Spline Collocation Method for Nonlinear Multi-Term Fractional Differential Equation

OpenAIRE

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...

GHM method for obtaining rationalsolutions of nonlinear differential equations.

Science.gov (United States)

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.

Direct application of Padé approximant for solving nonlinear differential equations.

Science.gov (United States)

Vazquez-Leal, Hector; Benhammouda, Brahim; Filobello-Nino, Uriel; Sarmiento-Reyes, Arturo; Jimenez-Fernandez, Victor Manuel; Garcia-Gervacio, Jose Luis; Huerta-Chua, Jesus; Morales-Mendoza, Luis Javier; Gonzalez-Lee, Mario

2014-01-01

This work presents a direct procedure to apply Padé method to find approximate solutions for nonlinear differential equations. Moreover, we present some cases study showing the strength of the method to generate highly accurate rational approximate solutions compared to other semi-analytical methods. The type of tested nonlinear equations are: a highly nonlinear boundary value problem, a differential-algebraic oscillator problem, and an asymptotic problem. The high accurate handy approximations obtained by the direct application of Padé method shows the high potential if the proposed scheme to approximate a wide variety of problems. What is more, the direct application of the Padé approximant aids to avoid the previous application of an approximative method like Taylor series method, homotopy perturbation method, Adomian Decomposition method, homotopy analysis method, variational iteration method, among others, as tools to obtain a power series solutions to post-treat with the Padé approximant. 34L30.

GDTM-Padé technique for the non-linear differential-difference equation

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Lu Jun-Feng

2013-01-01

Full Text Available This paper focuses on applying the GDTM-Padé technique to solve the non-linear differential-difference equation. The bell-shaped solitary wave solution of Belov-Chaltikian lattice equation is considered. Comparison between the approximate solutions and the exact ones shows that this technique is an efficient and attractive method for solving the differential-difference equations.

Solving Fuzzy Fractional Differential Equations Using Zadeh's Extension Principle

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

2013-01-01

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

Bäcklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations

International Nuclear Information System (INIS)

Lu, Bin

2012-01-01

In this Letter, the fractional derivatives in the sense of modified Riemann–Liouville derivative and the Bäcklund transformation of fractional Riccati equation are employed for constructing the exact solutions of nonlinear fractional partial differential equations. The power of this manageable method is presented by applying it to several examples. This approach can also be applied to other nonlinear fractional differential equations. -- Highlights: ► Backlund transformation of fractional Riccati equation is presented. ► A new method for solving nonlinear fractional differential equations is proposed. ► Three important fractional differential equations are solved successfully. ► Some new exact solutions of the fractional differential equations are obtained.

An ansatz for solving nonlinear partial differential equations in mathematical physics.

Science.gov (United States)

Akbar, M Ali; Ali, Norhashidah Hj Mohd

2016-01-01

In this article, we introduce an ansatz involving exact traveling wave solutions to nonlinear partial differential equations. To obtain wave solutions using direct method, the choice of an appropriate ansatz is of great importance. We apply this ansatz to examine new and further general traveling wave solutions to the (1+1)-dimensional modified Benjamin-Bona-Mahony equation. Abundant traveling wave solutions are derived including solitons, singular solitons, periodic solutions and general solitary wave solutions. The solutions emphasize the nobility of this ansatz in providing distinct solutions to various tangible phenomena in nonlinear science and engineering. The ansatz could be more efficient tool to deal with higher dimensional nonlinear evolution equations which frequently arise in many real world physical problems.

Maintaining the stability of nonlinear differential equations by the enhancement of HPM

International Nuclear Information System (INIS)

Hosein Nia, S.H.; Ranjbar, A.N.; Ganji, D.D.; Soltani, H.; Ghasemi, J.

2008-01-01

Homotopy perturbation method is an effective method to find a solution of a nonlinear differential equation. In this method, a nonlinear complex differential equation is transformed to a series of linear and nonlinear parts, almost simpler differential equations. These sets of equations are then solved iteratively. Finally, a linear series of the solutions completes the answer if the convergence is maintained. In this Letter, the need for stability verification is shown through some examples. Consequently, HPM is enhanced by a preliminary assumption. The idea is to keep the inherent stability of nonlinear dynamic, even the selected linear part is not

A procedure to construct exact solutions of nonlinear fractional differential equations.

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Güner, Özkan; Cevikel, Adem C

2014-01-01

We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions.

On Solving the Lorenz System by Differential Transformation Method

International Nuclear Information System (INIS)

Al-Sawalha, M. Mossa; Noorani, M. S. M.

