear initial value problem for delay differential equations. A numerical method is constructed for this problem which involves the appropriate Bakhvalov meshes on each time subinterval. The method is shown to be uniformly convergent with respect to the perturbation parameter. A numerical example is solved using the presented method, and the computed result is compared with exact solution of the problem.
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.
International Conference on Multiscale Methods and Partial Differential Equations.
Energy Technology Data Exchange (ETDEWEB)
Thomas Hou
2006-12-12
The International Conference on Multiscale Methods and Partial Differential Equations (ICMMPDE for short) was held at IPAM, UCLA on August 26-27, 2005. The conference brought together researchers, students and practitioners with interest in the theoretical, computational and practical aspects of multiscale problems and related partial differential equations. The conference provided a forum to exchange and stimulate new ideas from different disciplines, and to formulate new challenging multiscale problems that will have impact in applications.
Application of the bifurcation method to the modified Boussinesq equation
Directory of Open Access Journals (Sweden)
Shaoyong Li
2014-08-01
Firstly, we give a property of the solutions of the equation, that is, if $1+u(x, t$ is a solution, so is $1-u(x, t$. Secondly, by using the bifurcation method of dynamical systems we obtain some explicit expressions of solutions for the equation, which include kink-shaped solutions, blow-up solutions, periodic blow-up solutions and solitary wave solutions. Some previous results are extended.
KRYSI, Ordinary Differential Equations Solver with Sdirk Krylov Method
International Nuclear Information System (INIS)
Hindmarsh, A.C.; Norsett, S.P.
2001-01-01
1 - Description of program or function: KRYSI is a set of FORTRAN subroutines for solving ordinary differential equations initial value problems. It is suitable for both stiff and non-stiff systems. When solving the implicit stage equations in the stiff case, KRYSI uses a Krylov subspace iteration method called the SPIGMR (Scaled Preconditioned Incomplete Generalized Minimum Residual) method. No explicit Jacobian storage is required, except where used in pre- conditioning. A demonstration problem is included with a description of two pre-conditioners that are natural for its solution by KRYSI. 2 - Method of solution: KRYSI uses a three-stage, third-order singly diagonally implicit Runge-Kutta (SDIRK) method. In the stiff case, a preconditioned Krylov subspace iteration within a (so-called) inexact Newton iteration is used to solve the system of nonlinear algebraic equations
Finite element and discontinuous Galerkin methods for transient wave equations
Cohen, Gary
2017-01-01
This monograph presents numerical methods for solving transient wave equations (i.e. in time domain). More precisely, it provides an overview of continuous and discontinuous finite element methods for these equations, including their implementation in physical models, an extensive description of 2D and 3D elements with different shapes, such as prisms or pyramids, an analysis of the accuracy of the methods and the study of the Maxwell’s system and the important problem of its spurious free approximations. After recalling the classical models, i.e. acoustics, linear elastodynamics and electromagnetism and their variational formulations, the authors present a wide variety of finite elements of different shapes useful for the numerical resolution of wave equations. Then, they focus on the construction of efficient continuous and discontinuous Galerkin methods and study their accuracy by plane wave techniques and a priori error estimates. A chapter is devoted to the Maxwell’s system and the important problem ...
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)
GHM method for obtaining rationalsolutions of nonlinear differential equations.
Vazquez-Leal, Hector; Sarmiento-Reyes, Arturo
2015-01-01
In this paper, we propose the application of the general homotopy method (GHM) to obtain rational solutions of nonlinear differential equations. It delivers a high precision representation of the nonlinear differential equation using a few linear algebraic terms. In order to assess the benefits of this proposal, three nonlinear problems are solved and compared against other semi-analytic methods or numerical methods. The obtained results show that GHM is a powerful tool, capable to generate highly accurate rational solutions. AMS subject classification 34L30.
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
Comparative analysis of solution methods of the punctual kinetic equations
International Nuclear Information System (INIS)
Hernandez S, A.
2003-01-01
The following one written it presents a comparative analysis among different analytical solutions for the punctual kinetics equation, which present two variables of interest: a) the temporary behavior of the neutronic population, and b) The temporary behavior of the different groups of precursors of delayed neutrons. The first solution is based on a method that solves the transfer function of the differential equation for the neutronic population, in which intends to obtain the different poles that give the stability of this transfer function. In this section it is demonstrated that the temporary variation of the reactivity of the system can be managed as it is required, since the integration time for this method doesn't affect the result. However, the second solution is based on an iterative method like that of Runge-Kutta or the Euler method where the algorithm was only used to solve first order differential equations giving this way solution to each differential equation that conforms the equations of punctual kinetics. In this section it is demonstrated that only it can obtain a correct temporary behavior of the neutronic population when it is integrated on an interval of very short time, forcing to the temporary variation of the reactivity to change very quick way without one has some control about the time. In both methods the same change is used so much in the reactivity of the system like in the integration times, giving validity to the results graph the one the temporary behavior of the neutronic population vs. time. (Author)
Optimal analytic method for the nonlinear Hasegawa-Mima equation
Baxter, Mathew; Van Gorder, Robert A.; Vajravelu, Kuppalapalle
2014-05-01
The Hasegawa-Mima equation is a nonlinear partial differential equation that describes the electric potential due to a drift wave in a plasma. In the present paper, we apply the method of homotopy analysis to a slightly more general Hasegawa-Mima equation, which accounts for hyper-viscous damping or viscous dissipation. First, we outline the method for the general initial/boundary value problem over a compact rectangular spatial domain. We use a two-stage method, where both the convergence control parameter and the auxiliary linear operator are optimally selected to minimize the residual error due to the approximation. To do the latter, we consider a family of operators parameterized by a constant which gives the decay rate of the solutions. After outlining the general method, we consider a number of concrete examples in order to demonstrate the utility of this approach. The results enable us to study properties of the initial/boundary value problem for the generalized Hasegawa-Mima equation. In several cases considered, we are able to obtain solutions with extremely small residual errors after relatively few iterations are computed (residual errors on the order of 10-15 are found in multiple cases after only three iterations). The results demonstrate that selecting a parameterized auxiliary linear operator can be extremely useful for minimizing residual errors when used concurrently with the optimal homotopy analysis method, suggesting that this approach can prove useful for a number of nonlinear partial differential equations arising in physics and nonlinear mechanics.
International Nuclear Information System (INIS)
Wang Baodong; Song Lina; Zhang Hongqing
2007-01-01
In this paper, we present a new elliptic equation rational expansion method to uniformly construct a series of exact solutions for nonlinear partial differential equations. As an application of the method, we choose the (2 + 1)-dimensional Burgers equation to illustrate the method and successfully obtain some new and more general solutions
Solution of the porous media equation by Adomian's decomposition method
International Nuclear Information System (INIS)
Pamuk, Serdal
2005-01-01
The particular exact solutions of the porous media equation that usually occurs in nonlinear problems of heat and mass transfer, and in biological systems are obtained using Adomian's decomposition method. Also, numerical comparison of particular solutions in the decomposition method indicate that there is a very good agreement between the numerical solutions and particular exact solutions in terms of efficiency and accuracy
Higher-Order Integral Equation Methods in Computational Electromagnetics
DEFF Research Database (Denmark)
Jørgensen, Erik; Meincke, Peter
Higher-order integral equation methods have been investigated. The study has focused on improving the accuracy and efficiency of the Method of Moments (MoM) applied to electromagnetic problems. A new set of hierarchical Legendre basis functions of arbitrary order is developed. The new basis...
Efficient Numerical Methods for Stochastic Differential Equations in Computational Finance
Happola, Juho
2017-09-19
Stochastic Differential Equations (SDE) offer a rich framework to model the probabilistic evolution of the state of a system. Numerical approximation methods are typically needed in evaluating relevant Quantities of Interest arising from such models. In this dissertation, we present novel effective methods for evaluating Quantities of Interest relevant to computational finance when the state of the system is described by an SDE.
Multigrid methods for partial differential equations - a short introduction
International Nuclear Information System (INIS)
Linden, J.; Stueben, K.
1993-01-01
These notes summarize the multigrid methods and emphasis is laid on the algorithmic concepts of multigrid for solving linear and non-linear partial differential equations. In this paper there is brief description of the basic structure of multigrid methods. Detailed introduction is also contained with applications to VLSI process simulation. (A.B.)
Efficient Numerical Methods for Stochastic Differential Equations in Computational Finance
Happola, Juho
2017-01-01
Stochastic Differential Equations (SDE) offer a rich framework to model the probabilistic evolution of the state of a system. Numerical approximation methods are typically needed in evaluating relevant Quantities of Interest arising from such models. In this dissertation, we present novel effective methods for evaluating Quantities of Interest relevant to computational finance when the state of the system is described by an SDE.
Numerical simulation of GEW equation using RBF collocation method
Directory of Open Access Journals (Sweden)
Hamid Panahipour
2012-08-01
Full Text Available The generalized equal width (GEW equation is solved numerically by a meshless method based on a global collocation with standard types of radial basis functions (RBFs. Test problems including propagation of single solitons, interaction of two and three solitons, development of the Maxwellian initial condition pulses, wave undulation and wave generation are used to indicate the efficiency and accuracy of the method. Comparisons are made between the results of the proposed method and some other published numerical methods.
Application of Rational Expansion Method for Differential-Difference Equation
International Nuclear Information System (INIS)
Wang Qi
2011-01-01
In this paper, we applied the rational formal expansion method to construct a series of soliton-like and period-form solutions for nonlinear differential-difference equations. Compared with most existing methods, the proposed method not only recovers some known solutions, but also finds some new and more general solutions. The efficiency of the method can be demonstrated on Toda Lattice and Ablowitz-Ladik Lattice. (general)
High Weak Order Methods for Stochastic Differential Equations Based on Modified Equations
Abdulle, Assyr
2012-01-01
© 2012 Society for Industrial and Applied Mathematics. Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations. This approach is illustrated with the constructions of new methods of weak order two, in particular, semi-implicit integrators well suited for stiff (meansquare stable) stochastic problems, and implicit integrators that exactly conserve all quadratic first integrals of a stochastic dynamical system. Numerical examples confirm the theoretical results and show the versatility of our methodology.
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
Improved stochastic approximation methods for discretized parabolic partial differential equations
Guiaş, Flavius
2016-12-01
We present improvements of the stochastic direct simulation method, a known numerical scheme based on Markov jump processes which is used for approximating solutions of ordinary differential equations. This scheme is suited especially for spatial discretizations of evolution partial differential equations (PDEs). By exploiting the full path simulation of the stochastic method, we use this first approximation as a predictor and construct improved approximations by Picard iterations, Runge-Kutta steps, or a combination. This has as consequence an increased order of convergence. We illustrate the features of the improved method at a standard benchmark problem, a reaction-diffusion equation modeling a combustion process in one space dimension (1D) and two space dimensions (2D).
Solution of fractional differential equations by using differential transform method
International Nuclear Information System (INIS)
Arikoglu, Aytac; Ozkol, Ibrahim
2007-01-01
In this study, we implement a well known transformation technique, Differential Transform Method (DTM), to the area of fractional differential equations. Theorems that never existed before are introduced with their proofs. Also numerical examples are carried out for various types of problems, including the Bagley-Torvik, Ricatti and composite fractional oscillation equations for the application of the method. The results obtained are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, accurate and easy to apply
Solution of fractional differential equations by using differential transform method
Energy Technology Data Exchange (ETDEWEB)
Arikoglu, Aytac [Istanbul Technical University, Faculty of Aeronautics and Astronautics, Department of Aeronautical Engineering, Maslak, TR-34469 Istanbul (Turkey); Ozkol, Ibrahim [Istanbul Technical University, Faculty of Aeronautics and Astronautics, Department of Aeronautical Engineering, Maslak, TR-34469 Istanbul (Turkey)]. E-mail: ozkol@itu.edu.tr
2007-12-15
In this study, we implement a well known transformation technique, Differential Transform Method (DTM), to the area of fractional differential equations. Theorems that never existed before are introduced with their proofs. Also numerical examples are carried out for various types of problems, including the Bagley-Torvik, Ricatti and composite fractional oscillation equations for the application of the method. The results obtained are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, accurate and easy to apply.
Solving nonlinear evolution equation system using two different methods
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.
Error Analysis of Galerkin's Method for Semilinear Equations
Directory of Open Access Journals (Sweden)
Tadashi Kawanago
2012-01-01
Full Text Available We establish a general existence result for Galerkin's approximate solutions of abstract semilinear equations and conduct an error analysis. Our results may be regarded as some extension of a precedent work (Schultz 1969. The derivation of our results is, however, different from the discussion in his paper and is essentially based on the convergence theorem of Newton’s method and some techniques for deriving it. Some of our results may be applicable for investigating the quality of numerical verification methods for solutions of ordinary and partial differential equations.
An Approximate Method for the Acoustic Attenuating VTI Eikonal Equation
Hao, Q.
2017-05-26
We present an approximate method to solve the acoustic eikonal equation for attenuating transversely isotropic media with a vertical symmetry axis (VTI). A perturbation method is used to derive the perturbation formula for complex-valued traveltimes. The application of Shanks transform further enhances the accuracy of approximation. We derive both analytical and numerical solutions to the acoustic eikonal equation. The analytic solution is valid for homogeneous VTI media with moderate anellipticity and strong attenuation and attenuation-anisotropy. The numerical solution is applicable for inhomogeneous attenuating VTI media.
An Approximate Method for the Acoustic Attenuating VTI Eikonal Equation
Hao, Q.; Alkhalifah, Tariq Ali
2017-01-01
We present an approximate method to solve the acoustic eikonal equation for attenuating transversely isotropic media with a vertical symmetry axis (VTI). A perturbation method is used to derive the perturbation formula for complex-valued traveltimes. The application of Shanks transform further enhances the accuracy of approximation. We derive both analytical and numerical solutions to the acoustic eikonal equation. The analytic solution is valid for homogeneous VTI media with moderate anellipticity and strong attenuation and attenuation-anisotropy. The numerical solution is applicable for inhomogeneous attenuating VTI media.
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.
Directory of Open Access Journals (Sweden)
Nicola Koper
2012-03-01
Full Text Available Resource selection functions (RSF are often developed using satellite (ARGOS or Global Positioning System (GPS telemetry datasets, which provide a large amount of highly correlated data. We discuss and compare the use of generalized linear mixed-effects models (GLMM and generalized estimating equations (GEE for using this type of data to develop RSFs. GLMMs directly model differences among caribou, while GEEs depend on an adjustment of the standard error to compensate for correlation of data points within individuals. Empirical standard errors, rather than model-based standard errors, must be used with either GLMMs or GEEs when developing RSFs. There are several important differences between these approaches; in particular, GLMMs are best for producing parameter estimates that predict how management might influence individuals, while GEEs are best for predicting how management might influence populations. As the interpretation, value, and statistical significance of both types of parameter estimates differ, it is important that users select the appropriate analytical method. We also outline the use of k-fold cross validation to assess fit of these models. Both GLMMs and GEEs hold promise for developing RSFs as long as they are used appropriately.
Spectral methods for time dependent partial differential equations
Gottlieb, D.; Turkel, E.
1983-01-01
The theory of spectral methods for time dependent partial differential equations is reviewed. When the domain is periodic Fourier methods are presented while for nonperiodic problems both Chebyshev and Legendre methods are discussed. The theory is presented for both hyperbolic and parabolic systems using both Galerkin and collocation procedures. While most of the review considers problems with constant coefficients the extension to nonlinear problems is also discussed. Some results for problems with shocks are presented.
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 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.
Nonlinear ordinary differential equations analytical approximation and numerical methods
Hermann, Martin
2016-01-01
The book discusses the solutions to nonlinear ordinary differential equations (ODEs) using analytical and numerical approximation methods. Recently, analytical approximation methods have been largely used in solving linear and nonlinear lower-order ODEs. It also discusses using these methods to solve some strong nonlinear ODEs. There are two chapters devoted to solving nonlinear ODEs using numerical methods, as in practice high-dimensional systems of nonlinear ODEs that cannot be solved by analytical approximate methods are common. Moreover, it studies analytical and numerical techniques for the treatment of parameter-depending ODEs. The book explains various methods for solving nonlinear-oscillator and structural-system problems, including the energy balance method, harmonic balance method, amplitude frequency formulation, variational iteration method, homotopy perturbation method, iteration perturbation method, homotopy analysis method, simple and multiple shooting method, and the nonlinear stabilized march...
An alternative solver for the nodal expansion method equations - 106
International Nuclear Information System (INIS)
Carvalho da Silva, F.; Carlos Marques Alvim, A.; Senra Martinez, A.
2010-01-01
An automated procedure for nuclear reactor core design is accomplished by using a quick and accurate 3D nodal code, aiming at solving the diffusion equation, which describes the spatial neutron distribution in the reactor. This paper deals with an alternative solver for nodal expansion method (NEM), with only two inner iterations (mesh sweeps) per outer iteration, thus having the potential to reduce the time required to calculate the power distribution in nuclear reactors, but with accuracy similar to the ones found in conventional NEM. The proposed solver was implemented into a computational system which, besides solving the diffusion equation, also solves the burnup equations governing the gradual changes in material compositions of the core due to fuel depletion. Results confirm the effectiveness of the method for practical purposes. (authors)
Integral equation methods for vesicle electrohydrodynamics in three dimensions
Veerapaneni, Shravan
2016-12-01
In this paper, we develop a new boundary integral equation formulation that describes the coupled electro- and hydro-dynamics of a vesicle suspended in a viscous fluid and subjected to external flow and electric fields. The dynamics of the vesicle are characterized by a competition between the elastic, electric and viscous forces on its membrane. The classical Taylor-Melcher leaky-dielectric model is employed for the electric response of the vesicle and the Helfrich energy model combined with local inextensibility is employed for its elastic response. The coupled governing equations for the vesicle position and its transmembrane electric potential are solved using a numerical method that is spectrally accurate in space and first-order in time. The method uses a semi-implicit time-stepping scheme to overcome the numerical stiffness associated with the governing equations.
HAM-Based Adaptive Multiscale Meshless Method for Burgers Equation
Directory of Open Access Journals (Sweden)
Shu-Li Mei
2013-01-01
Full Text Available Based on the multilevel interpolation theory, we constructed a meshless adaptive multiscale interpolation operator (MAMIO with the radial basis function. Using this operator, any nonlinear partial differential equations such as Burgers equation can be discretized adaptively in physical spaces as a nonlinear matrix ordinary differential equation. In order to obtain the analytical solution of the system of ODEs, the homotopy analysis method (HAM proposed by Shijun Liao was developed to solve the system of ODEs by combining the precise integration method (PIM which can be employed to get the analytical solution of linear system of ODEs. The numerical experiences show that HAM is not sensitive to the time step, and so the arithmetic error is mainly derived from the discrete in physical space.
A nondissipative simulation method for the drift kinetic equation
International Nuclear Information System (INIS)
Watanabe, Tomo-Hiko; Sugama, Hideo; Sato, Tetsuya
2001-07-01
With the aim to study the ion temperature gradient (ITG) driven turbulence, a nondissipative kinetic simulation scheme is developed and comprehensively benchmarked. The new simulation method preserving the time-reversibility of basic kinetic equations can successfully reproduce the analytical solutions of asymmetric three-mode ITG equations which are extended to provide a more general reference for benchmarking than the previous work [T.-H. Watanabe, H. Sugama, and T. Sato: Phys. Plasmas 7 (2000) 984]. It is also applied to a dissipative three-mode system, and shows a good agreement with the analytical solution. The nondissipative simulation result of the ITG turbulence accurately satisfies the entropy balance equation. Usefulness of the nondissipative method for the drift kinetic simulations is confirmed in comparisons with other dissipative schemes. (author)
International Nuclear Information System (INIS)
Zhang Huiqun
2009-01-01
By using some exact solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact complex solutions for nonlinear partial differential equations. The method is implemented for the NLS equation, a new Hamiltonian amplitude equation, the coupled Schrodinger-KdV equations and the Hirota-Maccari equations. New exact complex solutions are obtained.
Approximate analytical methods for solving ordinary differential equations
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
Application of coarse-mesh methods to fluid dynamics equations
International Nuclear Information System (INIS)
Romstedt, P.; Werner, W.
1977-01-01
An Asymmetric Weighted Residual (ASWR) method for fluid dynamics equations is described. It leads to local operators with a 7-point Finite Difference (FD) structure, which is independent of the degree of the approximating polynomials. An 1-dimensional problem was solved by both this ASWR-method and a commonly used FD-method. The numerical results demonstrate that the ASWR-method combines high accuracy on a coarse computational mesh with short computing time per space point. The posibility of using fewer space points consequently brings about a considerable reduction in total running time for the ASWR-method as compared with conventional FD-methods. (orig.) [de
A Hennart nodal method for the diffusion equation
International Nuclear Information System (INIS)
Lesaint, P.; Noceir, S.; Verwaerde, D.
1995-01-01
A modification of the Hennart nodal method for neutron diffusion problems is presented. The final system of equations obtained by this method is not positive definite. However, a flux elimination technique leads to a simple positive definite system, which can be solved by the traditional iterative methods. Calculations of a two-dimensional International Atomic Energy Agency benchmark problem are performed and compared with results of the original Hennart nodal method and some finite element methods. The high computational efficiency of this modified nodal method is clearly demonstrated
Projection-iteration methods for solving nonlinear operator equations
International Nuclear Information System (INIS)
Nguyen Minh Chuong; Tran thi Lan Anh; Tran Quoc Binh
1989-09-01
In this paper, the authors investigate a nonlinear operator equation in uniformly convex Banach spaces as in metric spaces by using stationary and nonstationary generalized projection-iteration methods. Convergence theorems in the strong and weak sense were established. (author). 7 refs
An approximation method for nonlinear integral equations of Hammerstein type
International Nuclear Information System (INIS)
Chidume, C.E.; Moore, C.
1989-05-01
The solution of a nonlinear integral equation of Hammerstein type in Hilbert spaces is approximated by means of a fixed point iteration method. Explicit error estimates are given and, in some cases, convergence is shown to be at least as fast as a geometric progression. (author). 25 refs
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
Rothe's method for parabolic equations on non-cylindrical domains
Czech Academy of Sciences Publication Activity Database
Dasht, J.; Engström, J.; Kufner, Alois; Persson, L.E.
2006-01-01
Roč. 1, č. 1 (2006), s. 59-80 ISSN 0973-2306 Institutional research plan: CEZ:AV0Z10190503 Keywords : parabolic equations * non-cylindrical domains * Rothe's method * time-discretization Subject RIV: BA - General Mathematics
International Nuclear Information System (INIS)
Chen Yong; Yan Zhenya
2005-01-01
In this paper (2 + 1)-dimensional Gardner equation is investigated using a sine-Gordon equation expansion method, which was presented via a generalized sine-Gordon reduction equation and a new transformation. As a consequence, it is shown that the method is more powerful to obtain many types of new doubly periodic solutions of (2 + 1)-dimensional Gardner equation. In particular, solitary wave solutions are also given as simple limits of doubly periodic solutions
Dhage Iteration Method for Generalized Quadratic Functional Integral Equations
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Bapurao C. Dhage
2015-01-01
Full Text Available In this paper we prove the existence as well as approximations of the solutions for a certain nonlinear generalized quadratic functional integral equation. An algorithm for the solutions is developed and it is shown that the sequence of successive approximations starting at a lower or upper solution converges monotonically to the solutions of related quadratic functional integral equation under some suitable mixed hybrid conditions. We rely our main result on Dhage iteration method embodied in a recent hybrid fixed point theorem of Dhage (2014 in partially ordered normed linear spaces. An example is also provided to illustrate the abstract theory developed in the paper.
Numerical Simulation of Antennas with Improved Integral Equation Method
International Nuclear Information System (INIS)
Ma Ji; Fang Guang-You; Lu Wei
2015-01-01
Simulating antennas around a conducting object is a challenge task in computational electromagnetism, which is concerned with the behaviour of electromagnetic fields. To analyze this model efficiently, an improved integral equation-fast Fourier transform (IE-FFT) algorithm is presented in this paper. The proposed scheme employs two Cartesian grids with different size and location to enclose the antenna and the other object, respectively. On the one hand, IE-FFT technique is used to store matrix in a sparse form and accelerate the matrix-vector multiplication for each sub-domain independently. On the other hand, the mutual interaction between sub-domains is taken as the additional exciting voltage in each matrix equation. By updating integral equations several times, the whole electromagnetic system can achieve a stable status. Finally, the validity of the presented method is verified through the analysis of typical antennas in the presence of a conducting object. (paper)
Test equating, scaling, and linking methods and practices
Kolen, Michael J
2014-01-01
This book provides an introduction to test equating, scaling, and linking, including those concepts and practical issues that are critical for developers and all other testing professionals. In addition to statistical procedures, successful equating, scaling, and linking involves many aspects of testing, including procedures to develop tests, to administer and score tests, and to interpret scores earned on tests. Test equating methods are used with many standardized tests in education and psychology to ensure that scores from multiple test forms can be used interchangeably. Test scaling is the process of developing score scales that are used when scores on standardized tests are reported. In test linking, scores from two or more tests are related to one another. Linking has received much recent attention, due largely to investigations of linking similarly named tests from different test publishers or tests constructed for different purposes. In recent years, researchers from the education, psychology, and...
A mixed finite element method for nonlinear diffusion equations
Burger, Martin; Carrillo, José
2010-01-01
We propose a mixed finite element method for a class of nonlinear diffusion equations, which is based on their interpretation as gradient flows in optimal transportation metrics. We introduce an appropriate linearization of the optimal transport problem, which leads to a mixed symmetric formulation. This formulation preserves the maximum principle in case of the semi-discrete scheme as well as the fully discrete scheme for a certain class of problems. In addition solutions of the mixed formulation maintain exponential convergence in the relative entropy towards the steady state in case of a nonlinear Fokker-Planck equation with uniformly convex potential. We demonstrate the behavior of the proposed scheme with 2D simulations of the porous medium equations and blow-up questions in the Patlak-Keller-Segel model. © American Institute of Mathematical Sciences.
Fully Digital Chaotic Differential Equation-based Systems And Methods
Radwan, Ahmed Gomaa Ahmed
2012-09-06
Various embodiments are provided for fully digital chaotic differential equation-based systems and methods. In one embodiment, among others, a digital circuit includes digital state registers and one or more digital logic modules configured to obtain a first value from two or more of the digital state registers; determine a second value based upon the obtained first values and a chaotic differential equation; and provide the second value to set a state of one of the plurality of digital state registers. In another embodiment, a digital circuit includes digital state registers, digital logic modules configured to obtain outputs from a subset of the digital shift registers and to provide the input based upon a chaotic differential equation for setting a state of at least one of the subset of digital shift registers, and a digital clock configured to provide a clock signal for operating the digital shift registers.
Fully Digital Chaotic Differential Equation-based Systems And Methods
Radwan, Ahmed Gomaa Ahmed; Zidan, Mohammed A.; Salama, Khaled N.
2012-01-01
Various embodiments are provided for fully digital chaotic differential equation-based systems and methods. In one embodiment, among others, a digital circuit includes digital state registers and one or more digital logic modules configured to obtain a first value from two or more of the digital state registers; determine a second value based upon the obtained first values and a chaotic differential equation; and provide the second value to set a state of one of the plurality of digital state registers. In another embodiment, a digital circuit includes digital state registers, digital logic modules configured to obtain outputs from a subset of the digital shift registers and to provide the input based upon a chaotic differential equation for setting a state of at least one of the subset of digital shift registers, and a digital clock configured to provide a clock signal for operating the digital shift registers.
The integral equation method applied to eddy currents
International Nuclear Information System (INIS)
Biddlecombe, C.S.; Collie, C.J.; Simkin, J.; Trowbridge, C.W.
1976-04-01
An algorithm for the numerical solution of eddy current problems is described, based on the direct solution of the integral equation for the potentials. In this method only the conducting and iron regions need to be divided into elements, and there are no boundary conditions. Results from two computer programs using this method for iron free problems for various two-dimensional geometries are presented and compared with analytic solutions. (author)
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
Dirac equation in low dimensions: The factorization method
Energy Technology Data Exchange (ETDEWEB)
Sánchez-Monroy, J.A., E-mail: antosan@if.usp.br [Instituto de Física, Universidade de São Paulo, 05508-090, São Paulo, SP (Brazil); Quimbay, C.J., E-mail: cjquimbayh@unal.edu.co [Departamento de Física, Universidad Nacional de Colombia, Bogotá, D. C. (Colombia); CIF, Bogotá (Colombia)
2014-11-15
We present a general approach to solve the (1+1) and (2+1)-dimensional Dirac equations in the presence of static scalar, pseudoscalar and gauge potentials, for the case in which the potentials have the same functional form and thus the factorization method can be applied. We show that the presence of electric potentials in the Dirac equation leads to two Klein–Gordon equations including an energy-dependent potential. We then generalize the factorization method for the case of energy-dependent Hamiltonians. Additionally, the shape invariance is generalized for a specific class of energy-dependent Hamiltonians. We also present a condition for the absence of the Klein paradox (stability of the Dirac sea), showing how Dirac particles in low dimensions can be confined for a wide family of potentials. - Highlights: • The low-dimensional Dirac equation in the presence of static potentials is solved. • The factorization method is generalized for energy-dependent Hamiltonians. • The shape invariance is generalized for energy-dependent Hamiltonians. • The stability of the Dirac sea is related to the existence of supersymmetric partner Hamiltonians.
Numerical Methods for a Class of Differential Algebraic Equations
Directory of Open Access Journals (Sweden)
Lei Ren
2017-01-01
Full Text Available This paper is devoted to the study of some efficient numerical methods for the differential algebraic equations (DAEs. At first, we propose a finite algorithm to compute the Drazin inverse of the time varying DAEs. Numerical experiments are presented by Drazin inverse and Radau IIA method, which illustrate that the precision of the Drazin inverse method is higher than the Radau IIA method. Then, Drazin inverse, Radau IIA, and Padé approximation are applied to the constant coefficient DAEs, respectively. Numerical results demonstrate that the Padé approximation is powerful for solving constant coefficient DAEs.
Generalized multiscale finite element methods. nonlinear elliptic equations
Efendiev, Yalchin R.; Galvis, Juan; Li, Guanglian; Presho, Michael
2013-01-01
In this paper we use the Generalized Multiscale Finite Element Method (GMsFEM) framework, introduced in [26], in order to solve nonlinear elliptic equations with high-contrast coefficients. The proposed solution method involves linearizing the equation so that coarse-grid quantities of previous solution iterates can be regarded as auxiliary parameters within the problem formulation. With this convention, we systematically construct respective coarse solution spaces that lend themselves to either continuous Galerkin (CG) or discontinuous Galerkin (DG) global formulations. Here, we use Symmetric Interior Penalty Discontinuous Galerkin approach. Both methods yield a predictable error decline that depends on the respective coarse space dimension, and we illustrate the effectiveness of the CG and DG formulations by offering a variety of numerical examples. © 2014 Global-Science Press.
Iterative methods for compressible Navier-Stokes and Euler equations
Energy Technology Data Exchange (ETDEWEB)
Tang, W.P.; Forsyth, P.A.
1996-12-31
This workshop will focus on methods for solution of compressible Navier-Stokes and Euler equations. In particular, attention will be focused on the interaction between the methods used to solve the non-linear algebraic equations (e.g. full Newton or first order Jacobian) and the resulting large sparse systems. Various types of block and incomplete LU factorization will be discussed, as well as stability issues, and the use of Newton-Krylov methods. These techniques will be demonstrated on a variety of model transonic and supersonic airfoil problems. Applications to industrial CFD problems will also be presented. Experience with the use of C++ for solution of large scale problems will also be discussed. The format for this workshop will be four fifteen minute talks, followed by a roundtable discussion.
Numerical method for the nonlinear Fokker-Planck equation
International Nuclear Information System (INIS)
Zhang, D.S.; Wei, G.W.; Kouri, D.J.; Hoffman, D.K.
1997-01-01
A practical method based on distributed approximating functionals (DAFs) is proposed for numerically solving a general class of nonlinear time-dependent Fokker-Planck equations. The method relies on a numerical scheme that couples the usual path-integral concept to the DAF idea. The high accuracy and reliability of the method are illustrated by applying it to an exactly solvable nonlinear Fokker-Planck equation, and the method is compared with the accurate K-point Stirling interpolation formula finite-difference method. The approach is also used successfully to solve a nonlinear self-consistent dynamic mean-field problem for which both the cumulant expansion and scaling theory have been found by Drozdov and Morillo [Phys. Rev. E 54, 931 (1996)] to be inadequate to describe the occurrence of a long-lived transient bimodality. The standard interpretation of the transient bimodality in terms of the flat region in the kinetic potential fails for the present case. An alternative analysis based on the effective potential of the Schroedinger-like Fokker-Planck equation is suggested. Our analysis of the transient bimodality is strongly supported by two examples that are numerically much more challenging than other examples that have been previously reported for this problem. copyright 1997 The American Physical Society
International Nuclear Information System (INIS)
Yomba, Emmanuel
2005-01-01
By using a modified extended Fan's sub-equation method, we have obtained new and more general solutions including a series of non-travelling wave and coefficient function solutions namely: soliton-like solutions, triangular-like solutions, single and combined non-degenerative Jacobi elliptic wave function-like solutions for the (2 + 1)-dimensional dispersive long wave equation. The most important achievement of this method lies on the fact that, we have succeeded in one move to give all the solutions which can be previously obtained by application of at least four methods (method using Riccati equation, or first kind elliptic equation, or auxiliary ordinary equation, or generalized Riccati equation as mapping equation)
Runge-Kutta Methods for Linear Ordinary Differential Equations
Zingg, David W.; Chisholm, Todd T.
1997-01-01
Three new Runge-Kutta methods are presented for numerical integration of systems of linear inhomogeneous ordinary differential equations (ODES) with constant coefficients. Such ODEs arise in the numerical solution of the partial differential equations governing linear wave phenomena. The restriction to linear ODEs with constant coefficients reduces the number of conditions which the coefficients of the Runge-Kutta method must satisfy. This freedom is used to develop methods which are more efficient than conventional Runge-Kutta methods. A fourth-order method is presented which uses only two memory locations per dependent variable, while the classical fourth-order Runge-Kutta method uses three. This method is an excellent choice for simulations of linear wave phenomena if memory is a primary concern. In addition, fifth- and sixth-order methods are presented which require five and six stages, respectively, one fewer than their conventional counterparts, and are therefore more efficient. These methods are an excellent option for use with high-order spatial discretizations.
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)
Explicit Finite Difference Methods for the Delay Pseudoparabolic Equations
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I. Amirali
2014-01-01
Full Text Available Finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities two-level difference scheme is constructed. For the time integration the implicit rule is being used. Based on the method of energy estimates the fully discrete scheme is shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates are obtained in the discrete norm. Some numerical results confirming the expected behavior of the method are shown.
Finite element method for time-space-fractional Schrodinger equation
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Xiaogang Zhu
2017-07-01
Full Text Available In this article, we develop a fully discrete finite element method for the nonlinear Schrodinger equation (NLS with time- and space-fractional derivatives. The time-fractional derivative is described in Caputo's sense and the space-fractional derivative in Riesz's sense. Its stability is well derived; the convergent estimate is discussed by an orthogonal operator. We also extend the method to the two-dimensional time-space-fractional NLS and to avoid the iterative solvers at each time step, a linearized scheme is further conducted. Several numerical examples are implemented finally, which confirm the theoretical results as well as illustrate the accuracy of our methods.
A new method for the solution of the Schroedinger equation
International Nuclear Information System (INIS)
Amore, Paolo; Aranda, Alfredo; De Pace, Arturo
2004-01-01
We present a new method for the solution of the Schroedinger equation applicable to problems of a non-perturbative nature. The method works by identifying three different scales in the problem, which then are treated independently: an asymptotic scale, which depends uniquely on the form of the potential at large distances; an intermediate scale, still characterized by an exponential decay of the wavefunction; and, finally, a short distance scale, in which the wavefunction is sizable. The notion of optimized perturbation is then used in the last two regimes. We apply the method to the quantum anharmonic oscillator and find it suitable to treat both energy eigenvalues and wavefunctions, even for strong couplings
Support Operators Method for the Diffusion Equation in Multiple Materials
Energy Technology Data Exchange (ETDEWEB)
Winters, Andrew R. [Los Alamos National Laboratory; Shashkov, Mikhail J. [Los Alamos National Laboratory
2012-08-14
A second-order finite difference scheme for the solution of the diffusion equation on non-uniform meshes is implemented. The method allows the heat conductivity to be discontinuous. The algorithm is formulated on a one dimensional mesh and is derived using the support operators method. A key component of the derivation is that the discrete analog of the flux operator is constructed to be the negative adjoint of the discrete divergence, in an inner product that is a discrete analog of the continuum inner product. The resultant discrete operators in the fully discretized diffusion equation are symmetric and positive definite. The algorithm is generalized to operate on meshes with cells which have mixed material properties. A mechanism to recover intermediate temperature values in mixed cells using a limited linear reconstruction is introduced. The implementation of the algorithm is verified and the linear reconstruction mechanism is compared to previous results for obtaining new material temperatures.
International Nuclear Information System (INIS)
Kanki, Takashi; Uyama, Tadao; Tokuda, Shinji.
1995-07-01
In the numerical method to compute the matching data which are necessary for resistive MHD stability analyses, it is required to solve the eigenvalue problem and the associated singular equation. An iterative method is developed to solve the eigenvalue problem and the singular equation. In this method, the eigenvalue problem is replaced with an equivalent nonlinear equation and a singular equation is derived from Newton's method for the nonlinear equation. The multi-grid method (MGM), a high speed iterative method, can be applied to this method. The convergence of the eigenvalue and the eigenvector, and the CPU time in this method are investigated for a model equation. It is confirmed from the numerical results that this method is effective for solving the eigenvalue problem and the singular equation with numerical stability and high accuracy. It is shown by improving the MGM that the CPU time for this method is 50 times shorter than that of the direct method. (author)
Chen, Haiwen; Holland, Paul
2010-01-01
In this paper, we develop a new curvilinear equating for the nonequivalent groups with anchor test (NEAT) design under the assumption of the classical test theory model, that we name curvilinear Levine observed score equating. In fact, by applying both the kernel equating framework and the mean preserving linear transformation of…
Implicit methods for the Navier-Stokes equations
Yoon, S.; Kwak, D.
1990-01-01
Numerical solutions of the Navier-Stokes equations using explicit schemes can be obtained at the expense of efficiency. Conventional implicit methods which often achieve fast convergence rates suffer high cost per iteration. A new implicit scheme based on lower-upper factorization and symmetric Gauss-Seidel relaxation offers very low cost per iteration as well as fast convergence. High efficiency is achieved by accomplishing the complete vectorizability of the algorithm on oblique planes of sweep in three dimensions.
Unconditionally stable difference methods for delay partial differential equations
Huang, Chengming; Vandewalle, Stefan
2012-01-01
This paper is concerned with the numerical solution of parabolic partial differential equations with time-delay. We focus in particular on the delay dependent stability analysis of difference methods that use a non-constrained mesh, i.e., the time step-size is not required to be a submultiple of the delay. We prove that the fully discrete system unconditionally preserves the delay dependent asymptotic stability of the linear test problem under consideration, when the following discretizati...
Comparison of neutronic transport equation resolution nodal methods
International Nuclear Information System (INIS)
Zamonsky, O.M.; Gho, C.J.
1990-01-01
In this work, some transport equation resolution nodal methods are comparatively studied: the constant-constant (CC), linear-nodal (LN) and the constant-quadratic (CQ). A nodal scheme equivalent to finite differences has been used for its programming, permitting its inclusion in existing codes. Some bidimensional problems have been solved, showing that linear-nodal (LN) are, in general, obtained with accuracy in CPU shorter times. (Author) [es
Loop equations and bootstrap methods in the lattice
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Peter D. Anderson
2017-08-01
Full Text Available Pure gauge theories can be formulated in terms of Wilson Loops by means of the loop equation. In the large-N limit this equation closes in the expectation value of single loops. In particular, using the lattice as a regulator, it becomes a well defined equation for a discrete set of loops. In this paper we study different numerical approaches to solving this equation. Previous ideas gave good results in the strong coupling region. Here we propose an alternative method based on the observation that certain matrices ρˆ of Wilson loop expectation values are positive definite. They also have unit trace (ρˆ⪰0,Trρˆ=1, in fact they can be defined as reduced density matrices in the space of open loops after tracing over color indices and can be used to define an entropy associated with the loss of information due to such trace SWL=−Tr[ρˆlnρˆ]. The condition that such matrices are positive definite allows us to study the weak coupling region which is relevant for the continuum limit. In the exactly solvable case of two dimensions this approach gives very good results by considering just a few loops. In four dimensions it gives good results in the weak coupling region and therefore is complementary to the strong coupling expansion. We compare the results with standard Monte Carlo simulations.
Equation-Method for correcting clipping errors in OFDM signals.
Bibi, Nargis; Kleerekoper, Anthony; Muhammad, Nazeer; Cheetham, Barry
2016-01-01
Orthogonal frequency division multiplexing (OFDM) is the digital modulation technique used by 4G and many other wireless communication systems. OFDM signals have significant amplitude fluctuations resulting in high peak to average power ratios which can make an OFDM transmitter susceptible to non-linear distortion produced by its high power amplifiers (HPA). A simple and popular solution to this problem is to clip the peaks before an OFDM signal is applied to the HPA but this causes in-band distortion and introduces bit-errors at the receiver. In this paper we discuss a novel technique, which we call the Equation-Method, for correcting these errors. The Equation-Method uses the Fast Fourier Transform to create a set of simultaneous equations which, when solved, return the amplitudes of the peaks before they were clipped. We show analytically and through simulations that this method can, correct all clipping errors over a wide range of clipping thresholds. We show that numerical instability can be avoided and new techniques are needed to enable the receiver to differentiate between correctly and incorrectly received frequency-domain constellation symbols.
Ernst Equation and Riemann Surfaces: Analytical and Numerical Methods
International Nuclear Information System (INIS)
Ernst, Frederick J
2007-01-01
metric tensor components. The first two chapters of this book are devoted to some basic ideas: in the introductory chapter 1 the authors discuss the concept of integrability, comparing the integrability of the vacuum Ernst equation with the integrability of nonlinear equations of Korteweg-de Vries (KdV) type, while in chapter 2 they describe various circumstances in which the vacuum Ernst equation has been determined to be relevant, not only in connection with gravitation but also, for example, in the construction of solutions of the self-dual Yang-Mills equations. It is also in this chapter that one of several equivalent linear systems for the Ernst equation is described. The next two chapters are devoted to Dmitry Korotkin's concept of algebro-geometric solutions of a linear system: in chapter 3 the structure of such solutions of the vacuum Ernst equation, which involve Riemann theta functions of hyperelliptic algebraic curves of any genus, is contrasted with the periodic structure of such solutions of the KdV equation. How such solutions can be obtained, for example, by solving a matrix Riemann-Hilbert problem and how the metric tensor of the associated spacetime can be evaluated is described in detail. In chapter 4 the asymptotic behaviour and the similarity structure of the general algebro-geometric solutions of the Ernst equation are described, and the relationship of such solutions to the perhaps more familiar multi-soliton solutions is discussed. The next three chapters are based upon the authors' own published research: in chapter 5 it is shown that a problem involving counter-rotating infinitely thin disks of matter can be solved in terms of genus two Riemann theta functions, while in chapter 6 the authors describe numerical methods that facilitate the construction of such solutions, and in chapter 7 three-dimensional graphs are displayed that depict all metrical fields of the associated spacetime. Finally, in chapter 8, the difficulties associated with
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.
An Algebraic Method for Constructing Exact Solutions to Difference-Differential Equations
International Nuclear Information System (INIS)
Wang Zhen; Zhang Hongqing
2006-01-01
In this paper, we present a method to solve difference differential equation(s). As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation(s).
Optimal overlapping of waveform relaxation method for linear differential equations
International Nuclear Information System (INIS)
Yamada, Susumu; Ozawa, Kazufumi
2000-01-01
Waveform relaxation (WR) method is extremely suitable for solving large systems of ordinary differential equations (ODEs) on parallel computers, but the convergence of the method is generally slow. In order to accelerate the convergence, the methods which decouple the system into many subsystems with overlaps some of the components between the adjacent subsystems have been proposed. The methods, in general, converge much faster than the ones without overlapping, but the computational cost per iteration becomes larger due to the increase of the dimension of each subsystem. In this research, the convergence of the WR method for solving constant coefficients linear ODEs is investigated and the strategy to determine the number of overlapped components which minimizes the cost of the parallel computations is proposed. Numerical experiments on an SR2201 parallel computer show that the estimated number of the overlapped components by the proposed strategy is reasonable. (author)
A Meshfree Quasi-Interpolation Method for Solving Burgers’ Equation
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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.
Nonequilibrium Statistical Operator Method and Generalized Kinetic Equations
Kuzemsky, A. L.
2018-01-01
We consider some principal problems of nonequilibrium statistical thermodynamics in the framework of the Zubarev nonequilibrium statistical operator approach. We present a brief comparative analysis of some approaches to describing irreversible processes based on the concept of nonequilibrium Gibbs ensembles and their applicability to describing nonequilibrium processes. We discuss the derivation of generalized kinetic equations for a system in a heat bath. We obtain and analyze a damped Schrödinger-type equation for a dynamical system in a heat bath. We study the dynamical behavior of a particle in a medium taking the dissipation effects into account. We consider the scattering problem for neutrons in a nonequilibrium medium and derive a generalized Van Hove formula. We show that the nonequilibrium statistical operator method is an effective, convenient tool for describing irreversible processes in condensed matter.
International Nuclear Information System (INIS)
Kong Cuicui; Wang Dan; Song Lina; Zhang Hongqing
2009-01-01
In this paper, with the aid of symbolic computation and a general ansaetz, we presented a new extended rational expansion method to construct new rational formal exact solutions to nonlinear partial differential equations. In order to illustrate the effectiveness of this method, we apply it to the MKDV-Burgers equation and the (2 + 1)-dimensional dispersive long wave equation, then several new kinds of exact solutions are successfully obtained by using the new ansaetz. The method can also be applied to other nonlinear partial differential equations.
Acoustic 3D modeling by the method of integral equations
Malovichko, M.; Khokhlov, N.; Yavich, N.; Zhdanov, M.
2018-02-01
This paper presents a parallel algorithm for frequency-domain acoustic modeling by the method of integral equations (IE). The algorithm is applied to seismic simulation. The IE method reduces the size of the problem but leads to a dense system matrix. A tolerable memory consumption and numerical complexity were achieved by applying an iterative solver, accompanied by an effective matrix-vector multiplication operation, based on the fast Fourier transform (FFT). We demonstrate that, the IE system matrix is better conditioned than that of the finite-difference (FD) method, and discuss its relation to a specially preconditioned FD matrix. We considered several methods of matrix-vector multiplication for the free-space and layered host models. The developed algorithm and computer code were benchmarked against the FD time-domain solution. It was demonstrated that, the method could accurately calculate the seismic field for the models with sharp material boundaries and a point source and receiver located close to the free surface. We used OpenMP to speed up the matrix-vector multiplication, while MPI was used to speed up the solution of the system equations, and also for parallelizing across multiple sources. The practical examples and efficiency tests are presented as well.