2008-01-01

The differential transformation method (DTM) is employed to solve a nonlinear differential equation, namely the Lorenz system. Numerical results are compared to those obtained by the Runge–Kutta method to illustrate the preciseness and effectiveness of the proposed method. In particular, we examine the accuracy of the (DTM) as the Lorenz system changes from a non-chaotic system to a chaotic one. It is shown that the (DTM) is robust, accurate and easy to apply

Solving Differential Equations in R: Package deSolve

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In this paper we present the R package deSolve to solve initial value problems (IVP) written as ordinary differential equations (ODE), differential algebraic equations (DAE) of index 0 or 1 and partial differential equations (PDE), the latter solved using the method of lines appr...

Solving Differential Equations in R: Package deSolve

NARCIS (Netherlands)

Soetaert, K.E.R.; Petzoldt, T.; Setzer, R.W.

2010-01-01

In this paper we present the R package deSolve to solve initial value problems (IVP) written as ordinary differential equations (ODE), differential algebraic equations (DAE) of index 0 or 1 and partial differential equations (PDE), the latter solved using the method of lines approach. The

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

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.

Approximate Solutions of Nonlinear Partial Differential Equations by Modified q-Homotopy Analysis Method

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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.

Backward stochastic differential equations from linear to fully nonlinear theory

CERN Document Server

Zhang, Jianfeng

2017-01-01

This book provides a systematic and accessible approach to stochastic differential equations, backward stochastic differential equations, and their connection with partial differential equations, as well as the recent development of the fully nonlinear theory, including nonlinear expectation, second order backward stochastic differential equations, and path dependent partial differential equations. Their main applications and numerical algorithms, as well as many exercises, are included. The book focuses on ideas and clarity, with most results having been solved from scratch and most theories being motivated from applications. It can be considered a starting point for junior researchers in the field, and can serve as a textbook for a two-semester graduate course in probability theory and stochastic analysis. It is also accessible for graduate students majoring in financial engineering.

Convergence criteria for systems of nonlinear elliptic partial differential equations

International Nuclear Information System (INIS)

Sharma, R.K.

1986-01-01

This thesis deals with convergence criteria for a special system of nonlinear elliptic partial differential equations. A fixed-point algorithm is used, which iteratively solves one linearized elliptic partial differential equation at a time. Conditions are established that help foresee the convergence of the algorithm. Under reasonable hypotheses it is proved that the algorithm converges for such nonlinear elliptic systems. Extensive experimental results are reported and they show the algorithm converges in a wide variety of cases and the convergence is well correlated with the theoretical conditions introduced in this thesis

Application of differential transformation method for solving dengue transmission mathematical model

Science.gov (United States)

Ndii, Meksianis Z.; Anggriani, Nursanti; Supriatna, Asep K.

2018-03-01

The differential transformation method (DTM) is a semi-analytical numerical technique which depends on Taylor series and has application in many areas including Biomathematics. The aim of this paper is to employ the differential transformation method (DTM) to solve system of non-linear differential equations for dengue transmission mathematical model. Analytical and numerical solutions are determined and the results are compared to that of Runge-Kutta method. We found a good agreement between DTM and Runge-Kutta method.

Bright and dark soliton solutions for some nonlinear fractional differential equations

International Nuclear Information System (INIS)

Guner, Ozkan; Bekir, Ahmet

2016-01-01

In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space–time fractional modified Benjamin–Bona–Mahoney (mBBM) equation, the time fractional mKdV equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions: amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional derivatives are described in the modified Riemann–Liouville sense. (paper)

A multiple-scale power series method for solving nonlinear ordinary differential equations

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Chein-Shan Liu

2016-02-01

Full Text Available The power series solution is a cheap and effective method to solve nonlinear problems, like the Duffing-van der Pol oscillator, the Volterra population model and the nonlinear boundary value problems. A novel power series method by considering the multiple scales $R_k$ in the power term $(t/R_k^k$ is developed, which are derived explicitly to reduce the ill-conditioned behavior in the data interpolation. In the method a huge value times a tiny value is avoided, such that we can decrease the numerical instability and which is the main reason to cause the failure of the conventional power series method. The multiple scales derived from an integral can be used in the power series expansion, which provide very accurate numerical solutions of the problems considered in this paper.

A neuro approach to solve fuzzy Riccati differential equations

Energy Technology Data Exchange (ETDEWEB)

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

2015-10-22

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

Solving nonlinear evolution equation system using two different methods

Science.gov (United States)

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

2015-12-01

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

Iterative Methods for Solving Nonlinear Parabolic Problem in Pension Saving Management

Science.gov (United States)

Koleva, M. N.

2011-11-01

In this work we consider a nonlinear parabolic equation, obtained from Riccati like transformation of the Hamilton-Jacobi-Bellman equation, arising in pension saving management. We discuss two numerical iterative methods for solving the model problem—fully implicit Picard method and mixed Picard-Newton method, which preserves the parabolic characteristics of the differential problem. Numerical experiments for comparison the accuracy and effectiveness of the algorithms are discussed. Finally, observations are given.