Furihata, Daisuke
2010-01-01
Nonlinear Partial Differential Equations (PDEs) have become increasingly important in the description of physical phenomena. Unlike Ordinary Differential Equations, PDEs can be used to effectively model multidimensional systems. The methods put forward in Discrete Variational Derivative Method concentrate on a new class of ""structure-preserving numerical equations"" which improves the qualitative behaviour of the PDE solutions and allows for stable computing. The authors have also taken care to present their methods in an accessible manner, which means that the book will be useful to engineer
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)
Extensions of the auxiliary field method to solve Schroedinger equations
International Nuclear Information System (INIS)
Silvestre-Brac, Bernard; Semay, Claude; Buisseret, Fabien
2008-01-01
It has recently been shown that the auxiliary field method is an interesting tool to compute approximate analytical solutions of the Schroedinger equation. This technique can generate the spectrum associated with an arbitrary potential V(r) starting from the analytically known spectrum of a particular potential P(r). In the present work, general important properties of the auxiliary field method are proved, such as scaling laws and independence of the results on the choice of P(r). The method is extended in order to find accurate analytical energy formulae for radial potentials of the form aP(r) + V(r), and several explicit examples are studied. Connections existing between the perturbation theory and the auxiliary field method are also discussed
Extensions of the auxiliary field method to solve Schroedinger equations
Energy Technology Data Exchange (ETDEWEB)
Silvestre-Brac, Bernard [LPSC Universite Joseph Fourier, Grenoble 1, CNRS/IN2P3, Institut Polytechnique de Grenoble, Avenue des Martyrs 53, F-38026 Grenoble-Cedex (France); Semay, Claude; Buisseret, Fabien [Groupe de Physique Nucleaire Theorique, Universite de Mons-Hainaut, Academie universitaire Wallonie-Bruxelles, Place du Parc 20, B-7000 Mons (Belgium)], E-mail: silvestre@lpsc.in2p3.fr, E-mail: claude.semay@umh.ac.be, E-mail: fabien.buisseret@umh.ac.be
2008-10-24
It has recently been shown that the auxiliary field method is an interesting tool to compute approximate analytical solutions of the Schroedinger equation. This technique can generate the spectrum associated with an arbitrary potential V(r) starting from the analytically known spectrum of a particular potential P(r). In the present work, general important properties of the auxiliary field method are proved, such as scaling laws and independence of the results on the choice of P(r). The method is extended in order to find accurate analytical energy formulae for radial potentials of the form aP(r) + V(r), and several explicit examples are studied. Connections existing between the perturbation theory and the auxiliary field method are also discussed.
Preconditioned conjugate gradient methods for the Navier-Stokes equations
Ajmani, Kumud; Ng, Wing-Fai; Liou, Meng-Sing
1994-01-01
A preconditioned Krylov subspace method (GMRES) is used to solve the linear systems of equations formed at each time-integration step of the unsteady, two-dimensional, compressible Navier-Stokes equations of fluid flow. The Navier-Stokes equations are cast in an implicit, upwind finite-volume, flux-split formulation. Several preconditioning techniques are investigated to enhance the efficiency and convergence rate of the implicit solver based on the GMRES algorithm. The superiority of the new solver is established by comparisons with a conventional implicit solver, namely line Gauss-Seidel relaxation (LGSR). Computational test results for low-speed (incompressible flow over a backward-facing step at Mach 0.1), transonic flow (trailing edge flow in a transonic turbine cascade), and hypersonic flow (shock-on-shock interactions on a cylindrical leading edge at Mach 6.0) are presented. For the Mach 0.1 case, overall speedup factors of up to 17 (in terms of time-steps) and 15 (in terms of CPU time on a CRAY-YMP/8) are found in favor of the preconditioned GMRES solver, when compared with the LGSR solver. The corresponding speedup factors for the transonic flow case are 17 and 23, respectively. The hypersonic flow case shows slightly lower speedup factors of 9 and 13, respectively. The study of preconditioners conducted in this research reveals that a new LUSGS-type preconditioner is much more efficient than a conventional incomplete LU-type preconditioner.
International Nuclear Information System (INIS)
Yomba, Emmanuel
2008-01-01
With the aid of symbolic computation, a generalized auxiliary equation method is proposed to construct more general exact solutions to two types of NLPDEs. First, we present new family of solutions to a nonlinear Klein-Gordon equation, by using this auxiliary equation method including a new first-order nonlinear ODE with six-degree nonlinear term proposed by Sirendaoreji. Then, we apply an indirect F-function method very close to the F-expansion method to solve the generalized Camassa-Holm equation with fully nonlinear dispersion and fully nonlinear convection C(l,n,p). Taking advantage of the new first-order nonlinear ODE with six degree nonlinear term, this indirect F-function method is used to map the solutions of C(l,n,p) equations to those of that nonlinear ODE. As a result, we can successfully obtain in a unified way, many exact solutions
New finite volume methods for approximating partial differential equations on arbitrary meshes
International Nuclear Information System (INIS)
Hermeline, F.
2008-12-01
This dissertation presents some new methods of finite volume type for approximating partial differential equations on arbitrary meshes. The main idea lies in solving twice the problem to be dealt with. One addresses the elliptic equations with variable (anisotropic, antisymmetric, discontinuous) coefficients, the parabolic linear or non linear equations (heat equation, radiative diffusion, magnetic diffusion with Hall effect), the wave type equations (Maxwell, acoustics), the elasticity and Stokes'equations. Numerous numerical experiments show the good behaviour of this type of method. (author)
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
On numerical solution of Burgers' equation by homotopy analysis method
International Nuclear Information System (INIS)
Inc, Mustafa
2008-01-01
In this Letter, we present the Homotopy Analysis Method (shortly HAM) for obtaining the numerical solution of the one-dimensional nonlinear Burgers' equation. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Convergence of the solution and effects for the method is discussed. The comparison of the HAM results with the Homotopy Perturbation Method (HPM) and the results of [E.N. Aksan, Appl. Math. Comput. 174 (2006) 884; S. Kutluay, A. Esen, Int. J. Comput. Math. 81 (2004) 1433; S. Abbasbandy, M.T. Darvishi, Appl. Math. Comput. 163 (2005) 1265] are made. The results reveal that HAM is very simple and effective. The HAM contains the auxiliary parameter h, which provides us with a simple way to adjust and control the convergence region of solution series. The numerical solutions are compared with the known analytical and some numerical solutions
[Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (2)].
Murase, Kenya
2015-01-01
In this issue, symbolic methods for solving differential equations were firstly introduced. Of the symbolic methods, Laplace transform method was also introduced together with some examples, in which this method was applied to solving the differential equations derived from a two-compartment kinetic model and an equivalent circuit model for membrane potential. Second, series expansion methods for solving differential equations were introduced together with some examples, in which these methods were used to solve Bessel's and Legendre's differential equations. In the next issue, simultaneous differential equations and various methods for solving these differential equations will be introduced together with some examples in medical physics.
Hierarchical Matrices Method and Its Application in Electromagnetic Integral Equations
Directory of Open Access Journals (Sweden)
Han Guo
2012-01-01
Full Text Available Hierarchical (H- matrices method is a general mathematical framework providing a highly compact representation and efficient numerical arithmetic. When applied in integral-equation- (IE- based computational electromagnetics, H-matrices can be regarded as a fast algorithm; therefore, both the CPU time and memory requirement are reduced significantly. Its kernel independent feature also makes it suitable for any kind of integral equation. To solve H-matrices system, Krylov iteration methods can be employed with appropriate preconditioners, and direct solvers based on the hierarchical structure of H-matrices are also available along with high efficiency and accuracy, which is a unique advantage compared to other fast algorithms. In this paper, a novel sparse approximate inverse (SAI preconditioner in multilevel fashion is proposed to accelerate the convergence rate of Krylov iterations for solving H-matrices system in electromagnetic applications, and a group of parallel fast direct solvers are developed for dealing with multiple right-hand-side cases. Finally, numerical experiments are given to demonstrate the advantages of the proposed multilevel preconditioner compared to conventional “single level” preconditioners and the practicability of the fast direct solvers for arbitrary complex structures.
Generalized multiscale finite element method for elasticity equations
Chung, Eric T.
2014-10-05
In this paper, we discuss the application of generalized multiscale finite element method (GMsFEM) to elasticity equation in heterogeneous media. We consider steady state elasticity equations though some of our applications are motivated by elastic wave propagation in subsurface where the subsurface properties can be highly heterogeneous and have high contrast. We present the construction of main ingredients for GMsFEM such as the snapshot space and offline spaces. The latter is constructed using local spectral decomposition in the snapshot space. The spectral decomposition is based on the analysis which is provided in the paper. We consider both continuous Galerkin and discontinuous Galerkin coupling of basis functions. Both approaches have their cons and pros. Continuous Galerkin methods allow avoiding penalty parameters though they involve partition of unity functions which can alter the properties of multiscale basis functions. On the other hand, discontinuous Galerkin techniques allow gluing multiscale basis functions without any modifications. Because basis functions are constructed independently from each other, this approach provides an advantage. We discuss the use of oversampling techniques that use snapshots in larger regions to construct the offline space. We provide numerical results to show that one can accurately approximate the solution using reduced number of degrees of freedom.
Exact solitary wave solutions for some nonlinear evolution equations via Exp-function method
International Nuclear Information System (INIS)
Ebaid, A.
2007-01-01
Based on the Exp-function method, exact solutions for some nonlinear evolution equations are obtained. The KdV equation, Burgers' equation and the combined KdV-mKdV equation are chosen to illustrate the effectiveness of the method
The reduced basis method for the electric field integral equation
International Nuclear Information System (INIS)
Fares, M.; Hesthaven, J.S.; Maday, Y.; Stamm, B.
2011-01-01
We introduce the reduced basis method (RBM) as an efficient tool for parametrized scattering problems in computational electromagnetics for problems where field solutions are computed using a standard Boundary Element Method (BEM) for the parametrized electric field integral equation (EFIE). This combination enables an algorithmic cooperation which results in a two step procedure. The first step consists of a computationally intense assembling of the reduced basis, that needs to be effected only once. In the second step, we compute output functionals of the solution, such as the Radar Cross Section (RCS), independently of the dimension of the discretization space, for many different parameter values in a many-query context at very little cost. Parameters include the wavenumber, the angle of the incident plane wave and its polarization.
Reduced basis methods for partial differential equations an introduction
Quarteroni, Alfio; Negri, Federico
2016-01-01
This book provides a basic introduction to reduced basis (RB) methods for problems involving the repeated solution of partial differential equations (PDEs) arising from engineering and applied sciences, such as PDEs depending on several parameters and PDE-constrained optimization. The book presents a general mathematical formulation of RB methods, analyzes their fundamental theoretical properties, discusses the related algorithmic and implementation aspects, and highlights their built-in algebraic and geometric structures. More specifically, the authors discuss alternative strategies for constructing accurate RB spaces using greedy algorithms and proper orthogonal decomposition techniques, investigate their approximation properties and analyze offline-online decomposition strategies aimed at the reduction of computational complexity. Furthermore, they carry out both a priori and a posteriori error analysis. The whole mathematical presentation is made more stimulating by the use of representative examp...
A generalized trial solution method for solving the aerosol equation
International Nuclear Information System (INIS)
Simons, S.; Simpson, D.R.
1988-01-01
It is shown how the introduction of orthogonal functions together with a time-dependent scaling factor may be used to develop a generalized trial solution method for tackling the aerosol equation. The approach is worked out in detail for the case where the initial particle size spectrum follows a γ-distribution, and it is shown to be a viable technique as long as the initial volume fraction of particulate material is not too large. The method is applied to several situations of interest, and is shown to give more accurate results (with marginally shorter computing times) than are given by the three-parameter log-normal or γ distribution trial functions. (author)
Time evolution of the wave equation using rapid expansion method
Pestana, Reynam C.; Stoffa, Paul L.
2010-01-01
Forward modeling of seismic data and reverse time migration are based on the time evolution of wavefields. For the case of spatially varying velocity, we have worked on two approaches to evaluate the time evolution of seismic wavefields. An exact solution for the constant-velocity acoustic wave equation can be used to simulate the pressure response at any time. For a spatially varying velocity, a one-step method can be developed where no intermediate time responses are required. Using this approach, we have solved for the pressure response at intermediate times and have developed a recursive solution. The solution has a very high degree of accuracy and can be reduced to various finite-difference time-derivative methods, depending on the approximations used. Although the two approaches are closely related, each has advantages, depending on the problem being solved. © 2010 Society of Exploration Geophysicists.
Time evolution of the wave equation using rapid expansion method
Pestana, Reynam C.
2010-07-01
Forward modeling of seismic data and reverse time migration are based on the time evolution of wavefields. For the case of spatially varying velocity, we have worked on two approaches to evaluate the time evolution of seismic wavefields. An exact solution for the constant-velocity acoustic wave equation can be used to simulate the pressure response at any time. For a spatially varying velocity, a one-step method can be developed where no intermediate time responses are required. Using this approach, we have solved for the pressure response at intermediate times and have developed a recursive solution. The solution has a very high degree of accuracy and can be reduced to various finite-difference time-derivative methods, depending on the approximations used. Although the two approaches are closely related, each has advantages, depending on the problem being solved. © 2010 Society of Exploration Geophysicists.
[Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (1)].
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.
DEFF Research Database (Denmark)
Miansari, Mo; Miansari, Me; Barari, Amin
2009-01-01
In this article, He’s variational iteration method (VIM), is implemented to solve the linear Helmholtz partial differential equation and some nonlinear fifth-order Korteweg-de Vries (FKdV) partial differential equations with specified initial conditions. The initial approximations can be freely c...
Final Report: Symposium on Adaptive Methods for Partial Differential Equations
Energy Technology Data Exchange (ETDEWEB)
Pernice, M.; Johnson, C.R.; Smith, P.J.; Fogelson, A.
1998-12-10
OAK-B135 Final Report: Symposium on Adaptive Methods for Partial Differential Equations. Complex physical phenomena often include features that span a wide range of spatial and temporal scales. Accurate simulation of such phenomena can be difficult to obtain, and computations that are under-resolved can even exhibit spurious features. While it is possible to resolve small scale features by increasing the number of grid points, global grid refinement can quickly lead to problems that are intractable, even on the largest available computing facilities. These constraints are particularly severe for three dimensional problems that involve complex physics. One way to achieve the needed resolution is to refine the computational mesh locally, in only those regions where enhanced resolution is required. Adaptive solution methods concentrate computational effort in regions where it is most needed. These methods have been successfully applied to a wide variety of problems in computational science and engineering. Adaptive methods can be difficult to implement, prompting the development of tools and environments to facilitate their use. To ensure that the results of their efforts are useful, algorithm and tool developers must maintain close communication with application specialists. Conversely it remains difficult for application specialists who are unfamiliar with the methods to evaluate the trade-offs between the benefits of enhanced local resolution and the effort needed to implement an adaptive solution method.
Pseudospectral collocation methods for fourth order differential equations
Malek, Alaeddin; Phillips, Timothy N.
1994-01-01
Collocation schemes are presented for solving linear fourth order differential equations in one and two dimensions. The variational formulation of the model fourth order problem is discretized by approximating the integrals by a Gaussian quadrature rule generalized to include the values of the derivative of the integrand at the boundary points. Collocation schemes are derived which are equivalent to this discrete variational problem. An efficient preconditioner based on a low-order finite difference approximation to the same differential operator is presented. The corresponding multidomain problem is also considered and interface conditions are derived. Pseudospectral approximations which are C1 continuous at the interfaces are used in each subdomain to approximate the solution. The approximations are also shown to be C3 continuous at the interfaces asymptotically. A complete analysis of the collocation scheme for the multidomain problem is provided. The extension of the method to the biharmonic equation in two dimensions is discussed and results are presented for a problem defined in a nonrectangular domain.
Transmission problem for the Laplace equation and the integral equation method
Czech Academy of Sciences Publication Activity Database
Medková, Dagmar
2012-01-01
Roč. 387, č. 2 (2012), s. 837-843 ISSN 0022-247X Institutional research plan: CEZ:AV0Z10190503 Keywords : transmission problem * Laplace equation * boundary integral equation Subject RIV: BA - General Mathematics Impact factor: 1.050, year: 2012 http://www.sciencedirect.com/science/article/pii/S0022247X11008985
International Nuclear Information System (INIS)
Qin Maochang; Fan Guihong
2008-01-01
There are many interesting methods can be utilized to construct special solutions of nonlinear differential equations with constant coefficients. However, most of these methods are not applicable to nonlinear differential equations with variable coefficients. A new method is presented in this Letter, which can be used to find special solutions of nonlinear differential equations with variable coefficients. This method is based on seeking appropriate Bernoulli equation corresponding to the equation studied. Many well-known equations are chosen to illustrate the application of this method
The Mixed Finite Element Multigrid Method for Stokes Equations
Muzhinji, K.; Shateyi, S.; Motsa, S. S.
2015-01-01
The stable finite element discretization of the Stokes problem produces a symmetric indefinite system of linear algebraic equations. A variety of iterative solvers have been proposed for such systems in an attempt to construct efficient, fast, and robust solution techniques. This paper investigates one of such iterative solvers, the geometric multigrid solver, to find the approximate solution of the indefinite systems. The main ingredient of the multigrid method is the choice of an appropriate smoothing strategy. This study considers the application of different smoothers and compares their effects in the overall performance of the multigrid solver. We study the multigrid method with the following smoothers: distributed Gauss Seidel, inexact Uzawa, preconditioned MINRES, and Braess-Sarazin type smoothers. A comparative study of the smoothers shows that the Braess-Sarazin smoothers enhance good performance of the multigrid method. We study the problem in a two-dimensional domain using stable Hood-Taylor Q 2-Q 1 pair of finite rectangular elements. We also give the main theoretical convergence results. We present the numerical results to demonstrate the efficiency and robustness of the multigrid method and confirm the theoretical results. PMID:25945361
Fourier-Based Fast Multipole Method for the Helmholtz Equation
Cecka, Cris
2013-01-01
The fast multipole method (FMM) has had great success in reducing the computational complexity of solving the boundary integral form of the Helmholtz equation. We present a formulation of the Helmholtz FMM that uses Fourier basis functions rather than spherical harmonics. By modifying the transfer function in the precomputation stage of the FMM, time-critical stages of the algorithm are accelerated by causing the interpolation operators to become straightforward applications of fast Fourier transforms, retaining the diagonality of the transfer function, and providing a simplified error analysis. Using Fourier analysis, constructive algorithms are derived to a priori determine an integration quadrature for a given error tolerance. Sharp error bounds are derived and verified numerically. Various optimizations are considered to reduce the number of quadrature points and reduce the cost of computing the transfer function. © 2013 Society for Industrial and Applied Mathematics.
"Win with Katie McGee": The First Governor of Girls' State Looks Back.
Harsh, Sarah Wilmoth
2003-01-01
Rhododendron Girls' State, a 5-day West Virginia citizenship and leadership program, offers female high school students the opportunity to directly experience the campaign and electoral process. Katie McGee White discusses her experiences as the first Girls' State governor in 1941, her love of politics, and how she encouraged her students to be…
Soliton solution for nonlinear partial differential equations by cosine-function method
International Nuclear Information System (INIS)
Ali, A.H.A.; Soliman, A.A.; Raslan, K.R.
2007-01-01
In this Letter, we established a traveling wave solution by using Cosine-function algorithm for nonlinear partial differential equations. The method is used to obtain the exact solutions for five different types of nonlinear partial differential equations such as, general equal width wave equation (GEWE), general regularized long wave equation (GRLW), general Korteweg-de Vries equation (GKdV), general improved Korteweg-de Vries equation (GIKdV), and Coupled equal width wave equations (CEWE), which are the important soliton equations
Modulating functions method for parameters estimation in the fifth order KdV equation
Asiri, Sharefa M.; Liu, Da-Yan; Laleg-Kirati, Taous-Meriem
2017-01-01
In this work, the modulating functions method is proposed for estimating coefficients in higher-order nonlinear partial differential equation which is the fifth order Kortewegde Vries (KdV) equation. The proposed method transforms the problem into a
Efficient solution of parabolic equations by Krylov approximation methods
Gallopoulos, E.; Saad, Y.
1990-01-01
Numerical techniques for solving parabolic equations by the method of lines is addressed. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of a very small dimension to a known vector which is, in turn, computed accurately by exploiting well-known rational approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Some relevant approximation and stability issues are discussed. We present some numerical experiments with the method and compare its performance with a few explicit and implicit algorithms.
Method of mechanical quadratures for solving singular integral equations of various types
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.
Application of Exp-function method for (2 + 1)-dimensional nonlinear evolution equations
International Nuclear Information System (INIS)
Bekir, Ahmet; Boz, Ahmet
2009-01-01
In this paper, the Exp-function method is used to construct solitary and soliton solutions of (2 + 1)-dimensional nonlinear evolution equations. (2 + 1)-dimensional breaking soliton (Calogero) equation, modified Zakharov-Kuznetsov and Konopelchenko-Dubrovsky equations are chosen to illustrate the effectiveness of the method. The method is straightforward and concise, and its applications are promising. The Exp-function method presents a wider applicability for handling nonlinear wave 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.
High Weak Order Methods for Stochastic Differential Equations Based on Modified Equations
Abdulle, Assyr; Cohen, David; Vilmart, Gilles; Zygalakis, Konstantinos C.
2012-01-01
© 2012 Society for Industrial and Applied Mathematics. Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration
Directory of Open Access Journals (Sweden)
S. S. Motsa
2014-01-01
Full Text Available This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs. The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature.
Motsa, S S; Magagula, V M; Sibanda, P
2014-01-01
This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature.
New application of Exp-function method for improved Boussinesq equation
Energy Technology Data Exchange (ETDEWEB)
Abdou, M.A. [Theoretical Research Group, Physics Department, Faculty of Science, Mansoura University, 35516 Mansoura (Egypt); Department of Physics, Faculty of Education for Girls, Science Departments, King Khalid University, Bisha (Saudi Arabia)], E-mail: m_abdou_eg@yahoo.com; Soliman, A.A. [Department of Mathematics, Faculty of Education (AL-Arish) Suez Canal University, AL-Arish 45111 (Egypt); Department of Mathematics, Teacher' s College (Bisha), King Khalid University, Bisha, PO Box 551 (Saudi Arabia)], E-mail: asoliman_99@yahoo.com; El-Basyony, S.T. [Theoretical Research Group, Physics Department, Faculty of Science, Mansoura University, 35516 Mansoura (Egypt)
2007-10-01
The Exp-function method is used to obtain generalized solitary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics with the aid of symbolic computation method, namely, the improved Boussinesq equation. The method is straightforward and concise, and its applications is promising for other nonlinear evolution equations in mathematical physics.
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.
Grant, Mary C.; Zhang, Lilly; Damiano, Michele
2009-01-01
This study investigated kernel equating methods by comparing these methods to operational equatings for two tests in the SAT Subject Tests[TM] program. GENASYS (ETS, 2007) was used for all equating methods and scaled score kernel equating results were compared to Tucker, Levine observed score, chained linear, and chained equipercentile equating…
General method for reducing the two-body Dirac equation
International Nuclear Information System (INIS)
Galeao, A.P.; Ferreira, P.L.
1992-01-01
A semi relativistic two-body Dirac equation with an enlarged set of phenomenological potentials, including Breit-type terms, is investigated for the general case of unequal masses. Solutions corresponding to definite total angular momentum and parity are shown to fall into two classes, each one being obtained by solving a system of four coupled first-order radial differential equations. The reduction of each of these systems to a pair of coupled Schroedinger-type equations is also discussed. (author)
Integration of equations of parabolic type by the method of nets
Saul'Yev, V K; Stark, M; Ulam, S
1964-01-01
International Series of Monographs in Pure and Applied Mathematics, Volume 54: Integration of Equations of Parabolic Type by the Method of Nets deals with solving parabolic partial differential equations using the method of nets. The first part of this volume focuses on the construction of net equations, with emphasis on the stability and accuracy of the approximating net equations. The method of nets or method of finite differences (used to define the corresponding numerical method in ordinary differential equations) is one of many different approximate methods of integration of partial diff
A new RBF-Trefftz meshless method for partial differential equations
International Nuclear Information System (INIS)
Cao Leilei; Zhao Ning; Qin Qinghua
2010-01-01
Based on the radial basis functions (RBF) and T-Trefftz solution, this paper presents a new meshless method for numerically solving various partial differential equation systems. First, the analog equation method (AEM) is used to convert the original patial differential equation to an equivalent Poisson's equation. Then, the radial basis functions (RBF) are employed to approxiamate the inhomogeneous term, while the homogeneous solution is obtained by linear combination of a set of T-Trefftz solutions. The present scheme, named RBF-Trefftz has the advantage over the fundamental solution (MFS) method due to the use of nonsingular T-Trefftz solution rather than singular fundamental solutions, so it does not require the artificial boundary. The application and efficiency of the proposed method are validated through several examples which include different type of differential equations, such as Laplace equation, Hellmholtz equation, convectin-diffusion equation and time-dependent equation.
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.
On an improved method for solving evolution equations of higher ...
African Journals Online (AJOL)
In this paper we introduce a new algebraic procedure to compute new classes of solutions of (1+1)-nonlinear partial differential equations (nPDEs) both of physical and technical relevance. The basic assumption is that the unknown solution(s) of the nPDE under consideration satisfy an ordinary differential equation (ODE) of ...
Stochastic weighted particle methods for population balance equations
International Nuclear Information System (INIS)
Patterson, Robert I.A.; Wagner, Wolfgang; Kraft, Markus
2011-01-01
Highlights: → Weight transfer functions for Monte Carlo simulation of coagulation. → Efficient support for single-particle growth processes. → Comparisons to analytic solutions and soot formation problems. → Better numerical accuracy for less common particles. - Abstract: A class of coagulation weight transfer functions is constructed, each member of which leads to a stochastic particle algorithm for the numerical treatment of population balance equations. These algorithms are based on systems of weighted computational particles and the weight transfer functions are constructed such that the number of computational particles does not change during coagulation events. The algorithms also facilitate the simulation of physical processes that change single particles, such as growth, or other surface reactions. Four members of the algorithm family have been numerically validated by comparison to analytic solutions to simple problems. Numerical experiments have been performed for complex laminar premixed flame systems in which members of the class of stochastic weighted particle methods were compared to each other and to a direct simulation algorithm. Two of the weighted algorithms have been shown to offer performance advantages over the direct simulation algorithm in situations where interest is focused on the larger particles in a system. The extent of this advantage depends on the particular system and on the quantities of interest.
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....
Extended sine-Gordon Equation Method and Its Application to Maccari's System
International Nuclear Information System (INIS)
Song Lina; Zhang Hongqing
2005-01-01
An extended sine-Gordon equation method is proposed to construct exact travelling wave solutions to Maccari's equation based upon a generalized sine-Gordon equation. It is shown that more new travelling wave solutions can be found by this new method, which include bell-shaped soliton solutions, kink-shaped soliton solutions, periodic wave solution, and new travelling waves.
Nonlinear Fredholm Integral Equation of the Second Kind with Quadrature Methods
Directory of Open Access Journals (Sweden)
M. Jafari Emamzadeh
2010-06-01
Full Text Available In this paper, a numerical method for solving the nonlinear Fredholm integral equation is presented. We intend to approximate the solution of this equation by quadrature methods and by doing so, we solve the nonlinear Fredholm integral equation more accurately. Several examples are given at the end of this paper
The Closest Point Method and Multigrid Solvers for Elliptic Equations on Surfaces
Chen, Yujia; Macdonald, Colin B.
2015-01-01
© 2015 Society for Industrial and Applied Mathematics. Elliptic partial differential equations are important from both application and analysis points of view. In this paper we apply the closest point method to solve elliptic equations on general
The generalized tanh method to obtain exact solutions of nonlinear partial differential equation
Gómez, César
2007-01-01
In this paper, we present the generalized tanh method to obtain exact solutions of nonlinear partial differential equations, and we obtain solitons and exact solutions of some important equations of the mathematical physics.
Discontinuous Galerkin Subgrid Finite Element Method for Heterogeneous Brinkman’s Equations
Iliev, Oleg P.; Lazarov, Raytcho D.; Willems, Joerg
2010-01-01
We present a two-scale finite element method for solving Brinkman's equations with piece-wise constant coefficients. This system of equations model fluid flows in highly porous, heterogeneous media with complex topology of the heterogeneities. We
Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (3).
Murase, Kenya
2016-01-01
In this issue, simultaneous differential equations were introduced. These differential equations are often used in the field of medical physics. The methods for solving them were also introduced, which include Laplace transform and matrix methods. Some examples were also introduced, in which Laplace transform and matrix methods were applied to solving simultaneous differential equations derived from a three-compartment kinetic model for analyzing the glucose metabolism in tissues and Bloch equations for describing the behavior of the macroscopic magnetization in magnetic resonance imaging.In the next (final) issue, partial differential equations and various methods for solving them will be introduced together with some examples in medical physics.
Iterative method of the parameter variation for solution of nonlinear functional equations
International Nuclear Information System (INIS)
Davidenko, D.F.
1975-01-01
The iteration method of parameter variation is used for solving nonlinear functional equations in Banach spaces. The authors consider some methods for numerical integration of ordinary first-order differential equations and construct the relevant iteration methods of parameter variation, both one- and multifactor. They also discuss problems of mathematical substantiation of the method, study the conditions and rate of convergence, estimate the error. The paper considers the application of the method to specific functional equations
Construct solitary solutions of discrete hybrid equation by Adomian Decomposition Method
International Nuclear Information System (INIS)
Wang Zhen; Zhang Hongqing
2009-01-01
In this paper, we apply the Adomian Decomposition Method to solving the differential-difference equations. A typical example is applied to illustrate the validity and the great potential of the Adomian Decomposition Method in solving differential-difference equation. Kink shaped solitary solution and Bell shaped solitary solution are presented. Comparisons are made between the results of the proposed method and exact solutions. The results show that the Adomian Decomposition Method is an attractive method in solving the differential-difference equations.
Modified harmonic balance method for the solution of nonlinear jerk equations
Rahman, M. Saifur; Hasan, A. S. M. Z.
2018-03-01
In this paper, a second approximate solution of nonlinear jerk equations (third order differential equation) can be obtained by using modified harmonic balance method. The method is simpler and easier to carry out the solution of nonlinear differential equations due to less number of nonlinear equations are required to solve than the classical harmonic balance method. The results obtained from this method are compared with those obtained from the other existing analytical methods that are available in the literature and the numerical method. The solution shows a good agreement with the numerical solution as well as the analytical methods of the available literature.
Advanced methods for the solution of differential equations
Goldstein, M. E.; Braun, W. H.
1973-01-01
This book is based on a course presented at the Lewis Research Center for engineers and scientists who were interested in increasing their knowledge of differential equations. Those results which can actually be used to solve equations are therefore emphasized; and detailed proofs of theorems are, for the most part, omitted. However, the conclusions of the theorems are stated in a precise manner, and enough references are given so that the interested reader can find the steps of the proofs.
Fractional Complex Transform and exp-Function Methods for Fractional Differential Equations
Directory of Open Access Journals (Sweden)
Ahmet Bekir
2013-01-01
Full Text Available The exp-function method is presented for finding the exact solutions of nonlinear fractional equations. New solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie's modified Riemann-Liouville sense. We apply the exp-function method to both the nonlinear time and space fractional differential equations. As a result, some new exact solutions for them are successfully established.
A functional-analytic method for the study of difference equations
Directory of Open Access Journals (Sweden)
Siafarikas Panayiotis D
2004-01-01
Full Text Available We will give the generalization of a recently developed functional-analytic method for studying linear and nonlinear, ordinary and partial, difference equations in the and spaces, p∈ℕ, . The method will be illustrated by use of two examples concerning a nonlinear ordinary difference equation known as the Putnam equation, and a linear partial difference equation of three variables describing the discrete Newton law of cooling in three dimensions.
Exact solutions of nonlinear fractional differential equations by (G′/G)-expansion method
International Nuclear Information System (INIS)
Bekir Ahmet; Güner Özkan
2013-01-01
In this paper, we use the fractional complex transform and the (G′/G)-expansion method to study the nonlinear fractional differential equations and find the exact solutions. The fractional complex transform is proposed to convert a partial fractional differential equation with Jumarie's modified Riemann—Liouville derivative into its ordinary differential equation. It is shown that the considered transform and method are very efficient and powerful in solving wide classes of nonlinear fractional order equations
Directory of Open Access Journals (Sweden)
Wei Li
2014-01-01
Full Text Available Based on a general fractional Riccati equation and with Jumarie’s modified Riemann-Liouville derivative to an extended fractional Riccati expansion method for solving the time fractional Burgers equation and the space-time fractional Cahn-Hilliard equation, the exact solutions expressed by the hyperbolic functions and trigonometric functions are obtained. The obtained results show that the presented method is effective and appropriate for solving nonlinear fractional differential equations.
On Solution of a Fractional Diffusion Equation by Homotopy Transform Method
International Nuclear Information System (INIS)
Salah, A.; Hassan, S.S.A.
2012-01-01
The homotopy analysis transform method (HATM) is applied in this work in order to find the analytical solution of fractional diffusion equations (FDE). These equations are obtained from standard diffusion equations by replacing a second-order space derivative by a fractional derivative of order α and a first order time derivative by a fractional derivative. Furthermore, some examples are given. Numerical results show that the homotopy analysis transform method is easy to implement and accurate when applied to a fractional diffusion equations.
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)
Solving the interval type-2 fuzzy polynomial equation using the ranking method
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.
Directory of Open Access Journals (Sweden)
Shaheed N. Huseen
2013-01-01
Full Text Available A modified q-homotopy analysis method (mq-HAM was proposed for solving nth-order nonlinear differential equations. This method improves the convergence of the series solution in the nHAM which was proposed in (see Hassan and El-Tawil 2011, 2012. The proposed method provides an approximate solution by rewriting the nth-order nonlinear differential equation in the form of n first-order differential equations. The solution of these n differential equations is obtained as a power series solution. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed.
Wielandt method applied to the diffusion equations discretized by finite element nodal methods
International Nuclear Information System (INIS)
Mugica R, A.; Valle G, E. del
2003-01-01
Nowadays the numerical methods of solution to the diffusion equation by means of algorithms and computer programs result so extensive due to the great number of routines and calculations that should carry out, this rebounds directly in the execution times of this programs, being obtained results in relatively long times. This work shows the application of an acceleration method of the convergence of the classic method of those powers that it reduces notably the number of necessary iterations for to obtain reliable results, what means that the compute times they see reduced in great measure. This method is known in the literature like Wielandt method and it has incorporated to a computer program that is based on the discretization of the neutron diffusion equations in plate geometry and stationary state by polynomial nodal methods. In this work the neutron diffusion equations are described for several energy groups and their discretization by means of those called physical nodal methods, being illustrated in particular the quadratic case. It is described a model problem widely described in the literature which is solved for the physical nodal grade schemes 1, 2, 3 and 4 in three different ways: to) with the classic method of the powers, b) method of the powers with the Wielandt acceleration and c) method of the powers with the Wielandt modified acceleration. The results for the model problem as well as for two additional problems known as benchmark problems are reported. Such acceleration method can also be implemented to problems of different geometry to the proposal in this work, besides being possible to extend their application to problems in 2 or 3 dimensions. (Author)
Directory of Open Access Journals (Sweden)
Razieh Khajeh-Kazemi
2011-01-01
Full Text Available Background: The celebrated generalized estimating equations (GEE approach is often used in longitudinal data analysis While this method behaves robustly against misspecification of the working correlation structure, it has some limitations on efficacy of estimators, goodness-of-fit tests and model selection criteria The quadratic inference functions (QIF is a new statistical methodology that overcomes these limitations Methods : We administered the use of QIF and GEE in comparing the superior and inferior Ahmed glaucoma valve (AGV implantation, while our focus was on the efficiency of estimation and using model selection criteria, we compared the effect of implant location on intraocular pressure (IOP in refractory glaucoma patients We modeled the relationship between IOP and implant location, patient′s sex and age, best corrected visual acuity, history of cataract surgery, preoperative IOP and months after surgery with assuming unstructured working correlation Results : 63 eyes of 63 patients were included in this study, 28 eyes in inferior group and 35 eyes in superior group The GEE analysis revealed that preoperative IOP has a significant effect on IOP (p = 0 011 However, QIF showed that preoperative IOP, months after surgery and squared months are significantly associated with IOP after surgery (p < 0 05 Overall, estimates from QIF are more efficient than GEE (RE = 1 272 Conclusions : In the case of unstructured working correlation, the QIF is more efficient than GEE There were no considerable difference between these locations, our results confirmed previously published works which mentioned it is better that glaucoma patients undergo superior AGV implantation
Petersson, K J F; Friberg, L E; Karlsson, M O
2010-10-01
Computer models of biological systems grow more complex as computing power increase. Often these models are defined as differential equations and no analytical solutions exist. Numerical integration is used to approximate the solution; this can be computationally intensive, time consuming and be a large proportion of the total computer runtime. The performance of different integration methods depend on the mathematical properties of the differential equations system at hand. In this paper we investigate the possibility of runtime gains by calculating parts of or the whole differential equations system at given time intervals, outside of the differential equations solver. This approach was tested on nine models defined as differential equations with the goal to reduce runtime while maintaining model fit, based on the objective function value. The software used was NONMEM. In four models the computational runtime was successfully reduced (by 59-96%). The differences in parameter estimates, compared to using only the differential equations solver were less than 12% for all fixed effects parameters. For the variance parameters, estimates were within 10% for the majority of the parameters. Population and individual predictions were similar and the differences in OFV were between 1 and -14 units. When computational runtime seriously affects the usefulness of a model we suggest evaluating this approach for repetitive elements of model building and evaluation such as covariate inclusions or bootstraps.
Exact solutions for nonlinear evolution equations using Exp-function method
International Nuclear Information System (INIS)
Bekir, Ahmet; Boz, Ahmet
2008-01-01
In this Letter, the Exp-function method is used to construct solitary and soliton solutions of nonlinear evolution equations. The Klein-Gordon, Burger-Fisher and Sharma-Tasso-Olver equations are chosen to illustrate the effectiveness of the method. The method is straightforward and concise, and its applications are promising. The Exp-function method presents a wider applicability for handling nonlinear wave 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 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
International Nuclear Information System (INIS)
Yomba, Emmanuel
2005-01-01
An extended Fan's sub-equation method is used for constructing exact travelling wave solutions of nonlinear partial differential equations (NLPDEs). The key idea of this method is to take full advantage of the general elliptic equation involving five parameters which has more new solutions and whose degeneracies can lead to special sub-equations involving three parameters. More new solutions are obtained for KdV-MKdV, Broer-Kaup-Kupershmidt (BKK) and variant Boussinesq equations. Then we present a technique which not only gives us a clear relation among this general elliptic equation and other sub-equations involving three parameters (Riccati equation, first kind elliptic equation, auxiliary ordinary equation, generalized Riccati equation and so on), but also provides an approach to construct new exact solutions to NLPDEs
Calibration methods for the Hargreaves-Samani equation
Directory of Open Access Journals (Sweden)
Lucas Borges Ferreira
Full Text Available ABSTRACT The estimation of the reference evapotranspiration is an important factor for hydrological studies, design and management of irrigation systems, among others. The Penman Monteith equation presents high precision and accuracy in the estimation of this variable. However, its use becomes limited due to the large number of required meteorological data. In this context, the Hargreaves-Samani equation could be used as alternative, although, for a better performance a local calibration is required. Thus, the aim was to compare the calibration process of the Hargreaves-Samani equation by linear regression, by adjustment of the coefficients (A and B and exponent (C of the equation and by combinations of the two previous alternatives. Daily data from 6 weather stations, located in the state of Minas Gerais, from the period 1997 to 2016 were used. The calibration of the Hargreaves-Samani equation was performed in five ways: calibration by linear regression, adjustment of parameter “A”, adjustment of parameters “A” and “C”, adjustment of parameters “A”, “B” and “C” and adjustment of parameters “A”, “B” and “C” followed by calibration by linear regression. The performances of the models were evaluated based on the statistical indicators mean absolute error, mean bias error, Willmott’s index of agreement, correlation coefficient and performance index. All the studied methodologies promoted better estimations of reference evapotranspiration. The simultaneous adjustment of the empirical parameters “A”, “B” and “C” was the best alternative for calibration of the Hargreaves-Samani equation.
A New Pseudoinverse Matrix Method For Balancing Chemical Equations And Their Stability
International Nuclear Information System (INIS)
Risteski, Ice B.
2008-01-01
In this work is given a new pseudoniverse matrix method for balancing chemical equations. Here offered method is founded on virtue of the solution of a Diophantine matrix equation by using of a Moore-Penrose pseudoinverse matrix. The method has been tested on several typical chemical equations and found to be very successful for the all equations in our extensive balancing research. This method, which works successfully without any limitations, also has the capability to determine the feasibility of a new chemical reaction, and if it is feasible, then it will balance the equation. Chemical equations treated here possess atoms with fractional oxidation numbers. Also, in the present work are introduced necessary and sufficient criteria for stability of chemical equations over stability of their extended matrices
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)
Implicit Boundary Integral Methods for the Helmholtz Equation in Exterior Domains
2016-06-01
solve the Helmholtz equation as ∂Ω goes through significant change in its shape and topology — applications for which implicit representation of the...boundary-value problems for the wave equation and maxwell’s equations. Russian Math . Surv., 1965. [16] S. Reutskiy. The method of fundamental
Derivation of equation of quasipotential type using the method of Fock-- Podolsky
Energy Technology Data Exchange (ETDEWEB)
Blokhintsev, D I; Rizov, V A; Todorov, I T
1975-12-31
A quasipotential equation is derived for the relativistic Coulomb problem from the equations of motion of quantum electrodynamics using the method of Fock-- Podolsky (Tamm-Dancoff). Relation with an inhomogeneous equation for the 4-point retarded function is exhibited. (auth)
He, Yong
2013-01-01
Common test items play an important role in equating multiple test forms under the common-item nonequivalent groups design. Inconsistent item parameter estimates among common items can lead to large bias in equated scores for IRT true score equating. Current methods extensively focus on detection and elimination of outlying common items, which…
Using a Linear Regression Method to Detect Outliers in IRT Common Item Equating
He, Yong; Cui, Zhongmin; Fang, Yu; Chen, Hanwei
2013-01-01
Common test items play an important role in equating alternate test forms under the common item nonequivalent groups design. When the item response theory (IRT) method is applied in equating, inconsistent item parameter estimates among common items can lead to large bias in equated scores. It is prudent to evaluate inconsistency in parameter…
The application of He's exp-function method to a nonlinear differential-difference equation
International Nuclear Information System (INIS)
Dai Chaoqing; Cen Xu; Wu Shengsheng
2009-01-01
This paper applies He's exp-function method, which was originally proposed to find new exact travelling wave solutions of nonlinear partial differential equations (NPDEs) or coupled nonlinear partial differential equations (CNPDEs), to a nonlinear differential-difference equation, and some new travelling wave solutions are obtained.
The Navier-Stokes Equations Theory and Numerical Methods
Masuda, Kyûya; Rautmann, Reimund; Solonnikov, Vsevolod
1990-01-01
These proceedings contain original (refereed) research articles by specialists from many countries, on a wide variety of aspects of Navier-Stokes equations. Additionally, 2 survey articles intended for a general readership are included: one surveys the present state of the subject via open problems, and the other deals with the interplay between theory and numerical analysis.
High-order finite-difference methods for Poisson's equation
van Linde, Hendrik Jan
1971-01-01
In this thesis finite-difference approximations to the three boundary value problems for Poisson’s equation are given, with discretization errors of O(H^3) for the mixed boundary value problem, O(H^3 |ln(h)| for the Neumann problem and O(H^4)for the Dirichlet problem respectively . First an operator
Axisymmetric multiphase lattice Boltzmann method for generic equations of state
Reijers, S.A.; Gelderblom, H.; Toschi, F.
2016-01-01
We present an axisymmetric lattice Boltzmann model based on the Kupershtokh et al. multiphase model that is capable of solving liquid–gas density ratios up to 103. Appropriate source terms are added to the lattice Boltzmann evolution equation to fully recover the axisymmetric multiphase conservation
Fringe integral equation method for a truncated grounded dielectric slab
DEFF Research Database (Denmark)
Jørgensen, Erik; Maci, S.; Toccafondi, A.
2001-01-01
The problem of scattering by a semi-infinite grounded dielectric slab illuminated by an arbitrary incident TMz polarized electric field is studied by solving a new set of “fringe” integral equations (F-IEs), whose functional unknowns are physically associated to the wave diffraction processes...
International Nuclear Information System (INIS)
Edgar, S.B.
1990-01-01
The structures of the N.P. and G.H.P formalisms are reviewed in order to understand and demonstrate the important role played by the commutator equations in the associated integration procedures. Particular attention is focused on how the commutator equations are to be satisfied, or checked for consistency. It is shown that Held's integration method will only guarantee genuine solutions of Einstein's equations when all the commutator equations are correctly and completely satisfied. (authors)
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...
On a computer implementation of the block Gauss–Seidel method for normal systems of equations
Alexander I. Zhdanov; Ekaterina Yu. Bogdanova
2016-01-01
This article focuses on the modification of the block option Gauss-Seidel method for normal systems of equations, which is a sufficiently effective method of solving generally overdetermined, systems of linear algebraic equations of high dimensionality. The main disadvantage of methods based on normal equations systems is the fact that the condition number of the normal system is equal to the square of the condition number of the original problem. This fact has a negative impact on the rate o...
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.
International Nuclear Information System (INIS)
Jun-Mao, Wang; Miao, Zhang; Wen-Liang, Zhang; Rui, Zhang; Jia-Hua, Han
2008-01-01
We present a new method to find the exact travelling wave solutions of nonlinear evolution equations, with the aid of the symbolic computation. Based on this method, we successfully solve the modified Benjamin–Bona–Mahoney equation, and obtain some new solutions which can be expressed by trigonometric functions and hyperbolic functions. It is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics. (general)
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
Wills, John M.; Mattsson, Ann E.
2012-02-01
Density functional theory (DFT) provides a formally predictive base for equation of state properties. Available approximations to the exchange/correlation functional provide accurate predictions for many materials in the periodic table. For heavy materials however, DFT calculations, using available functionals, fail to provide quantitative predictions, and often fail to be even qualitative. This deficiency is due both to the lack of the appropriate confinement physics in the exchange/correlation functional and to approximations used to evaluate the underlying equations. In order to assess and develop accurate functionals, it is essential to eliminate all other sources of error. In this talk we describe an efficient first-principles electronic structure method based on the Dirac equation and compare the results obtained with this method with other methods generally used. Implications for high-pressure equation of state of relativistic materials are demonstrated in application to Ce and the light actinides. Sandia National Laboratories is a multi-program laboratory managed andoperated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.
Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (4).
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.
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
A second order discontinuous Galerkin fast sweeping method for Eikonal equations
Li, Fengyan; Shu, Chi-Wang; Zhang, Yong-Tao; Zhao, Hongkai
2008-09-01
In this paper, we construct a second order fast sweeping method with a discontinuous Galerkin (DG) local solver for computing viscosity solutions of a class of static Hamilton-Jacobi equations, namely the Eikonal equations. Our piecewise linear DG local solver is built on a DG method developed recently [Y. Cheng, C.-W. Shu, A discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations, Journal of Computational Physics 223 (2007) 398-415] for the time-dependent Hamilton-Jacobi equations. The causality property of Eikonal equations is incorporated into the design of this solver. The resulting local nonlinear system in the Gauss-Seidel iterations is a simple quadratic system and can be solved explicitly. The compactness of the DG method and the fast sweeping strategy lead to fast convergence of the new scheme for Eikonal equations. Extensive numerical examples verify efficiency, convergence and second order accuracy of the proposed method.
Application of the Exp-function method to the equal-width wave equation
International Nuclear Information System (INIS)
Biazar, J; Ayati, Z
2008-01-01
In this paper, the Exp-function method is used to find an exact solution of the equal-width wave (EW) equation. The method is straightforward and concise, and its applications are promising. It is shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful mathematical tool for solving the EW equation.
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.
Stochastic fractional differential equations: Modeling, method and analysis
International Nuclear Information System (INIS)
Pedjeu, Jean-C.; Ladde, Gangaram S.
2012-01-01
By introducing a concept of dynamic process operating under multi-time scales in sciences and engineering, a mathematical model described by a system of multi-time scale stochastic differential equations is formulated. The classical Picard–Lindelöf successive approximations scheme is applied to the model validation problem, namely, existence and uniqueness of solution process. Naturally, this leads to the problem of finding closed form solutions of both linear and nonlinear multi-time scale stochastic differential equations of Itô–Doob type. Finally, to illustrate the scope of ideas and presented results, multi-time scale stochastic models for ecological and epidemiological processes in population dynamic are outlined.
A general method for enclosing solutions of interval linear equations
Czech Academy of Sciences Publication Activity Database
Rohn, Jiří
2012-01-01
Roč. 6, č. 4 (2012), s. 709-717 ISSN 1862-4472 R&D Projects: GA ČR GA201/09/1957; GA ČR GC201/08/J020 Institutional research plan: CEZ:AV0Z10300504 Keywords : interval linear equations * solution set * enclosure * absolute value inequality Subject RIV: BA - General Mathematics Impact factor: 1.654, year: 2012
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
Directory of Open Access Journals (Sweden)
Murat Osmanoglu
2013-01-01
Full Text Available We have considered linear partial differential algebraic equations (LPDAEs of the form , which has at least one singular matrix of . We have first introduced a uniform differential time index and a differential space index. The initial conditions and boundary conditions of the given system cannot be prescribed for all components of the solution vector here. To overcome this, we introduced these indexes. Furthermore, differential transform method has been given to solve LPDAEs. We have applied this method to a test problem, and numerical solution of the problem has been compared with analytical solution.
Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations
International Nuclear Information System (INIS)
Hong Jialin; Li Chun
2006-01-01
In this paper, we consider the multi-symplectic Runge-Kutta (MSRK) methods applied to the nonlinear Dirac equation in relativistic quantum physics, based on a discovery of the multi-symplecticity of the equation. In particular, the conservation of energy, momentum and charge under MSRK discretizations is investigated by means of numerical experiments and numerical comparisons with non-MSRK methods. Numerical experiments presented reveal that MSRK methods applied to the nonlinear Dirac equation preserve exactly conservation laws of charge and momentum, and conserve the energy conservation in the corresponding numerical accuracy to the method utilized. It is verified numerically that MSRK methods are stable and convergent with respect to the conservation laws of energy, momentum and charge, and MSRK methods preserve not only the inner geometric structure of the equation, but also some crucial conservative properties in quantum physics. A remarkable advantage of MSRK methods applied to the nonlinear Dirac equation is the precise preservation of charge conservation law
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
Directory of Open Access Journals (Sweden)
Hasibun Naher
2014-10-01
Full Text Available In this article, new extension of the generalized and improved (G′/G-expansion method is proposed for constructing more general and a rich class of new exact traveling wave solutions of nonlinear evolution equations. To demonstrate the novelty and motivation of the proposed method, we implement it to the Korteweg-de Vries (KdV equation. The new method is oriented toward the ease of utilize and capability of computer algebraic system and provides a more systematic, convenient handling of the solution process of nonlinear equations. Further, obtained solutions disclose a wider range of applicability for handling a large variety of nonlinear partial differential equations.
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.
Spline Collocation Method for Nonlinear Multi-Term Fractional Differential Equation
Choe, Hui-Chol; Kang, Yong-Suk
2013-01-01
We study an approximation method to solve nonlinear multi-term fractional differential equations with initial conditions or boundary conditions. First, we transform the nonlinear multi-term fractional differential equations with initial conditions and boundary conditions to nonlinear fractional integral equations and consider the relations between them. We present a Spline Collocation Method and prove the existence, uniqueness and convergence of approximate solution as well as error estimatio...
Nuclear methods and the nuclear equation of state
1999-01-01
The theoretical study of the nuclear equation of state (EOS) is a field of research which deals with most of the fundamental problems of nuclear physics. This book gives an overview of the present status of the microscopic theory of the nuclear EOS. Its aim is essentially twofold: first, to serve as a textbook for students entering the field, by covering the different subjects as exhaustively and didactically as possible; second, to be a reference book for all researchers active in the theory of nuclear matter, by providing a report on the latest developments. Special emphasis is given to the
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
Khan, Sami Ullah; Ali, Ishtiaq
2018-03-01
Explicit solutions to delay differential equation (DDE) and stochastic delay differential equation (SDDE) can rarely be obtained, therefore numerical methods are adopted to solve these DDE and SDDE. While on the other hand due to unstable nature of both DDE and SDDE numerical solutions are also not straight forward and required more attention. In this study, we derive an efficient numerical scheme for DDE and SDDE based on Legendre spectral-collocation method, which proved to be numerical methods that can significantly speed up the computation. The method transforms the given differential equation into a matrix equation by means of Legendre collocation points which correspond to a system of algebraic equations with unknown Legendre coefficients. The efficiency of the proposed method is confirmed by some numerical examples. We found that our numerical technique has a very good agreement with other methods with less computational effort.
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.
Numerical methods for the solution of ordinary differential equations
International Nuclear Information System (INIS)
Azeem, M.
1999-01-01
The ode 113 code solves non-stiff differential equations and is a fully variable step, variable order, PECE implementation in terms of modified divided differences of Adams-Bashforth-Moulton family of formulas of order 1-12. The main objectives of this project were to modify PECE mode of ode 113 into PEC mode, study the variable step size and variable order strategy of both the modes and finally, develop the switching strategy between both PECE and PEC modes to minimize the cost of solving the ordinary differential equations. Using some test problems (including stiff, mild stiff and non-stiff), it was found that the PEC mode was more efficient for non-stiff problems at crude and intermediate tolerances and the PECE mode for all problems at the stringent tolerance. An automatic switching strategy was developed using the results observed from the step size and order plots of all the test problems for both the modes and gave the optimum results. (author)
Leapfrog variants of iterative methods for linear algebra equations
Saylor, Paul E.
1988-01-01
Two iterative methods are considered, Richardson's method and a general second order method. For both methods, a variant of the method is derived for which only even numbered iterates are computed. The variant is called a leapfrog method. Comparisons between the conventional form of the methods and the leapfrog form are made under the assumption that the number of unknowns is large. In the case of Richardson's method, it is possible to express the final iterate in terms of only the initial approximation, a variant of the iteration called the grand-leap method. In the case of the grand-leap variant, a set of parameters is required. An algorithm is presented to compute these parameters that is related to algorithms to compute the weights and abscissas for Gaussian quadrature. General algorithms to implement the leapfrog and grand-leap methods are presented. Algorithms for the important special case of the Chebyshev method are also given.
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)
Solution and study of nodal neutron transport equation applying the LTSN-DiagExp method
International Nuclear Information System (INIS)
Hauser, Eliete Biasotto; Pazos, Ruben Panta; Vilhena, Marco Tullio de; Barros, Ricardo Carvalho de
2003-01-01
In this paper we report advances about the three-dimensional nodal discrete-ordinates approximations of neutron transport equation for Cartesian geometry. We use the combined collocation method of the angular variables and nodal approach for the spatial variables. By nodal approach we mean the iterated transverse integration of the S N equations. This procedure leads to the set of one-dimensional averages angular fluxes in each spatial variable. The resulting system of equations is solved with the LTS N method, first applying the Laplace transform to the set of the nodal S N equations and then obtained the solution by symbolic computation. We include the LTS N method by diagonalization to solve the nodal neutron transport equation and then we outline the convergence of these nodal-LTS N approximations with the help of a norm associated to the quadrature formula used to approximate the integral term of the neutron transport equation. (author)
International Nuclear Information System (INIS)
Shestakov, A.I.; Mirin, A.A.
1984-01-01
A numerical method based on Fourier expansions and finite differences is presented. The method is demonstrated by solving a scalar, three-dimensional elliptic equation arising in MFE research, but has applicability to a wider class of problems. The scheme solves equations whose solutions are expected to be periodic in one or more of the independent variables
Directory of Open Access Journals (Sweden)
Sheng-Ping Yan
2014-01-01
Full Text Available We perform a comparison between the local fractional Adomian decomposition and local fractional function decomposition methods applied to the Laplace equation. The operators are taken in the local sense. The results illustrate the significant features of the two methods which are both very effective and straightforward for solving the differential equations with local fractional derivative.
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.
Solving Fokker-Planck Equations on Cantor Sets Using Local Fractional Decomposition Method
Directory of Open Access Journals (Sweden)
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.
Q-Step methods for Newton-Jacobi operator equation | Uwasmusi ...
African Journals Online (AJOL)
The paper considers the Newton-Jacobi operator equation for the solution of nonlinear systems of equations. Special attention is paid to the computational part of this method with particular reference to the q-step methods. Journal of the Nigerian Association of Mathematical Physics Vol. 8 2004: pp. 237-241 ...
International Nuclear Information System (INIS)
Zhong, Z.
1985-01-01
A new approach to the solution of certain differential equations, the double complex function method, is developed, combining ordinary complex numbers and hyperbolic complex numbers. This method is applied to the theory of stationary axisymmetric Einstein equations in general relativity. A family of exact double solutions, double transformation groups, and n-soliton double solutions are obtained
Numerical Solution of Fuzzy Differential Equations by Runge-Kutta Verner Method
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T. Jayakumar
2015-01-01
Full Text Available In this paper we study the numerical methods for Fuzzy Differential equations by an application of the Runge-Kutta Verner method for fuzzy differential equations. We prove a convergence result and give numerical examples to illustrate the theory.
Approximate Analytic and Numerical Solutions to Lane-Emden Equation via Fuzzy Modeling Method
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De-Gang Wang
2012-01-01
Full Text Available A novel algorithm, called variable weight fuzzy marginal linearization (VWFML method, is proposed. This method can supply approximate analytic and numerical solutions to Lane-Emden equations. And it is easy to be implemented and extended for solving other nonlinear differential equations. Numerical examples are included to demonstrate the validity and applicability of the developed technique.
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)
A study on linear and nonlinear Schrodinger equations by the variational iteration method
International Nuclear Information System (INIS)
Wazwaz, Abdul-Majid
2008-01-01
In this work, we introduce a framework to obtain exact solutions to linear and nonlinear Schrodinger equations. The He's variational iteration method (VIM) is used for analytic treatment of these equations. Numerical examples are tested to show the pertinent features of this method
A polynomial expansion method and its application in the coupled Zakharov-Kuznetsov equations
International Nuclear Information System (INIS)
Huang Wenhua
2006-01-01
A polynomial expansion method is presented to solve nonlinear evolution equations. Applying this method, the coupled Zakharov-Kuznetsov equations in fluid system are studied and many exact travelling wave solutions are obtained. These solutions include solitary wave solutions, periodic solutions and rational type solutions
Directory of Open Access Journals (Sweden)
Wu Guo-Cheng
2012-01-01
Full Text Available This note presents a Laplace transform approach in the determination of the Lagrange multiplier when the variational iteration method is applied to time fractional heat diffusion equation. The presented approach is more straightforward and allows some simplification in application of the variational iteration method to fractional differential equations, thus improving the convergence of the successive iterations.
Stability of generalized Runge-Kutta methods for stiff kinetics coupled differential equations
International Nuclear Information System (INIS)
Aboanber, A E
2006-01-01
A stability and efficiency improved class of generalized Runge-Kutta methods of order 4 are developed for the numerical solution of stiff system kinetics equations for linear and/or nonlinear coupled differential equations. The determination of the coefficients required by the method is precisely obtained from the so-called equations of condition which in turn are derived by an approach based on Butcher series. Since the equations of condition are fewer in number, free parameters can be chosen for optimizing any desired feature of the process. A further related coefficient set with different values of these parameters and the region of absolute stability of the method have been introduced. In addition, the A(α) stability properties of the method are investigated. Implementing the method in a personal computer estimated the accuracy and speed of calculations and verified the good performances of the proposed new schemes for several sample problems of the stiff system point kinetics equations with reactivity feedback
Directory of Open Access Journals (Sweden)
Özkan Güner
2014-01-01
Full Text Available We apply the functional variable method, exp-function method, and (G′/G-expansion method to establish the exact solutions of the nonlinear fractional partial differential equation (NLFPDE in the sense of the modified Riemann-Liouville derivative. As a result, some new exact solutions for them are obtained. The results show that these methods are very effective and powerful mathematical tools for solving nonlinear fractional equations arising in mathematical physics. As a result, these methods can also be applied to other nonlinear fractional differential equations.
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.
Integrable discretizations and self-adaptive moving mesh method for a coupled short pulse equation
International Nuclear Information System (INIS)
Feng, Bao-Feng; Chen, Junchao; Chen, Yong; Maruno, Ken-ichi; Ohta, Yasuhiro
2015-01-01
In the present paper, integrable semi-discrete and fully discrete analogues of a coupled short pulse (CSP) equation are constructed. The key to the construction are the bilinear forms and determinant structure of the solutions of the CSP equation. We also construct N-soliton solutions for the semi-discrete and fully discrete analogues of the CSP equations in the form of Casorati determinants. In the continuous limit, we show that the fully discrete CSP equation converges to the semi-discrete CSP equation, then further to the continuous CSP equation. Moreover, the integrable semi-discretization of the CSP equation is used as a self-adaptive moving mesh method for numerical simulations. The numerical results agree with the analytical results very well. (paper)
Green`s function of Maxwell`s equations and corresponding implications for iterative methods
Energy Technology Data Exchange (ETDEWEB)
Singer, B.S. [Macquarie Univ., Sydney (Australia); Fainberg, E.B. [Inst. of Physics of the Earth, Moscow (Russian Federation)
1996-12-31
Energy conservation law imposes constraints on the norm and direction of the Hilbert space vector representing a solution of Maxwell`s equations. In this paper, we derive these constrains and discuss the corresponding implications for the Green`s function of Maxwell`s equations in a dissipative medium. It is shown that Maxwell`s equations can be reduced to an integral equation with a contracting kernel. The equation can be solved using simple iterations. Software based on this algorithm have successfully been applied to a wide range of problems dealing with high contrast models. The matrix corresponding to the integral equation has a well defined spectrum. The equation can be symmetrized and solved using different approaches, for instance one of the conjugate gradient methods.
Theory of difference equations numerical methods and applications
Lakshmikantham, Vangipuram
1988-01-01
In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank mat
International Nuclear Information System (INIS)
Zhi Hongyan
2009-01-01
In this paper, based on the symbolic computing system Maple, the direct method for Lie symmetry groups presented by Sen-Yue Lou [J. Phys. A: Math. Gen. 38 (2005) L129] is extended from the continuous differential equations to the differential-difference equations. With the extended method, we study the well-known differential-difference KP equation, KZ equation and (2+1)-dimensional ANNV system, and both the Lie point symmetry groups and the non-Lie symmetry groups are obtained.
International Nuclear Information System (INIS)
Feng Qinghua
2013-01-01
In this paper, an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann—Liouville derivative. By a fractional complex transformation, a given fractional differential-difference equation can be turned into another differential-difference equation of integer order. The validity of the method is illustrated by applying it to solve the fractional Hybrid lattice equation and the fractional relativistic Toda lattice system. As a result, some new exact solutions including hyperbolic function solutions, trigonometric function solutions and rational solutions are established. (general)
A functional-analytic method for the study of difference equations
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Panayiotis D. Siafarikas
2004-07-01
Full Text Available We will give the generalization of a recently developed functional-analytic method for studying linear and nonlinear, ordinary and partial, difference equations in the Ã¢Â„Â“p1 and Ã¢Â„Â“p2 spaces, pÃ¢ÂˆÂˆÃ¢Â„Â•, pÃ¢Â‰Â¥1. The method will be illustrated by use of two examples concerning a nonlinear ordinary difference equation known as the Putnam equation, and a linear partial difference equation of three variables describing the discrete Newton law of cooling in three dimensions.
International Nuclear Information System (INIS)
Sanchez, Richard.
1980-11-01
This work is divided into two part the first part (note CEA-N-2165) deals with the solution of complex two-dimensional transport problems, the second one treats the critically mixed methods of resolution. These methods are applied for one-dimensional geometries with highly anisotropic scattering. In order to simplify the set of integral equation provided by the integral transport equation, the integro-differential equation is used to obtain relations that allow to lower the number of integral equation to solve; a general mathematical and numerical study is presented [fr
International Nuclear Information System (INIS)
Love, J.C.; Demas, J.N.
1983-01-01
The Foerster equation describes excited-state decay curves involving resonance intermolecular energy transfer. A linearized solution based on the phase-plane method has been developed. The new method is quick, insensitive to the fitting region, accurate, and precise
Local defect correction for boundary integral equation methods
Kakuba, G.; Anthonissen, M.J.H.
2014-01-01
The aim in this paper is to develop a new local defect correction approach to gridding for problems with localised regions of high activity in the boundary element method. The technique of local defect correction has been studied for other methods as finite difference methods and finite volume
Local defect correction for boundary integral equation methods
Kakuba, G.; Anthonissen, M.J.H.
2013-01-01
This paper presents a new approach to gridding for problems with localised regions of high activity. The technique of local defect correction has been studied for other methods as ¿nite difference methods and ¿nite volume methods. In this paper we develop the technique for the boundary element
International Nuclear Information System (INIS)
Saurel, Richard; Franquet, Erwin; Daniel, Eric; Le Metayer, Olivier
2007-01-01
A new projection method is developed for the Euler equations to determine the thermodynamic state in computational cells. It consists in the resolution of a mechanical relaxation problem between the various sub-volumes present in a computational cell. These sub-volumes correspond to the ones traveled by the various waves that produce states with different pressures, velocities, densities and temperatures. Contrarily to Godunov type schemes the relaxed state corresponds to mechanical equilibrium only and remains out of thermal equilibrium. The pressure computation with this relaxation process replaces the use of the conventional equation of state (EOS). A simplified relaxation method is also derived and provides a specific EOS (named the Numerical EOS). The use of the Numerical EOS gives a cure to spurious pressure oscillations that appear at contact discontinuities for fluids governed by real gas EOS. It is then extended to the computation of interface problems separating fluids with different EOS (liquid-gas interface for example) with the Euler equations. The resulting method is very robust, accurate, oscillation free and conservative. For the sake of simplicity and efficiency the method is developed in a Lagrange-projection context and is validated over exact solutions. In a companion paper [F. Petitpas, E. Franquet, R. Saurel, A relaxation-projection method for compressible flows. Part II: computation of interfaces and multiphase mixtures with stiff mechanical relaxation. J. Comput. Phys. (submitted for publication)], the method is extended to the numerical approximation of a non-conservative hyperbolic multiphase flow model for interface computation and shock propagation into mixtures
Solution of two group neutron diffusion equation by using homotopy analysis method
International Nuclear Information System (INIS)
Cavdar, S.
2010-01-01
The Homotopy Analysis Method (HAM), proposed in 1992 by Shi Jun Liao and has been developed since then, is based on differential geometry as well as homotopy which is a fundamental concept in topology. It has proved to be useful for obtaining series solutions of many such problems involving algebraic, linear/non-linear, ordinary/partial differential equations, differential-integral equations, differential-difference equations, and coupled equations of them. Briefly, through HAM, it is possible to construct a continuous mapping of an initial guess approximation to the exact solution of the equation of concern. An auxiliary linear operator is chosen to construct such kind of a continuous mapping and an auxiliary parameter is used to ensure the convergence of series solution. We present the solutions of two-group neutron diffusion equation through HAM in this work. We also compare the results with that obtained by other well-known solution analytical and numeric methods.
International Nuclear Information System (INIS)
Prastyaningrum, I.; Cari, C.; Suparmi, A.
2016-01-01
The approximation analytical solution of Dirac equation for Modified Poschl Teller plus Trigonometric Scarf Potential are investigated numerically in terms of finite Romanovsky Polynomial. The combination of two potentials are substituted into Dirac Equation then the variables are separated into radial and angular parts. The Dirac equation is solved by using Romanovsky Polynomial Method. The equation that can reduce from the second order of differential equation into the differential equation of hypergeometry type by substituted variable method. The energy spectrum is numerically solved using Matlab 2011. Where the increase in the radial quantum number nr and variable of modified Poschl Teller Potential causes the energy to decrease. The radial and the angular part of the wave function also visualized with Matlab 2011. The results show, by the disturbance of a combination between this potential can change the wave function of the radial and angular part. (paper)
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.
DEFF Research Database (Denmark)
Mejlbro, Leif
1997-01-01
An alternative formula for the solution of linear differential equations of order n is suggested. When applicable, the suggested method requires fewer and simpler computations than the well-known method using Wronskians.......An alternative formula for the solution of linear differential equations of order n is suggested. When applicable, the suggested method requires fewer and simpler computations than the well-known method using Wronskians....
Archaeological survey of the McGee Ranch vicinity, Hanford Site, Washington
Energy Technology Data Exchange (ETDEWEB)
Gard, H.A.; Poet, R.M.
1992-09-01
In response to a request for a cultural resources review from Westinghouse Hanford Company for the Action Plan for Characterization of McGee Ranch Soil, Pacific Northwest Laboratory's Hanford Cultural Resources Laboratory (HCRL) conducted an archaeological survey of the McGee Ranch vicinity, located in the northwest portion of the Hanford Site. Staff members covered 8.4 km{sup 2} and recorded 42 cultural resources; 22 sites, and 20 isolated artifacts. Only 2 sites and 3 isolates were attributed to a prehistoric Native American occupation. The historic sites date from the turn of the century to the 1940s and are representative of the settlement patterns that occurred throughout the Columbia Basin. In addition to an archaeological pedestrian survey of the project area, we conducted literature and records searches and examined available aerial photographs. Records kept at HCRL were reviewed to determine if any archaeological survey had been conducted previously within the project area. Although no survey had been conducted, portions of the area adjacent to project boundaries were surveyed in 1988 and 1990. During those surveys, historic and prehistoric cultural resources were observed, increasing the possibility that similar land usage had taken place within the current project boundaries. Literature searches established a general historical sequence for this area. Aerial photographs alerted researchers to homesteads and linear features, such as roads and irrigation ditches, that might not be apparent from ground level.
Archaeological survey of the McGee Ranch vicinity, Hanford Site, Washington
Energy Technology Data Exchange (ETDEWEB)
Gard, H.A.; Poet, R.M.
1992-09-01
In response to a request for a cultural resources review from Westinghouse Hanford Company for the Action Plan for Characterization of McGee Ranch Soil, Pacific Northwest Laboratory`s Hanford Cultural Resources Laboratory (HCRL) conducted an archaeological survey of the McGee Ranch vicinity, located in the northwest portion of the Hanford Site. Staff members covered 8.4 km{sup 2} and recorded 42 cultural resources; 22 sites, and 20 isolated artifacts. Only 2 sites and 3 isolates were attributed to a prehistoric Native American occupation. The historic sites date from the turn of the century to the 1940s and are representative of the settlement patterns that occurred throughout the Columbia Basin. In addition to an archaeological pedestrian survey of the project area, we conducted literature and records searches and examined available aerial photographs. Records kept at HCRL were reviewed to determine if any archaeological survey had been conducted previously within the project area. Although no survey had been conducted, portions of the area adjacent to project boundaries were surveyed in 1988 and 1990. During those surveys, historic and prehistoric cultural resources were observed, increasing the possibility that similar land usage had taken place within the current project boundaries. Literature searches established a general historical sequence for this area. Aerial photographs alerted researchers to homesteads and linear features, such as roads and irrigation ditches, that might not be apparent from ground level.
Comparative analysis among several methods used to solve the point kinetic equations
International Nuclear Information System (INIS)
Nunes, Anderson L.; Goncalves, Alessandro da C.; Martinez, Aquilino S.; Silva, Fernando Carvalho da
2007-01-01
The main objective of this work consists on the methodology development for comparison of several methods for the kinetics equations points solution. The evaluated methods are: the finite differences method, the stiffness confinement method, improved stiffness confinement method and the piecewise constant approximations method. These methods were implemented and compared through a systematic analysis that consists basically of confronting which one of the methods consume smaller computational time with higher precision. It was calculated the relative which function is to combine both criteria in order to reach the goal. Through the analyses of the performance factor it is possible to choose the best method for the solution of point kinetics equations. (author)
Comparative analysis among several methods used to solve the point kinetic equations
Energy Technology Data Exchange (ETDEWEB)
Nunes, Anderson L.; Goncalves, Alessandro da C.; Martinez, Aquilino S.; Silva, Fernando Carvalho da [Universidade Federal, Rio de Janeiro, RJ (Brazil). Coordenacao dos Programas de Pos-graduacao de Engenharia. Programa de Engenharia Nuclear; E-mails: alupo@if.ufrj.br; agoncalves@con.ufrj.br; aquilino@lmp.ufrj.br; fernando@con.ufrj.br
2007-07-01
The main objective of this work consists on the methodology development for comparison of several methods for the kinetics equations points solution. The evaluated methods are: the finite differences method, the stiffness confinement method, improved stiffness confinement method and the piecewise constant approximations method. These methods were implemented and compared through a systematic analysis that consists basically of confronting which one of the methods consume smaller computational time with higher precision. It was calculated the relative which function is to combine both criteria in order to reach the goal. Through the analyses of the performance factor it is possible to choose the best method for the solution of point kinetics equations. (author)
LeMesurier, Brenton
2012-01-01
A new approach is described for generating exactly energy-momentum conserving time discretizations for a wide class of Hamiltonian systems of DEs with quadratic momenta, including mechanical systems with central forces; it is well-suited in particular to the large systems that arise in both spatial discretizations of nonlinear wave equations and lattice equations such as the Davydov System modeling energetic pulse propagation in protein molecules. The method is unconditionally stable, making it well-suited to equations of broadly “Discrete NLS form”, including many arising in nonlinear optics. Key features of the resulting discretizations are exact conservation of both the Hamiltonian and quadratic conserved quantities related to continuous linear symmetries, preservation of time reversal symmetry, unconditional stability, and respecting the linearity of certain terms. The last feature allows a simple, efficient iterative solution of the resulting nonlinear algebraic systems that retain unconditional stability, avoiding the need for full Newton-type solvers. One distinction from earlier work on conservative discretizations is a new and more straightforward nearly canonical procedure for constructing the discretizations, based on a “discrete gradient calculus with product rule” that mimics the essential properties of partial derivatives. This numerical method is then used to study the Davydov system, revealing that previously conjectured continuum limit approximations by NLS do not hold, but that sech-like pulses related to NLS solitons can nevertheless sometimes arise.
International Nuclear Information System (INIS)
Freed, K.F.; Herman, M.F.; Yeager, D.L.
1980-01-01
A description is provided of the common conceptual origins of many-body equations of motion and Green's function methods in Liouville operator formulations of the quantum mechanics of atomic and molecular electronic structure. Numerical evidence is provided to show the inadequacies of the traditional strictly perturbative approaches to these methods. Nonperturbative methods are introduced by analogy with techniques developed for handling large configuration interaction calculations and by evaluating individual matrix elements to higher accuracy. The important role of higher excitations is exhibited by the numerical calculations, and explicit comparisons are made between converged equations of motion and configuration interaction calculations for systems where a fundamental theorem requires the equality of the energy differences produced by these different approaches. (Auth.)
Solution of systems of linear algebraic equations by the method of summation of divergent series
International Nuclear Information System (INIS)
Kirichenko, G.A.; Korovin, Ya.S.; Khisamutdinov, M.V.; Shmojlov, V.I.
2015-01-01
A method for solving systems of linear algebraic equations has been proposed on the basis on the summation of the corresponding continued fractions. The proposed algorithm for solving systems of linear algebraic equations is classified as direct algorithms providing an exact solution in a finite number of operations. Examples of solving systems of linear algebraic equations have been presented and the effectiveness of the algorithm has been estimated [ru
New Iterative Method for Fractional Gas Dynamics and Coupled Burger’s Equations
Directory of Open Access Journals (Sweden)
Mohamed S. Al-luhaibi
2015-01-01
Full Text Available This paper presents the approximate analytical solutions to solve the nonlinear gas dynamics and coupled Burger’s equations with fractional time derivative. By using initial values, the explicit solutions of the equations are solved by using a reliable algorithm. Numerical results show that the new iterative method is easy to implement and accurate when applied to time-fractional partial differential equations.
Multi-symplectic Preissmann methods for generalized Zakharov-Kuznetsov equation
International Nuclear Information System (INIS)
Wang Junjie; Yang Kuande; Wang Liantang
2012-01-01
Generalized Zakharov-Kuznetsov equation, a typical nonlinear wave equation, was studied based on the multi-symplectic theory in Hamilton space. The multi-symplectic formulations of generalized Zakharov-Kuznetsov equation with several conservation laws are presented. The multi-symplectic Preissmann method is used to discretize the formulations. The numerical experiment is given, and the results verify the efficiency of the multi-symplectic scheme. (authors)
Dinar, N.
1978-01-01
Several aspects of multigrid methods are briefly described. The main subjects include the development of very efficient multigrid algorithms for systems of elliptic equations (Cauchy-Riemann, Stokes, Navier-Stokes), as well as the development of control and prediction tools (based on local mode Fourier analysis), used to analyze, check and improve these algorithms. Preliminary research on multigrid algorithms for time dependent parabolic equations is also described. Improvements in existing multigrid processes and algorithms for elliptic equations were studied.
Application of the B-Determining Equations Method to One Problem of Free Turbulence
Directory of Open Access Journals (Sweden)
Oleg V. Kaptsov
2012-10-01
Full Text Available A three-dimensional model of the far turbulent wake behind a self-propelled body in a passively stratified medium is considered. The model is reduced to a system of ordinary differential equations by a similarity reduction and the B-determining equations method. The system of ordinary differential equations satisfying natural boundary conditions is solved numerically. The solutions obtained here are in close agreement with experimental data.
Solution of second order linear fuzzy difference equation by Lagrange's multiplier method
Directory of Open Access Journals (Sweden)
Sankar Prasad Mondal
2016-06-01
Full Text Available In this paper we execute the solution procedure for second order linear fuzzy difference equation by Lagrange's multiplier method. In crisp sense the difference equation are easy to solve, but when we take in fuzzy sense it forms a system of difference equation which is not so easy to solve. By the help of Lagrange's multiplier we can solved it easily. The results are illustrated by two different numerical examples and followed by two applications.
A Marker Method for the Solution of the Damped Burgers' Equation
International Nuclear Information System (INIS)
Jerome L.V. Lewandowski
2005-01-01
A new method for the solution of the damped Burgers equation is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details. The marker method is applicable to a general class of nonlinear dispersive partial differential equations
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.
Low-mode truncation methods in the sine-Gordon equation
International Nuclear Information System (INIS)
Xiong Chuyu.
1991-01-01
In this dissertation, the author studies the chaotic and coherent motions (i.e., low-dimensional chaotic attractor) in some near integrable partial differential equations, particularly the sine-Gordon equation and the nonlinear Schroedinger equation. In order to study the motions, he uses low mode truncation methods to reduce these partial differential equations to some truncated models (low-dimensional ordinary differential equations). By applying many methods available to low-dimensional ordinary differential equations, he can understand the low-dimensional chaotic attractor of PDE's much better. However, there are two important questions one needs to answer: (1) How many modes is good enough for the low mode truncated models to capture the dynamics uniformly? (2) Is the chaotic attractor in a low mode truncated model close to the chaotic attractor in the original PDE? And how close is? He has developed two groups of powerful methods to help to answer these two questions. They are the computation methods of continuation and local bifurcation, and local Lyapunov exponents and Lyapunov exponents. Using these methods, he concludes that the 2N-nls ODE is a good model for the sine-Gordon equation and the nonlinear Schroedinger equation provided one chooses a 'good' basis and uses 'enough' modes (where 'enough' depends on the parameters of the system but is small for the parameter studied here). Therefore, one can use 2N-nls ODE to study the chaos of PDE's in more depth
Combined methods for elliptic equations with singularities, interfaces and infinities
Li, Zi Cai
1998-01-01
In this book the author sets out to answer two important questions: 1. Which numerical methods may be combined together? 2. How can different numerical methods be matched together? In doing so the author presents a number of useful combinations, for instance, the combination of various FEMs, the combinations of FEM-FDM, REM-FEM, RGM-FDM, etc. The combined methods have many advantages over single methods: high accuracy of solutions, less CPU time, less computer storage, easy coupling with singularities as well as the complicated boundary conditions. Since coupling techniques are essential to combinations, various matching strategies among different methods are carefully discussed. The author provides the matching rules so that optimal convergence, even superconvergence, and optimal stability can be achieved, and also warns of the matching pitfalls to avoid. Audience: The book is intended for both mathematicians and engineers and may be used as text for advanced students.
Perfectly Matched Layer for the Wave Equation Finite Difference Time Domain Method
Miyazaki, Yutaka; Tsuchiya, Takao
2012-07-01
The perfectly matched layer (PML) is introduced into the wave equation finite difference time domain (WE-FDTD) method. The WE-FDTD method is a finite difference method in which the wave equation is directly discretized on the basis of the central differences. The required memory of the WE-FDTD method is less than that of the standard FDTD method because no particle velocity is stored in the memory. In this study, the WE-FDTD method is first combined with the standard FDTD method. Then, Berenger's PML is combined with the WE-FDTD method. Some numerical demonstrations are given for the two- and three-dimensional sound fields.
EQUITY SHARES EQUATING THE RESULTS OF FCFF AND FCFE METHODS
Directory of Open Access Journals (Sweden)
Bartłomiej Cegłowski
2012-06-01
Full Text Available The aim of the article is to present the method of establishing equity shares in weight average cost of capital (WACC, in which the value of loan capital results from the fixed assumptions accepted in the financial plan (for example a schedule of loan repayment and own equity is evaluated by means of a discount method. The described method causes that, regardless of whether cash flows are calculated as FCFF or FCFE, the result of the company valuation will be identical.
Energy Technology Data Exchange (ETDEWEB)
Saha Ray, S., E-mail: santanusaharay@yahoo.com; Patra, A.
2014-10-15
Highlights: • A stationary transport equation has been solved using the technique of Haar wavelet collocation method. • This paper intends to provide the great utility of Haar wavelets to nuclear science problem. • In the present paper, two-dimensional Haar wavelets are applied. • The proposed method is mathematically very simple, easy and fast. - Abstract: In this paper the numerical solution for the fractional order stationary neutron transport equation is presented using Haar wavelet Collocation Method (HWCM). Haar wavelet collocation method is efficient and powerful in solving wide class of linear and nonlinear differential equations. This paper intends to provide an application of Haar wavelets to nuclear science problems. This paper describes the application of Haar wavelets for the numerical solution of fractional order stationary neutron transport equation in homogeneous medium with isotropic scattering. The proposed method is mathematically very simple, easy and fast. To demonstrate about the efficiency and applicability of the method, two test problems are discussed.
Inverse operator method for solutions of nonlinear dynamical equations and some typical applications
International Nuclear Information System (INIS)
Fang Jinqing; Yao Weiguang
1993-01-01
The inverse operator method (IOM) is described briefly. We have realized the IOM for the solutions of nonlinear dynamical equations by the mathematics-mechanization (MM) with computers. They can then offer a new and powerful method applicable to many areas of physics. We have applied them successfully to study the chaotic behaviors of some nonlinear dynamical equations. As typical examples, the well-known Lorentz equation, generalized Duffing equation and two coupled generalized Duffing equations are investigated by using the IOM and the MM. The results are in good agreement with those given by Runge-Kutta method. So the IOM realized by the MM is of potential application valuable in nonlinear physics and many other fields
Analysis of a time fractional wave-like equation with the homotopy analysis method
International Nuclear Information System (INIS)
Xu Hang; Cang Jie
2008-01-01
The time fractional wave-like differential equation with a variable coefficient is studied analytically. By using a simple transformation, the governing equation is reduced to two fractional ordinary differential equations. Then the homotopy analysis method is employed to derive the solutions of these equations. The accurate series solutions are obtained. Especially, when h f =h g =-1, these solutions are exactly the same as those results given by the Adomian decomposition method. The present work shows the validity and great potential of the homotopy analysis method for solving nonlinear fractional differential equations. The basic idea described in this Letter is expected to be further employed to solve other similar nonlinear problems in fractional calculus
Evaluation of Maryland abutment scour equation through selected threshold velocity methods
Benedict, S.T.
2010-01-01
The U.S. Geological Survey, in cooperation with the Maryland State Highway Administration, used field measurements of scour to evaluate the sensitivity of the Maryland abutment scour equation to the critical (or threshold) velocity variable. Four selected methods for estimating threshold velocity were applied to the Maryland abutment scour equation, and the predicted scour to the field measurements were compared. Results indicated that performance of the Maryland abutment scour equation was sensitive to the threshold velocity with some threshold velocity methods producing better estimates of predicted scour than did others. In addition, results indicated that regional stream characteristics can affect the performance of the Maryland abutment scour equation with moderate-gradient streams performing differently from low-gradient streams. On the basis of the findings of the investigation, guidance for selecting threshold velocity methods for application to the Maryland abutment scour equation are provided, and limitations are noted.
Directory of Open Access Journals (Sweden)
Ji Juan-Juan
2017-01-01
Full Text Available A table lookup method for solving nonlinear fractional partial differential equations (fPDEs is proposed in this paper. Looking up the corresponding tables, we can quickly obtain the exact analytical solutions of fPDEs by using this method. To illustrate the validity of the method, we apply it to construct the exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the (2+1-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation. As a result, many new types of exact analytical solutions are obtained including triangular periodic solution, hyperbolic function solution, singular solution, multiple solitary wave solution, and Jacobi elliptic function solution.
A simple method for estimating the convection- dispersion equation ...
African Journals Online (AJOL)
Jane
2011-08-31
Aug 31, 2011 ... approach of modeling solute transport in porous media uses the deterministic ... Methods of estimating CDE transport parameters can be divided into statistical ..... diffusion-type model for longitudinal mixing of fluids in flow.
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.
Approximate solution of the transport equation by methods of Galerkin type
International Nuclear Information System (INIS)
Pitkaranta, J.
1977-01-01
Questions of the existence, uniqueness, and convergence of approximate solutions of transport equations by methods of the Galerkin type (where trial and weighting functions are the same) are discussed. The results presented do not exclude the infinite-dimensional case. Two strategies can be followed in the variational approximation of the transport operator: one proceeds from the original form of the transport equation, while the other is based on the partially symmetrized equation. Both principles are discussed in this paper. The transport equation is assumed in a discretized multigroup form
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.
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A. A. Hemeda
2013-01-01
Full Text Available An extension of the so-called new iterative method (NIM has been used to handle linear and nonlinear fractional partial differential equations. The main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently. Therefore, a general framework of the NIM is presented for analytical treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense. Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation, fractional Klein-Gordon equation, and fractional Boussinesq-like equation are investigated to show the pertinent features of the technique. Comparison of the results obtained by the NIM with those obtained by both Adomian decomposition method (ADM and the variational iteration method (VIM reveals that the NIM is very effective and convenient. The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus.
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
Using direct algebraic method,exact solitary wave solutions are performed for a class of third order nonlinear dispersive disipative partial differential equations. These solutions are obtained under certain conditions for the relationship between the coefficients of the equation. The exact solitary waves of this class are rational functions of real exponentials of kink-type solutions.
A Simple Method to Find out when an Ordinary Differential Equation Is Separable
Cid, Jose Angel
2009-01-01
We present an alternative method to that of Scott (D. Scott, "When is an ordinary differential equation separable?", "Amer. Math. Monthly" 92 (1985), pp. 422-423) to teach the students how to discover whether a differential equation y[prime] = f(x,y) is separable or not when the nonlinearity f(x, y) is not explicitly factorized. Our approach is…
Variational formulation and projectional methods for the second order transport equation
International Nuclear Information System (INIS)
Borysiewicz, M.; Stankiewicz, R.
1979-01-01
Herein the variational problem for a second-order boundary value problem for the neutron transport equation is formulated. The projectional methods solving the problem are examined. The approach is compared with that based on the original untransformed form of the neutron transport equation
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
International Nuclear Information System (INIS)
Chaichian, M.; Kulish, P. P.
1978-04-01
Supersymmetric Liouville and sine-Gordon equations are studied. We write down for these models the system of linear equations for which the method of inverse scattering problem should be applicable. Expressions for an infinite set of conserved currents are explicitly given. Supersymmetric Baecklund transformations and generalized conservation laws are constructed. (author)
A Method for Solving the Voltage and Torque Equations of the Split ...
African Journals Online (AJOL)
Akorede
v′ Voltage applied across the d – axis rotor winding referred ... The embedded MATLAB function and other useful blocks from the ... III. EQUATIONS OF THE SPLIT PHASE INDUCTION MOTOR. The voltage, flux and electromagnetic torque equations are ..... of single phase induction motor using frequency control method ...
Soliton-like solutions to the GKdV equation by extended mapping method
International Nuclear Information System (INIS)
Wu Ranchao; Sun Jianhua
2007-01-01
In this note, many new exact solutions of the generalized KdV equation, such as rational solutions, periodic solutions like Jacobian elliptic and triangular functions, soliton-like solutions, are constructed by symbolic computation and the extended mapping method, with the auxiliary ordinary equation replaced by a more general one
Directory of Open Access Journals (Sweden)
Hossein Jafari
2016-04-01
Full Text Available The non-differentiable solution of the linear and non-linear partial differential equations on Cantor sets is implemented in this article. The reduced differential transform method is considered in the local fractional operator sense. The four illustrative examples are given to show the efficiency and accuracy features of the presented technique to solve local fractional partial differential equations.
A note on the convergence of the Zakharov-Kuznetsov equation by homotopy analysis method
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Amir Fallahzadeh
2014-07-01
Full Text Available In this paper, the convergence of Zakharov-Kuznetsov (ZK equation by homotopy analysis method (HAM is investigated. A theorem is proved to guarantee the convergence of HAMand to find the series solution of this equation via a reliable algorithm.
An Evaluation of Five Linear Equating Methods for the NEAT Design
Mroch, Andrew A.; Suh, Youngsuk; Kane, Michael T.; Ripkey, Douglas R.
2009-01-01
This study uses the results of two previous papers (Kane, Mroch, Suh, & Ripkey, this issue; Suh, Mroch, Kane, & Ripkey, this issue) and the literature on linear equating to evaluate five linear equating methods along several dimensions, including the plausibility of their assumptions and their levels of bias and root mean squared difference…
New Quasi-Newton Method for Solving Systems of Nonlinear Equations
Czech Academy of Sciences Publication Activity Database
Lukšan, Ladislav; Vlček, Jan
2017-01-01
Roč. 62, č. 2 (2017), s. 121-134 ISSN 0862-7940 R&D Projects: GA ČR GA13-06684S Institutional support: RVO:67985807 Keywords : nonlinear equations * systems of equations * trust-region methods * quasi-Newton methods * adjoint Broyden methods * numerical algorithms * numerical experiments Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics Impact factor: 0.618, year: 2016 http://hdl.handle.net/10338.dmlcz/146699
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
Modeling Solution of Nonlinear Dispersive Partial Differential Equations using the Marker Method
International Nuclear Information System (INIS)
Lewandowski, Jerome L.V.
2005-01-01
A new method for the solution of nonlinear dispersive partial differential equations is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details
Ullah, Hakeem; Islam, Saeed; Khan, Ilyas; Shafie, Sharidan; Fiza, Mehreen
2015-01-01
In this paper we applied a new analytic approximate technique Optimal Homotopy Asymptotic Method (OHAM) for treatment of coupled differential- difference equations (DDEs). To see the efficiency and reliability of the method, we consider Relativistic Toda coupled nonlinear differential-difference equation. It provides us a convenient way to control the convergence of approximate solutions when it is compared with other methods of solution found in the literature. The obtained solutions show that OHAM is effective, simpler, easier and explicit. PMID:25874457
A constrained Hartree-Fock-Bogoliubov equation derived from the double variational method
International Nuclear Information System (INIS)
Onishi, Naoki; Horibata, Takatoshi.
1980-01-01
The double variational method is applied to the intrinsic state of the generalized BCS wave function. A constrained Hartree-Fock-Bogoliubov equation is derived explicitly in the form of an eigenvalue equation. A method of obtaining approximate overlap and energy overlap integrals is proposed. This will help development of numerical calculations of the angular momentum projection method, especially for general intrinsic wave functions without any symmetry restrictions. (author)
The (′/-Expansion Method for Abundant Traveling Wave Solutions of Caudrey-Dodd-Gibbon Equation
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Hasibun Naher
2011-01-01
Full Text Available We construct the traveling wave solutions of the fifth-order Caudrey-Dodd-Gibbon (CDG equation by the (/-expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, the trigonometric, and the rational functions. It is shown that the (/-expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations.
International Nuclear Information System (INIS)
Mokhtari, R.; Toodar, A. Samadi; Chegini, N. G.
2011-01-01
We the extend application of the generalized differential quadrature method (GDQM) to solve some coupled nonlinear Schrödinger equations. The cosine-based GDQM is employed and the obtained system of ordinary differential equations is solved via the fourth order Runge—Kutta method. The numerical solutions coincide with the exact solutions in desired machine precision and invariant quantities are conserved sensibly. Some comparisons with the methods applied in the literature are carried out. (general)
Ullah, Hakeem; Islam, Saeed; Khan, Ilyas; Shafie, Sharidan; Fiza, Mehreen
2015-01-01
In this paper we applied a new analytic approximate technique Optimal Homotopy Asymptotic Method (OHAM) for treatment of coupled differential-difference equations (DDEs). To see the efficiency and reliability of the method, we consider Relativistic Toda coupled nonlinear differential-difference equation. It provides us a convenient way to control the convergence of approximate solutions when it is compared with other methods of solution found in the literature. The obtained solutions show that OHAM is effective, simpler, easier and explicit.
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.
Methods for the solution of the two-dimensional radiation-transfer equation
International Nuclear Information System (INIS)
Weaver, R.; Mihalas, D.; Olson, G.
1982-01-01
We use the variable Eddington factor (VEF) approximation to solve the time-dependent two-dimensional radiation transfer equation. The transfer equation and its moments are derived for an inertial frame of reference in cylindrical geometry. Using the VEF tensor to close the moment equations, we manipulate them into a combined moment equation that results in an energy equation, which is automatically flux limited. There are two separable facets in this method of solution. First, given the variable Eddington tensor, we discuss the efficient solution of the combined moment matrix equation. The second facet of the problem is the calculation of the variable Eddington tensor. Several options for this calculation, as well as physical limitations on the use of locally-calculated Eddington factors, are discussed
Study on state equation for hydrogen storage measurement by volumetric method
International Nuclear Information System (INIS)
Dai Wei; Xu Jiajing; Wang Chaoyang; Tang Yongjian
2014-01-01
Volumetric measurement technique is one of the most popular methods for determining the amount of hydrogen storage. A new state equation was established which extended the limitations from the ideal gas state equation, the van der Waals equation and the Gou equation. The new state equation was then employed to describe the p-V-T character of hydrogen and investigate the adsorption quantity of hydrogen storage in resorcin-formaldehyde aerogel under different temperatures and pressures. The new equation was used to describe the density of hydrogen under different temperatures and pressures. The results are in good agreement with the experimental data. The differences arising from various underlying physics were carefully analyzed. (authors)
Final Report: Symposium on Adaptive Methods for Partial Differential Equations
Energy Technology Data Exchange (ETDEWEB)
Pernice, Michael; Johnson, Christopher R.; Smith, Philip J.; Fogelson, Aaron
1998-12-08
Complex physical phenomena often include features that span a wide range of spatial and temporal scales. Accurate simulation of such phenomena can be difficult to obtain, and computations that are under-resolved can even exhibit spurious features. While it is possible to resolve small scale features by increasing the number of grid points, global grid refinement can quickly lead to problems that are intractable, even on the largest available computing facilities. These constraints are particularly severe for three dimensional problems that involve complex physics. One way to achieve the needed resolution is to refine the computational mesh locally, in only those regions where enhanced resolution is required. Adaptive solution methods concentrate computational effort in regions where it is most needed. These methods have been successfully applied to a wide variety of problems in computational science and engineering. Adaptive methods can be difficult to implement, prompting the development of tools and environments to facilitate their use. To ensure that the results of their efforts are useful, algorithm and tool developers must maintain close communication with application specialists. Conversely it remains difficult for application specialists who are unfamiliar with the methods to evaluate the trade-offs between the benefits of enhanced local resolution and the effort needed to implement an adaptive solution method.
Modulating functions method for parameters estimation in the fifth order KdV equation
Asiri, Sharefa M.
2017-07-25
In this work, the modulating functions method is proposed for estimating coefficients in higher-order nonlinear partial differential equation which is the fifth order Kortewegde Vries (KdV) equation. The proposed method transforms the problem into a system of linear algebraic equations of the unknowns. The statistical properties of the modulating functions solution are described in this paper. In addition, guidelines for choosing the number of modulating functions, which is an important design parameter, are provided. The effectiveness and robustness of the proposed method are shown through numerical simulations in both noise-free and noisy cases.
Directory of Open Access Journals (Sweden)
Şuayip Yüzbaşı
2017-03-01
Full Text Available In this paper, we suggest a matrix method for obtaining the approximate solutions of the delay linear Fredholm integro-differential equations with constant coefficients using the shifted Legendre polynomials. The problem is considered with mixed conditions. Using the required matrix operations, the delay linear Fredholm integro-differential equation is transformed into a matrix equation. Additionally, error analysis for the method is presented using the residual function. Illustrative examples are given to demonstrate the efficiency of the method. The results obtained in this study are compared with the known results.
Fibonacci-regularization method for solving Cauchy integral equations of the first kind
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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.
On a computer implementation of the block Gauss–Seidel method for normal systems of equations
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Alexander I. Zhdanov
2016-12-01
Full Text Available This article focuses on the modification of the block option Gauss-Seidel method for normal systems of equations, which is a sufficiently effective method of solving generally overdetermined, systems of linear algebraic equations of high dimensionality. The main disadvantage of methods based on normal equations systems is the fact that the condition number of the normal system is equal to the square of the condition number of the original problem. This fact has a negative impact on the rate of convergence of iterative methods based on normal equations systems. To increase the speed of convergence of iterative methods based on normal equations systems, for solving ill-conditioned problems currently different preconditioners options are used that reduce the condition number of the original system of equations. However, universal preconditioner for all applications does not exist. One of the effective approaches that improve the speed of convergence of the iterative Gauss–Seidel method for normal systems of equations, is to use its version of the block. The disadvantage of the block Gauss–Seidel method for production systems is the fact that it is necessary to calculate the pseudoinverse matrix for each iteration. We know that finding the pseudoinverse is a difficult computational procedure. In this paper, we propose a procedure to replace the matrix pseudo-solutions to the problem of normal systems of equations by Cholesky. Normal equations arising at each iteration of Gauss–Seidel method, have a relatively low dimension compared to the original system. The results of numerical experimentation demonstrating the effectiveness of the proposed approach are given.
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.
Fast Numerical Methods for Stochastic Partial Differential Equations
2016-04-15
Particle Swarm Optimization (PSO) method. Inspired by the social behavior of the bird flocking or fish schooling, the particle swarm optimization (PSO...Weerasinghe, Hongmei Chi and Yanzhao Cao, Particle Swarm Optimization Simulation via Optimal Halton Sequences, accepted by Procedia Computer Science (2016...Optimization Simulation via Optimal Halton Sequences, accepted by Procedia Computer Science (2016). 2. Haiyan Tian, Hongmei Chi and Yanzhao Cao
The Embedding Method for Linear Partial Differential Equations
Indian Academy of Sciences (India)
The recently suggested embedding method to solve linear boundary value problems is here extended to cover situations where the domain of interest is unbounded or multiply connected. The extensions involve the use of complete sets of exterior and interior eigenfunctions on canonical domains. Applications to typical ...
Energy Technology Data Exchange (ETDEWEB)
Benoist, P; Kavenoky, A [Commissariat a l' Energie Atomique, Saclay (France). Centre d' Etudes Nucleaires
1968-01-15
In a new method of approximation of the Boltzmann equation, one starts from a particular form of the equation which involves only the angular flux at the boundary of the considered medium and where the space variable does not appear explicitly. Expanding in orthogonal polynomials the angular flux of neutrons leaking from the medium and making no assumption about the angular flux within the medium, very good approximations to several classical plane geometry problems, i.e. the albedo of slabs and the transmission by slabs, the extrapolation length of the Milne problem, the spectrum of neutrons reflected by a semi-infinite slowing down medium. The method can be extended to other geometries. (authors) [French] On etablit une nouvelle methode d'approximation pour l'equation de Boltzmann en partant d'une forme particuliere de cette equation qui n'implique que le flux angulaire a la frontiere du milieu et ou les variables d'espace n'apparaissent pas explicitement. Par un developpement en polynomes orthogonaux du flux angulaire sortant du milieu et sans faire d'hypothese sur le flux angulaire a l'interieur du milieu, on obtient de tres bonnes approximations pour plusieurs problemes classiques en geometrie plane: l'albedo et le facteur de transmission des plaques, la longueur d'extrapolation du probleme de Milne, le spectre des neutrons reflechis par un milieu semi-infini ralentisseur. La methode se generalise a d'autres geometries. (auteurs)
The generalized approximation method and nonlinear heat transfer equations
Directory of Open Access Journals (Sweden)
Rahmat Khan
2009-01-01
Full Text Available Generalized approximation technique for a solution of one-dimensional steady state heat transfer problem in a slab made of a material with temperature dependent thermal conductivity, is developed. The results obtained by the generalized approximation method (GAM are compared with those studied via homotopy perturbation method (HPM. For this problem, the results obtained by the GAM are more accurate as compared to the HPM. Moreover, our (GAM generate a sequence of solutions of linear problems that converges monotonically and rapidly to a solution of the original nonlinear problem. Each approximate solution is obtained as the solution of a linear problem. We present numerical simulations to illustrate and confirm the theoretical results.
Lattice Boltzmann method for solving the bioheat equation
International Nuclear Information System (INIS)
Zhang Haifeng
2008-01-01
In this work, we develop the lattice Boltzmann method (LBM) as a potential solver for the bioheat problems. The accuracy of the present LBM algorithm is validated through comparison with the analytical solution and the finite element simulation. The results show that the LBM can give a precise prediction of the temperature distribution, and it is efficient to deal with the space- and time-dependent heat source, which are often encountered in the treatment planning of tumor hyperthermia. (note)
Wavelet Methods for Solving Fractional Order Differential Equations
A. K. Gupta; S. Saha Ray
2014-01-01
Fractional calculus is a field of applied mathematics which deals with derivatives and integrals of arbitrary orders. The fractional calculus has gained considerable importance during the past decades mainly due to its application in diverse fields of science and engineering such as viscoelasticity, diffusion of biological population, signal processing, electromagnetism, fluid mechanics, electrochemistry, and many more. In this paper, we review different wavelet methods for solving both linea...
Application of the N/D-method to the Roy equations
International Nuclear Information System (INIS)
Heemskerk, A.C.
1978-01-01
The subject of this thesis is the study and numerical application of the Roy equations for elastic pion-pion scattering. These equations were introduced by S.M. Roy in 1971 and express the minimal requirements any ππ partial wave amplitude should satisfy, such as analyticity and crossing symmetry. When combined with the unitarity relations for the partial waves a complete system of equations (the Roy system) results valid in the low-energy region. The author has studied the existance and uniqueness question for the Roy system and developed successful strategies to solve the system by means of iteration. Several solutions are presented and discussed. The main ingredients in the present approach are a) the use of the N/D-method to regularize the singular Roy equations and implement unitarity and b) the introduction of various iteration methods which do not need any derivatives to solve the resulting nonlinear system of equations numerically. (Auth.)
Ebert, Marcelo R
2018-01-01
This book provides an overview of different topics related to the theory of partial differential equations. Selected exercises are included at the end of each chapter to prepare readers for the “research project for beginners” proposed at the end of the book. It is a valuable resource for advanced graduates and undergraduate students who are interested in specializing in this area. The book is organized in five parts: In Part 1 the authors review the basics and the mathematical prerequisites, presenting two of the most fundamental results in the theory of partial differential equations: the Cauchy-Kovalevskaja theorem and Holmgren's uniqueness theorem in its classical and abstract form. It also introduces the method of characteristics in detail and applies this method to the study of Burger's equation. Part 2 focuses on qualitative properties of solutions to basic partial differential equations, explaining the usual properties of solutions to elliptic, parabolic and hyperbolic equations for the archetypes...
International Nuclear Information System (INIS)
Ixaru, G.L.
1978-03-01
The method developed in the previous paper (preprint, C.I.Ph. (Bucharest), MC-2-78, 1978) is here investigated from computational point of view. Special emphasis is paid to the two basic descriptors of the efficiency: the volume of memory required and the computational effort (timing). Next, two experimental cases are reported. They (i) confirm the theoretical estimates for the rate cf convergence of each version of the present method and (ii) show that the present method is substantially faster than the others. Specifically, it is found that for typical physical problems it is faster by a factor of ten up to twenty than the methods commonly used, viz. Numerov and de Vogelaere. The data reported also allow an inUirect comparison with the method of Gordon. I l/ allow an indirect comparison with the method of Gordon. It is shown that, while this exhibits the same rate as our basic, lowest order version, the computational effort for the latter is, in case of systems with nine equations, only half than for the method of Gordon. At the end of the paper some types of physical problems are suggested which should be the most benefitting if solved numerically with the present method. (author)
Ghisletta, Paolo; Spini, Dario
2004-01-01
Correlated data are very common in the social sciences. Most common applications include longitudinal and hierarchically organized (or clustered) data. Generalized estimating equations (GEE) are a convenient and general approach to the analysis of several kinds of correlated data. The main advantage of GEE resides in the unbiased estimation of…
Rosenbaum, J. S.
1976-01-01
If a system of ordinary differential equations represents a property conserving system that can be expressed linearly (e.g., conservation of mass), it is then desirable that the numerical integration method used conserve the same quantity. It is shown that both linear multistep methods and Runge-Kutta methods are 'conservative' and that Newton-type methods used to solve the implicit equations preserve the inherent conservation of the numerical method. It is further shown that a method used by several authors is not conservative.
Methods for solving the stochastic point reactor kinetic equations
International Nuclear Information System (INIS)
Quabili, E.R.; Karasulu, M.
1979-01-01
Two new methods are presented for analysis of the statistical properties of nonlinear outputs of a point reactor to stochastic non-white reactivity inputs. They are Bourret's approximation and logarithmic linearization. The results have been compared with the exact results, previously obtained in the case of Gaussian white reactivity input. It was found that when the reactivity noise has short correlation time, Bourret's approximation should be recommended because it yields results superior to those yielded by logarithmic linearization. When the correlation time is long, Bourret's approximation is not valid, but in that case, if one can assume the reactivity noise to be Gaussian, one may use the logarithmic linearization. (author)
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.
Czech Academy of Sciences Publication Activity Database
Lukšan, Ladislav; Vlček, Jan
1998-01-01
Roč. 8, č. 3-4 (1998), s. 201-223 ISSN 1055-6788 R&D Projects: GA ČR GA201/96/0918 Keywords : nonlinear equations * Armijo-type descent methods * Newton-like methods * truncated methods * global convergence * nonsymmetric linear systems * conjugate gradient -type methods * residual smoothing * computational experiments Subject RIV: BB - Applied Statistics, Operational Research
Directory of Open Access Journals (Sweden)
Guo Zheng-Hong
2016-01-01
Full Text Available In this article, the Sumudu transform series expansion method is used to handle the local fractional Laplace equation arising in the steady fractal heat-transfer problem via local fractional calculus.
Iterative observer based method for source localization problem for Poisson equation in 3D
Majeed, Muhammad Usman; Laleg-Kirati, Taous-Meriem
2017-01-01
A state-observer based method is developed to solve point source localization problem for Poisson equation in a 3D rectangular prism with available boundary data. The technique requires a weighted sum of solutions of multiple boundary data
Bonito, Andrea; Guermond, Jean-Luc; Popov, Bojan
2013-01-01
We establish the L2-stability of an entropy viscosity technique applied to nonlinear scalar conservation equations. First-and second-order explicit time-stepping techniques using continuous finite elements in space are considered. The method
Application of Exp-function method to potential Kadomtsev-Petviashvili equation
International Nuclear Information System (INIS)
Xian Daquan; Dai Zhengde
2009-01-01
Exact periodic kink-wave solution, periodic soliton and doubly periodic solutions for the potential Kadomtsev-Petviashvii (PKP) equation are obtained using Exp-function method with the help of Maple computation.
A new integral method for solving the point reactor neutron kinetics equations
International Nuclear Information System (INIS)
Li Haofeng; Chen Wenzhen; Luo Lei; Zhu Qian
2009-01-01
A numerical integral method that efficiently provides the solution of the point kinetics equations by using the better basis function (BBF) for the approximation of the neutron density in one time step integrations is described and investigated. The approach is based on an exact analytic integration of the neutron density equation, where the stiffness of the equations is overcome by the fully implicit formulation. The procedure is tested by using a variety of reactivity functions, including step reactivity insertion, ramp input and oscillatory reactivity changes. The solution of the better basis function method is compared to other analytical and numerical solutions of the point reactor kinetics equations. The results show that selecting a better basis function can improve the efficiency and accuracy of this integral method. The better basis function method can be used in real time forecasting for power reactors in order to prevent reactivity accidents.
Method for solving the periodic problem for integro-differential equations
Directory of Open Access Journals (Sweden)
Snezhana G. Hristova
1989-05-01
Full Text Available In the paper a monotone-iterative method for approximate finding a couple of minimal and maximal quasisolutions of the periodic problem for a system of integro-differential equations of Volterra type is justified.
Stability analysis of Runge-Kutta methods for nonlinear neutral delay integro-differential equations
Institute of Scientific and Technical Information of China (English)
2007-01-01
The sufficient conditions for the stability and asymptotic stability of Runge-Kutta methods for nonlinear neutral delay integro-differential equations are derived. A numerical test that confirms the theoretical results is given in the end.
A stochastic collocation method for the second order wave equation with a discontinuous random speed
Motamed, Mohammad; Nobile, Fabio; Tempone, Raul
2012-01-01
In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical
The Full—Discrete Mixed Finite Element Methods for Nonlinear Hyperbolic Equations
Institute of Scientific and Technical Information of China (English)
YanpingCHEN; YunqingHUANG
1998-01-01
This article treats mixed finite element methods for second order nonlinear hyperbolic equations.A fully discrete scheme is presented and improved L2-error estimates are established.The convergence of both the function value andthe flux is demonstrated.
Fixed Point Methods in the Stability of the Cauchy Functional Equations
Directory of Open Access Journals (Sweden)
Z. Dehvari
2013-03-01
Full Text Available By using the fixed point methods, we prove some generalized Hyers-Ulam stability of homomorphisms for Cauchy and CauchyJensen functional equations on the product algebras and on the triple systems.
An Equal-Order DG Method for the Incompressible Navier-Stokes Equations
Cockburn, Bernardo; Kanschat, Guido; Schö tzau, Dominik
2008-01-01
We introduce and analyze a discontinuous Galerkin method for the incompressible Navier-Stokes equations that is based on finite element spaces of the same polynomial order for the approximation of the velocity and the pressure. Stability
Traveling Wave Solutions of ZK-BBM Equation Sine-Cosine Method
Directory of Open Access Journals (Sweden)
Sadaf Bibi
2014-03-01
Full Text Available Travelling wave solutions are obtained by using a relatively new technique which is called sine-cosine method for ZK-BBM equations. Solution procedure and obtained results re-confirm the efficiency of the proposed scheme.
Sound field computations in the Bay of Bengal using parabolic equation method
Digital Repository Service at National Institute of Oceanography (India)
Navelkar, G.S.; Somayajulu, Y.K.; Murty, C.S.
Effect of the cold core eddy in the Bay of Bengal on acoustic propagation was analysed by parabolic equation (PE) method. Source depth, frequency and propagation range considered respectively for the two numerical experiments are 150 m, 400 Hz, 650...
Xing, Yanyuan; Yan, Yubin
2018-03-01
Gao et al. [11] (2014) introduced a numerical scheme to approximate the Caputo fractional derivative with the convergence rate O (k 3 - α), 0 equation is sufficiently smooth, Lv and Xu [20] (2016) proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate O (k 3 - α), 0 equation has low regularity and in this case the numerical method fails to have the convergence rate O (k 3 - α), 0 quadratic interpolation polynomials. Based on this scheme, we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using Laplace transform methods that the time discretization scheme has the convergence rate O (k 3 - α), 0 0 for smooth and nonsmooth data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the theoretical results are consistent with the numerical results.
Wu, Zedong; Alkhalifah, Tariq Ali
2018-01-01
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
International Nuclear Information System (INIS)
Killingbeck, J.
1979-01-01
By using the methods of perturbation theory it is possible to construct simple formulae for the numerical integration of the Schroedinger equation, and also to calculate expectation values solely by means of simple eigenvalue calculations. (Auth.)
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.
Energy Technology Data Exchange (ETDEWEB)
Holden, Helge; Karlsen, Kenneth H.; Lie, Knut-Andreas
1999-10-01
We present and analyze a numerical method for the solution of a class of scalar, multi-dimensional, nonlinear degenerate convection-diffusion equations. The method is based on operator splitting to separate the convective and the diffusive terms in the governing equation. The nonlinear, convective part is solved using front tracking and dimensional splitting, while the nonlinear diffusion equation is solved by a suitable difference scheme. We verify L{sup 1} compactness of the corresponding set of approximate solutions and derive precise entropy estimates. In particular, these results allow us to pass to the limit in our approximations and recover an entropy solution of the problem in question. The theory presented covers a large class of equations. Important subclasses are hyperbolic conservation laws, porous medium type equations, two-phase reservoir flow equations, and strongly degenerate equations coming from the recent theory of sedimentation-consolidation processes. A thorough numerical investigation of the method analyzed in this paper (and similar methods) is presented in a companion paper. (author)
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.
International Nuclear Information System (INIS)
Yang, Xiao-Jun; Srivastava, H.M.; He, Ji-Huan; Baleanu, Dumitru
2013-01-01
In this Letter, we propose to use the Cantor-type cylindrical-coordinate method in order to investigate a family of local fractional differential operators on Cantor sets. Some testing examples are given to illustrate the capability of the proposed method for the heat-conduction equation on a Cantor set and the damped wave equation in fractal strings. It is seen to be a powerful tool to convert differential equations on Cantor sets from Cantorian-coordinate systems to Cantor-type cylindrical-coordinate systems.
Novel method for solution of coupled radial Schrödinger equations
International Nuclear Information System (INIS)
Ershov, S. N.; Vaagen, J. S.; Zhukov, M. V.
2011-01-01
One of the major problems in numerical solution of coupled differential equations is the maintenance of linear independence for different sets of solution vectors. A novel method for solution of radial Schrödinger equations is suggested. It consists of rearrangement of coupled equations in a way that is appropriate to avoid usual numerical instabilities associated with components of the wave function in their classically forbidden regions. Applications of the new method for nuclear structure calculations within the hyperspherical harmonics approach are given.
Solution of the generalized Emden-Fowler equations by the hybrid functions method
International Nuclear Information System (INIS)
Tabrizidooz, H R; Marzban, H R; Razzaghi, M
2009-01-01
In this paper, we present a numerical algorithm for solving the generalized Emden-Fowler equations, which have many applications in mathematical physics and astrophysics. The method is based on hybrid functions approximations. The properties of hybrid functions, which consist of block-pulse functions and Lagrange interpolating polynomials, are presented. These properties are then utilized to reduce the computation of the generalized Emden-Fowler equations to a system of nonlinear equations. The method is easy to implement and yields very accurate results.
New exact solutions of coupled Boussinesq–Burgers equations by Exp-function method
Directory of Open Access Journals (Sweden)
L.K. Ravi
2017-03-01
Full Text Available In the present paper, we build the new analytical exact solutions of a nonlinear differential equation, specifically, coupled Boussinesq–Burgers equations by means of Exp-function method. Then, we analyze the results by plotting the three dimensional soliton graphs for each case, which exhibit the simplicity and effectiveness of the proposed method. The primary purpose of this paper is to employ a new approach, which allows us victorious and efficient derivation of the new analytical exact solutions for the coupled Boussinesq–Burgers equations.
A Novel Method for Analytical Solutions of Fractional Partial Differential Equations
Directory of Open Access Journals (Sweden)
Mehmet Ali Akinlar
2013-01-01
Full Text Available A new solution technique for analytical solutions of fractional partial differential equations (FPDEs is presented. The solutions are expressed as a finite sum of a vector type functional. By employing MAPLE software, it is shown that the solutions might be extended to an arbitrary degree which makes the present method not only different from the others in the literature but also quite efficient. The method is applied to special Bagley-Torvik and Diethelm fractional differential equations as well as a more general fractional differential equation.
Rani, Monika; Bhatti, Harbax S.; Singh, Vikramjeet
2017-11-01
In optical communication, the behavior of the ultrashort pulses of optical solitons can be described through nonlinear Schrodinger equation. This partial differential equation is widely used to contemplate a number of physically important phenomena, including optical shock waves, laser and plasma physics, quantum mechanics, elastic media, etc. The exact analytical solution of (1+n)-dimensional higher order nonlinear Schrodinger equation by He's variational iteration method has been presented. Our proposed solutions are very helpful in studying the solitary wave phenomena and ensure rapid convergent series and avoid round off errors. Different examples with graphical representations have been given to justify the capability of the method.
Interacting multiagent systems kinetic equations and Monte Carlo methods
Pareschi, Lorenzo
2014-01-01
The description of emerging collective phenomena and self-organization in systems composed of large numbers of individuals has gained increasing interest from various research communities in biology, ecology, robotics and control theory, as well as sociology and economics. Applied mathematics is concerned with the construction, analysis and interpretation of mathematical models that can shed light on significant problems of the natural sciences as well as our daily lives. To this set of problems belongs the description of the collective behaviours of complex systems composed by a large enough number of individuals. Examples of such systems are interacting agents in a financial market, potential voters during political elections, or groups of animals with a tendency to flock or herd. Among other possible approaches, this book provides a step-by-step introduction to the mathematical modelling based on a mesoscopic description and the construction of efficient simulation algorithms by Monte Carlo methods. The ar...
Coarse Analysis of Microscopic Models using Equation-Free Methods
DEFF Research Database (Denmark)
Marschler, Christian
of these models might be high-dimensional, the properties of interest are usually macroscopic and lowdimensional in nature. Examples are numerous and not necessarily restricted to computer models. For instance, the power output, energy consumption and temperature of engines are interesting quantities....... Applications include the learning behavior in the barn owl’s auditory system, traffic jam formation in an optimal velocity model for circular car traffic and oscillating behavior of pedestrian groups in a counter-flow through a corridor with narrow door. The methods do not only quantify interesting properties...... in these models (learning outcome, traffic jam density, oscillation period), but also allow to investigate unstable solutions, which are important information to determine basins of attraction of stable solutions and thereby reveal information on the long-term behavior of an initial state....
Numerical methods for solving the governing equations for a seriated continuum
International Nuclear Information System (INIS)
Narum, R.E.; Noble, C.; Mortensen, G.A.; McFadden, J.H.
1976-09-01
A desire to more accurately predict the behavior of transient two-phase flows has resulted in an investigation of the feasibility of computing unequal phase velocities and unequal phase temperatures. The finite difference forms of a set of equations governing a seriated continuum are presented along with two methods developed for solving the resulting systems of simultaneous nonlinear equations. Results from a one-dimensional computer code are presented to illustrate the capabilities of one of the solution methods
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.
A Novel Method for Analytical Solutions of Fractional Partial Differential Equations
Mehmet Ali Akinlar; Muhammet Kurulay
2013-01-01
A new solution technique for analytical solutions of fractional partial differential equations (FPDEs) is presented. The solutions are expressed as a finite sum of a vector type functional. By employing MAPLE software, it is shown that the solutions might be extended to an arbitrary degree which makes the present method not only different from the others in the literature but also quite efficient. The method is applied to special Bagley-Torvik and Diethelm fractional differential equations as...
Exp-function method for constructing exact solutions of Sharma-Tasso-Olver equation
International Nuclear Information System (INIS)
Erbas, Baris; Yusufoglu, Elcin
2009-01-01
In this paper we use the Exp-function method for the analytic treatment of Sharma-Tasso-Olver equation. New solitonary solutions are formally derived. Change of parameters, which drastically changes the characteristics of the equations, is examined. It is shown that the Exp-function method provides a powerful mathematical tool for solving high-dimensional nonlinear evolutions in mathematical physics. The proposed schemes are reliable and manageable.
Partial differential equations with variable exponents variational methods and qualitative analysis
Radulescu, Vicentiu D
2015-01-01
Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis provides researchers and graduate students with a thorough introduction to the theory of nonlinear partial differential equations (PDEs) with a variable exponent, particularly those of elliptic type. The book presents the most important variational methods for elliptic PDEs described by nonhomogeneous differential operators and containing one or more power-type nonlinearities with a variable exponent. The authors give a systematic treatment of the basic mathematical theory and constructive meth
Inverse Free Iterative Methods for Nonlinear Ill-Posed Operator Equations
Directory of Open Access Journals (Sweden)
Ioannis K. Argyros
2014-01-01
ill-posed operator equation F(x=y. The proposed method is a modified form of Tikhonov gradient (TIGRA method considered by Ramlau (2003. The regularization parameter is chosen according to the balancing principle considered by Pereverzev and Schock (2005. The error estimate is derived under a general source condition and is of optimal order. Some numerical examples involving integral equations are also given in this paper.
International Nuclear Information System (INIS)
Dehghan, Mehdi; Shakeri, Fatemeh
2007-01-01
In this work, the solution of an inverse problem concerning a diffusion equation with source control parameters is presented. The homotopy perturbation method is employed to solve this equation. This method changes a difficult problem into a simple problem which can be easily solved. In this procedure, according to the homotopy technique, a homotopy with an embedding parameter p element of [0,1] is constructed, and this parameter is considered a 'small parameter', so the method is called the homotopy perturbation method, which can take full advantage of the traditional perturbation method and homotopy technique. The approximations obtained by the proposed method are uniformly valid not only for small parameters, but also for very large parameters. The fact that this technique, in contrast to the traditional perturbation methods, does not require a small parameter in the system, leads to wide applications in nonlinear equations
A Globally Convergent Matrix-Free Method for Constrained Equations and Its Linear Convergence Rate
Directory of Open Access Journals (Sweden)
Min Sun
2014-01-01
Full Text Available A matrix-free method for constrained equations is proposed, which is a combination of the well-known PRP (Polak-Ribière-Polyak conjugate gradient method and the famous hyperplane projection method. The new method is not only derivative-free, but also completely matrix-free, and consequently, it can be applied to solve large-scale constrained equations. We obtain global convergence of the new method without any differentiability requirement on the constrained equations. Compared with the existing gradient methods for solving such problem, the new method possesses linear convergence rate under standard conditions, and a relax factor γ is attached in the update step to accelerate convergence. Preliminary numerical results show that it is promising in practice.
A multilevel correction adaptive finite element method for Kohn-Sham equation
Hu, Guanghui; Xie, Hehu; Xu, Fei
2018-02-01
In this paper, an adaptive finite element method is proposed for solving Kohn-Sham equation with the multilevel correction technique. In the method, the Kohn-Sham equation is solved on a fixed and appropriately coarse mesh with the finite element method in which the finite element space is kept improving by solving the derived boundary value problems on a series of adaptively and successively refined meshes. A main feature of the method is that solving large scale Kohn-Sham system is avoided effectively, and solving the derived boundary value problems can be handled efficiently by classical methods such as the multigrid method. Hence, the significant acceleration can be obtained on solving Kohn-Sham equation with the proposed multilevel correction technique. The performance of the method is examined by a variety of numerical experiments.
Flux weighted method for solution of stiff neutron dynamic equations and its application
International Nuclear Information System (INIS)
Li Huiyun; Jiao Huixian
1987-12-01
To analyze reactivity event for nuclear power plants, it is necessary to solve the neutron dynamic equations, which is a group of typical stiff constant differential equations. Very small time steps could only be adopted when the group of equations is solved by common methods. However, a large time steps might be selected if the Flux Weighted Medthod introduced in this paper is used. Generally, weighted factor θ i1 is set as a constant. Naturally, this treatment method can decrease the accuracy of calculation for the increase of the steadiness of solving the equations. An accurate theoretical formula of 4 x 4 matrix of θ i1 is rigorously derived so that the accuracy of calculation is ensured, as well as the steadiness of solved equations is increased. This method have the advantage over classical Runge-kutta Method and other methods. The time steps could be increased by a factor of 1 ∼ 3 orders of magnitude so as to save a lot of computating time. The programe solving neutron dynamic equation, which is prepared by using Flux Weighted Method, could be sued for real time analog of training simulator, as well as for analysis and computation of reactivity event (including rod jumping out event)
New analytical exact solutions of time fractional KdV-KZK equation by Kudryashov methods
S Saha, Ray
2016-04-01
In this paper, new exact solutions of the time fractional KdV-Khokhlov-Zabolotskaya-Kuznetsov (KdV-KZK) equation are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann-Liouville derivative is used to convert the nonlinear time fractional KdV-KZK equation into the nonlinear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV-KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics. The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV-KZK equation.
New analytical exact solutions of time fractional KdV–KZK equation by Kudryashov methods
International Nuclear Information System (INIS)
Saha Ray, S
2016-01-01
In this paper, new exact solutions of the time fractional KdV–Khokhlov–Zabolotskaya–Kuznetsov (KdV–KZK) equation are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann–Liouville derivative is used to convert the nonlinear time fractional KdV–KZK equation into the nonlinear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV–KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics. The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV–KZK equation. (paper)
Spectral element method for vector radiative transfer equation
International Nuclear Information System (INIS)
Zhao, J.M.; Liu, L.H.; Hsu, P.-F.; Tan, J.Y.
2010-01-01
A spectral element method (SEM) is developed to solve polarized radiative transfer in multidimensional participating medium. The angular discretization is based on the discrete-ordinates approach, and the spatial discretization is conducted by spectral element approach. Chebyshev polynomial is used to build basis function on each element. Four various test problems are taken as examples to verify the performance of the SEM. The effectiveness of the SEM is demonstrated. The h and the p convergence characteristics of the SEM are studied. The convergence rate of p-refinement follows the exponential decay trend and is superior to that of h-refinement. The accuracy and efficiency of the higher order approximation in the SEM is well demonstrated for the solution of the VRTE. The predicted angular distribution of brightness temperature and Stokes vector by the SEM agree very well with the benchmark solutions in references. Numerical results show that the SEM is accurate, flexible and effective to solve multidimensional polarized radiative transfer problems.
A self-consistent nodal method in response matrix formalism for the multigroup diffusion equations
International Nuclear Information System (INIS)
Malambu, E.M.; Mund, E.H.
1996-01-01
We develop a nodal method for the multigroup diffusion equations, based on the transverse integration procedure (TIP). The efficiency of the method rests upon the convergence properties of a high-order multidimensional nodal expansion and upon numerical implementation aspects. The discrete 1D equations are cast in response matrix formalism. The derivation of the transverse leakage moments is self-consistent i.e. does not require additional assumptions. An outstanding feature of the method lies in the linear spatial shape of the local transverse leakage for the first-order scheme. The method is described in the two-dimensional case. The method is validated on some classical benchmark problems. (author)
Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations
Abdulle, Assyr
2013-01-01
We introduce a new family of explicit integrators for stiff Itô stochastic differential equations (SDEs) of weak order two. These numerical methods belong to the class of one-step stabilized methods with extended stability domains and do not suffer from the step size reduction faced by standard explicit methods. The family is based on the standard second order orthogonal Runge-Kutta-Chebyshev (ROCK2) methods for deterministic problems. The convergence, meansquare, and asymptotic stability properties of the methods are analyzed. Numerical experiments, including applications to nonlinear SDEs and parabolic stochastic partial differential equations are presented and confirm the theoretical results. © 2013 Society for Industrial and Applied Mathematics.
The Closest Point Method and Multigrid Solvers for Elliptic Equations on Surfaces
Chen, Yujia
2015-01-01
© 2015 Society for Industrial and Applied Mathematics. Elliptic partial differential equations are important from both application and analysis points of view. In this paper we apply the closest point method to solve elliptic equations on general curved surfaces. Based on the closest point representation of the underlying surface, we formulate an embedding equation for the surface elliptic problem, then discretize it using standard finite differences and interpolation schemes on banded but uniform Cartesian grids. We prove the convergence of the difference scheme for the Poisson\\'s equation on a smooth closed curve. In order to solve the resulting large sparse linear systems, we propose a specific geometric multigrid method in the setting of the closest point method. Convergence studies in both the accuracy of the difference scheme and the speed of the multigrid algorithm show that our approaches are effective.
Jiang, Lijian; Efendiev, Yalchin; Ginting, Victor
2010-01-01
In this paper, we discuss a numerical multiscale approach for solving wave equations with heterogeneous coefficients. Our interest comes from geophysics applications and we assume that there is no scale separation with respect to spatial variables. To obtain the solution of these multiscale problems on a coarse grid, we compute global fields such that the solution smoothly depends on these fields. We present a Galerkin multiscale finite element method using the global information and provide a convergence analysis when applied to solve the wave equations. We investigate the relation between the smoothness of the global fields and convergence rates of the global Galerkin multiscale finite element method for the wave equations. Numerical examples demonstrate that the use of global information renders better accuracy for wave equations with heterogeneous coefficients than the local multiscale finite element method. © 2010 IMACS.
Numerical solution of multi group-Two dimensional- Adjoint equation with finite element method
International Nuclear Information System (INIS)
Poursalehi, N.; Khalafi, H.; Shahriari, M.; Minoochehr
2008-01-01
Adjoint equation is used for perturbation theory in nuclear reactor design. For numerical solution of adjoint equation, usually two methods are applied. These are Finite Element and Finite Difference procedures. Usually Finite Element Procedure is chosen for solving of adjoint equation, because it is more use able in variety of geometries. In this article, Galerkin Finite Element method is discussed. This method is applied for numerical solving multi group, multi region and two dimensional (X, Y) adjoint equation. Typical reactor geometry is partitioned with triangular meshes and boundary condition for adjoint flux is considered zero. Finally, for a case of defined parameters, Finite Element Code was applied and results were compared with Citation Code
International Nuclear Information System (INIS)
Shang Yadong
2008-01-01
The extended hyperbolic functions method for nonlinear wave equations is presented. Based on this method, we obtain a multiple exact explicit solutions for the nonlinear evolution equations which describe the resonance interaction between the long wave and the short wave. The solutions obtained in this paper include (a) the solitary wave solutions of bell-type for S and L, (b) the solitary wave solutions of kink-type for S and bell-type for L, (c) the solitary wave solutions of a compound of the bell-type and the kink-type for S and L, (d) the singular travelling wave solutions, (e) periodic travelling wave solutions of triangle function types, and solitary wave solutions of rational function types. The variety of structure to the exact solutions of the long-short wave equation is illustrated. The methods presented here can also be used to obtain exact solutions of nonlinear wave equations in n dimensions
Jiang, Lijian
2010-08-01
In this paper, we discuss a numerical multiscale approach for solving wave equations with heterogeneous coefficients. Our interest comes from geophysics applications and we assume that there is no scale separation with respect to spatial variables. To obtain the solution of these multiscale problems on a coarse grid, we compute global fields such that the solution smoothly depends on these fields. We present a Galerkin multiscale finite element method using the global information and provide a convergence analysis when applied to solve the wave equations. We investigate the relation between the smoothness of the global fields and convergence rates of the global Galerkin multiscale finite element method for the wave equations. Numerical examples demonstrate that the use of global information renders better accuracy for wave equations with heterogeneous coefficients than the local multiscale finite element method. © 2010 IMACS.
Mathematical and computational methods for semiclassical Schrödinger equations
Jin, Shi
2011-04-28
We consider time-dependent (linear and nonlinear) Schrödinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high-frequency oscillations. The design of efficient numerical methods which produce an accurate approximation of the solutions, or at least of the associated physical observables, is a formidable mathematical challenge. In this article we shall review the basic analytical methods for dealing with such equations, including WKB asymptotics, Wigner measure techniques and Gaussian beams. Moreover, we shall give an overview of the current state of the art of numerical methods (most of which are based on the described analytical techniques) for the Schrödinger equation in the semiclassical regime. © 2011 Cambridge University Press.
Adjustment technique without explicit formation of normal equations /conjugate gradient method/
Saxena, N. K.
1974-01-01
For a simultaneous adjustment of a large geodetic triangulation system, a semiiterative technique is modified and used successfully. In this semiiterative technique, known as the conjugate gradient (CG) method, original observation equations are used, and thus the explicit formation of normal equations is avoided, 'huge' computer storage space being saved in the case of triangulation systems. This method is suitable even for very poorly conditioned systems where solution is obtained only after more iterations. A detailed study of the CG method for its application to large geodetic triangulation systems was done that also considered constraint equations with observation equations. It was programmed and tested on systems as small as two unknowns and three equations up to those as large as 804 unknowns and 1397 equations. When real data (573 unknowns, 965 equations) from a 1858-km-long triangulation system were used, a solution vector accurate to four decimal places was obtained in 2.96 min after 1171 iterations (i.e., 2.0 times the number of unknowns).
International Nuclear Information System (INIS)
Arum Sari, Resita; Suparmi, A; Cari, C
2016-01-01
The Dirac equation for Eckart potential and trigonometric Manning Rosen potential with exact spin symmetry is obtained using an asymptotic iteration method. The combination of the two potentials is substituted into the Dirac equation, then the variables are separated into radial and angular parts. The Dirac equation is solved by using an asymptotic iteration method that can reduce the second order differential equation into a differential equation with substitution variables of hypergeometry type. The relativistic energy is calculated using Matlab 2011. This study is limited to the case of spin symmetry. With the asymptotic iteration method, the energy spectra of the relativistic equations and equations of orbital quantum number l can be obtained, where both are interrelated between quantum numbers. The energy spectrum is also numerically solved using the Matlab software, where the increase in the radial quantum number n r causes the energy to decrease. The radial part and the angular part of the wave function are defined as hypergeometry functions and visualized with Matlab 2011. The results show that the disturbance of a combination of the Eckart potential and trigonometric Manning Rosen potential can change the radial part and the angular part of the wave function. (paper)
Application of an analytical method for solution of thermal hydraulic conservation equations
Energy Technology Data Exchange (ETDEWEB)
Fakory, M.R. [Simulation, Systems & Services Technologies Company (S3 Technologies), Columbia, MD (United States)
1995-09-01
An analytical method has been developed and applied for solution of two-phase flow conservation equations. The test results for application of the model for simulation of BWR transients are presented and compared with the results obtained from application of the explicit method for integration of conservation equations. The test results show that with application of the analytical method for integration of conservation equations, the Courant limitation associated with explicit Euler method of integration was eliminated. The results obtained from application of the analytical method (with large time steps) agreed well with the results obtained from application of explicit method of integration (with time steps smaller than the size imposed by Courant limitation). The results demonstrate that application of the analytical approach significantly improves the numerical stability and computational efficiency.
Khater method for nonlinear Sharma Tasso-Olever (STO) equation of fractional order
Bibi, Sadaf; Mohyud-Din, Syed Tauseef; Khan, Umar; Ahmed, Naveed
In this work, we have implemented a direct method, known as Khater method to establish exact solutions of nonlinear partial differential equations of fractional order. Number of solutions provided by this method is greater than other traditional methods. Exact solutions of nonlinear fractional order Sharma Tasso-Olever (STO) equation are expressed in terms of kink, travelling wave, periodic and solitary wave solutions. Modified Riemann-Liouville derivative and Fractional complex transform have been used for compatibility with fractional order sense. Solutions have been graphically simulated for understanding the physical aspects and importance of the method. A comparative discussion between our established results and the results obtained by existing ones is also presented. Our results clearly reveal that the proposed method is an effective, powerful and straightforward technique to work out new solutions of various types of differential equations of non-integer order in the fields of applied sciences and engineering.
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
International Nuclear Information System (INIS)
Park, Yujin; Kazantzis, Nikolaos; Parlos, Alexander G.; Chong, Kil To
2013-01-01
Highlights: • Numerical solution for stiff differential equations using matrix exponential method. • The approximation is based on First Order Hold assumption. • Various input examples applied to the point kinetics equations. • The method shows superior useful and effective activity. - Abstract: A system of nonlinear differential equations is derived to model the dynamics of neutron density and the delayed neutron precursors within a point kinetics equation modeling framework for a nuclear reactor. The point kinetic equations are mathematically characterized as stiff, occasionally nonlinear, ordinary differential equations, posing significant challenges when numerical solutions are sought and traditionally resulting in the need for smaller time step intervals within various computational schemes. In light of the above realization, the present paper proposes a new discretization method inspired by system-theoretic notions and technically based on a combination of the matrix exponential method (MEM) and the First-Order Hold (FOH) assumption. Under the proposed time discretization structure, the sampled-data representation of the nonlinear point kinetic system of equations is derived. The performance of the proposed time discretization procedure is evaluated using several case studies with sinusoidal reactivity profiles and multiple input examples (reactivity and neutron source function). It is shown, that by applying the proposed method under a First-Order Hold for the neutron density and the precursor concentrations at each time step interval, the stiffness problem associated with the point kinetic equations can be adequately addressed and resolved. Finally, as evidenced by the aforementioned detailed simulation studies, the proposed method retains its validity and accuracy for a wide range of reactor operating conditions, including large sampling periods dictated by physical and/or technical limitations associated with the current state of sensor and
Asiri, Sharefa M.; Laleg-Kirati, Taous-Meriem
2016-01-01
In this paper, modulating functions-based method is proposed for estimating space–time-dependent unknowns in one-dimensional partial differential equations. The proposed method simplifies the problem into a system of algebraic equations linear
Longitudinal beta-binomial modeling using GEE for overdispersed binomial data.
Wu, Hongqian; Zhang, Ying; Long, Jeffrey D
2017-03-15
Longitudinal binomial data are frequently generated from multiple questionnaires and assessments in various scientific settings for which the binomial data are often overdispersed. The standard generalized linear mixed effects model may result in severe underestimation of standard errors of estimated regression parameters in such cases and hence potentially bias the statistical inference. In this paper, we propose a longitudinal beta-binomial model for overdispersed binomial data and estimate the regression parameters under a probit model using the generalized estimating equation method. A hybrid algorithm of the Fisher scoring and the method of moments is implemented for computing the method. Extensive simulation studies are conducted to justify the validity of the proposed method. Finally, the proposed method is applied to analyze functional impairment in subjects who are at risk of Huntington disease from a multisite observational study of prodromal Huntington disease. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.
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.
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)
International Nuclear Information System (INIS)
Biazar, J.; Eslami, M.; Aminikhah, H.
2009-01-01
In this article, an application of He's homotopy perturbation method is applied to solve systems of Volterra integral equations of the first kind. Some non-linear examples are prepared to illustrate the efficiency and simplicity of the method. Applying the method for linear systems is so easily that it does not worth to have any example.
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.
Steady-state transport equation resolution by particle methods, and numerical results
International Nuclear Information System (INIS)
Mercier, B.
1985-10-01
A method to solve steady-state transport equation has been given. Principles of the method are given. The method is studied in two different cases; estimations given by the theory are compared to numerical results. Results got in 1-D (spherical geometry) and in 2-D (axisymmetric geometry) are given [fr
A Special Variant of the Moment Method for Fredholm Integral Equations of the Second Kind
Directory of Open Access Journals (Sweden)
S. A. Solov’eva
2015-01-01
Full Text Available We consider the linear Fredholm integral equation of the second kind, where the kernel and the free term are smooth functions. We find the unknown function in this class as well.Exact and approximate methods for the solution of linear Fredholm integral equations of the second kind are well developed. However, classical methods do not take into account the structural properties of the kernel and the free term of equation.In this paper we develop and justify a special variant of the moment method to solve this equation, which takes into account the differential properties of initial data. The proposed paper furthers studies of N.S Gabbasov, I.P. Kasakina, and S.A Solov’eva. We use approximation theory, version of the general theory of approximate methods of analysis that Gabdulkhayev B.G suggested, and methods of functional analysis to prove theorems. In addition, we use N.S. Gabbasov’s ideas and methods in papers that are devoted to the Fredholm equations of the first kind, as well as N.S. Gabbasov and S.A Solov’eva’s investigations on the Fredholm equations of the third kind in the space of distributions.The first part of the paper provides a description of the basic function space and elements of the theory of approximation in it.In the second part we propose and theoretically justify a generalized moment method. We have demonstrated that the improvement of differential properties of the initial data improves the approximation accuracy. Since, in practice, the approximate equations are solved, as a rule, only approximately, we prove the stability and causality of the proposed method. The resulting estimate of the paper is in good agreement with the estimate for the ordinary moment method for equations of the second kind in the space of continuous functions.In the final section we have shown that a developed method is optimal in order of accuracy among all polynomial projection methods to solve Fredholm integral equations of the second
International Nuclear Information System (INIS)
Yurov, A.V.; Yurova, A.A.
2006-01-01
The simple algebraic method for construction of exact solutions of two-dimensional hydrodynamic equations of incompressible flow is proposed. This method can be applied both to nonviscous flow (Euler equations) and to viscous flow (Navier-Stokes equations). In the case of nonviscous flow, the problem is reduced to sequential solving of three linear partial differential equations. In the case of viscous flow, the Navier-Stokes equations are reduced to three linear partial differential equations and one differential equation of the first order [ru
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 Telegraph Equation and Its Solution by Reduced Differential Transform Method
Directory of Open Access Journals (Sweden)
Vineet K. Srivastava
2013-01-01
Full Text Available One-dimensional second-order hyperbolic telegraph equation was formulated using Ohm’s law and solved by a recent and reliable semianalytic method, namely, the reduced differential transform method (RDTM. Using this method, it is possible to find the exact solution or a closed approximate solution of a differential equation. Three numerical examples have been carried out in order to check the effectiveness, the accuracy, and convergence of the method. The RDTM is a powerful mathematical technique for solving wide range of problems arising in science and engineering fields.
Zhong, Jiaqi; Zeng, Cheng; Yuan, Yupeng; Zhang, Yuzhe; Zhang, Ye
2018-04-01
The aim of this paper is to present an explicit numerical algorithm based on improved spectral Galerkin method for solving the unsteady diffusion-convection-reaction equation. The principal characteristics of this approach give the explicit eigenvalues and eigenvectors based on the time-space separation method and boundary condition analysis. With the help of Fourier series and Galerkin truncation, we can obtain the finite-dimensional ordinary differential equations which facilitate the system analysis and controller design. By comparing with the finite element method, the numerical solutions are demonstrated via two examples. It is shown that the proposed method is effective.
International Nuclear Information System (INIS)
Gamba, Irene M.; Haack, Jeffrey R.
2014-01-01
We present the formulation of a conservative spectral method for the Boltzmann collision operator with anisotropic scattering cross-sections. The method is an extension of the conservative spectral method of Gamba and Tharkabhushanam [17,18], which uses the weak form of the collision operator to represent the collisional term as a weighted convolution in Fourier space. The method is tested by computing the collision operator with a suitably cut-off angular cross section and comparing the results with the solution of the Landau equation. We analytically study the convergence rate of the Fourier transformed Boltzmann collision operator in the grazing collisions limit to the Fourier transformed Landau collision operator under the assumption of some regularity and decay conditions of the solution to the Boltzmann equation. Our results show that the angular singularity which corresponds to the Rutherford scattering cross section is the critical singularity for which a grazing collision limit exists for the Boltzmann operator. Additionally, we numerically study the differences between homogeneous solutions of the Boltzmann equation with the Rutherford scattering cross section and an artificial cross section, which give convergence to solutions of the Landau equation at different asymptotic rates. We numerically show the rate of the approximation as well as the consequences for the rate of entropy decay for homogeneous solutions of the Boltzmann equation and Landau equation
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.
DeVille, R. E. Lee; Harkin, Anthony; Holzer, Matt; Josić, Krešimir; Kaper, Tasso J.
2008-06-01
For singular perturbation problems, the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. E. 49 (1994) 4502-4511] has been shown to be an effective general approach for deriving reduced or amplitude equations that govern the long time dynamics of the system. It has been applied to a variety of problems traditionally analyzed using disparate methods, including the method of multiple scales, boundary layer theory, the WKBJ method, the Poincaré-Lindstedt method, the method of averaging, and others. In this article, we show how the RG method may be used to generate normal forms for large classes of ordinary differential equations. First, we apply the RG method to systems with autonomous perturbations, and we show that the reduced or amplitude equations generated by the RG method are equivalent to the classical Poincaré-Birkhoff normal forms for these systems up to and including terms of O(ɛ2), where ɛ is the perturbation parameter. This analysis establishes our approach and generalizes to higher order. Second, we apply the RG method to systems with nonautonomous perturbations, and we show that the reduced or amplitude equations so generated constitute time-asymptotic normal forms, which are based on KBM averages. Moreover, for both classes of problems, we show that the main coordinate changes are equivalent, up to translations between the spaces in which they are defined. In this manner, our results show that the RG method offers a new approach for deriving normal forms for nonautonomous systems, and it offers advantages since one can typically more readily identify resonant terms from naive perturbation expansions than from the nonautonomous vector fields themselves. Finally, we establish how well the solution to the RG equations approximates the solution of the original equations on time scales of O(1/ɛ).