Two-dimensional differential transform method for solving linear and non-linear Schroedinger equations

International Nuclear Information System (INIS)

Ravi Kanth, A.S.V.; Aruna, K.

2009-01-01

In this paper, we propose a reliable algorithm to develop exact and approximate solutions for the linear and nonlinear Schroedinger equations. The approach rest mainly on two-dimensional differential transform method which is one of the approximate methods. The method can easily be applied to many linear and nonlinear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. Several illustrative examples are given to demonstrate the effectiveness of the present method.

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

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Inci Cilingir Sungu

2015-01-01

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

On multilevel RBF collocation to solve nonlinear PDEs arising from endogenous stochastic volatility models

Science.gov (United States)

Bastani, Ali Foroush; Dastgerdi, Maryam Vahid; Mighani, Abolfazl

2018-06-01

The main aim of this paper is the analytical and numerical study of a time-dependent second-order nonlinear partial differential equation (PDE) arising from the endogenous stochastic volatility model, introduced in [Bensoussan, A., Crouhy, M. and Galai, D., Stochastic equity volatility related to the leverage effect (I): equity volatility behavior. Applied Mathematical Finance, 1, 63-85, 1994]. As the first step, we derive a consistent set of initial and boundary conditions to complement the PDE, when the firm is financed by equity and debt. In the sequel, we propose a Newton-based iteration scheme for nonlinear parabolic PDEs which is an extension of a method for solving elliptic partial differential equations introduced in [Fasshauer, G. E., Newton iteration with multiquadrics for the solution of nonlinear PDEs. Computers and Mathematics with Applications, 43, 423-438, 2002]. The scheme is based on multilevel collocation using radial basis functions (RBFs) to solve the resulting locally linearized elliptic PDEs obtained at each level of the Newton iteration. We show the effectiveness of the resulting framework by solving a prototypical example from the field and compare the results with those obtained from three different techniques: (1) a finite difference discretization; (2) a naive RBF collocation and (3) a benchmark approximation, introduced for the first time in this paper. The numerical results confirm the robustness, higher convergence rate and good stability properties of the proposed scheme compared to other alternatives. We also comment on some possible research directions in this field.

Nonlinear partial differential equations of second order

CERN Document Server

Dong, Guangchang

1991-01-01

This book addresses a class of equations central to many areas of mathematics and its applications. Although there is no routine way of solving nonlinear partial differential equations, effective approaches that apply to a wide variety of problems are available. This book addresses a general approach that consists of the following: Choose an appropriate function space, define a family of mappings, prove this family has a fixed point, and study various properties of the solution. The author emphasizes the derivation of various estimates, including a priori estimates. By focusing on a particular approach that has proven useful in solving a broad range of equations, this book makes a useful contribution to the literature.

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

International Nuclear Information System (INIS)

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

1996-01-01

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

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

KAUST Repository

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

2014-01-01

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

A new multi-step technique with differential transform method for analytical solution of some nonlinear variable delay differential equations.

Science.gov (United States)

Benhammouda, Brahim; Vazquez-Leal, Hector

2016-01-01

This work presents an analytical solution of some nonlinear delay differential equations (DDEs) with variable delays. Such DDEs are difficult to treat numerically and cannot be solved by existing general purpose codes. A new method of steps combined with the differential transform method (DTM) is proposed as a powerful tool to solve these DDEs. This method reduces the DDEs to ordinary differential equations that are then solved by the DTM. Furthermore, we show that the solutions can be improved by Laplace-Padé resummation method. Two examples are presented to show the efficiency of the proposed technique. The main advantage of this technique is that it possesses a simple procedure based on a few straight forward steps and can be combined with any analytical method, other than the DTM, like the homotopy perturbation method.

Nonlinear differential equations

Energy Technology Data Exchange (ETDEWEB)

Dresner, L.

1988-01-01

This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.

Nonlinear differential equations

International Nuclear Information System (INIS)

Dresner, L.

1988-01-01

This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics

Lattice Boltzmann model for high-order nonlinear partial differential equations.

Science.gov (United States)

Chai, Zhenhua; He, Nanzhong; Guo, Zhaoli; Shi, Baochang

2018-01-01

In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂_{t}ϕ+∑_{k=1}^{m}α_{k}∂_{x}^{k}Π_{k}(ϕ)=0 (1≤k≤m≤6), α_{k} are constant coefficients, Π_{k}(ϕ) are some known differential functions of ϕ. As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K(n,n)-Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009)1672-179910.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009)PHYADX0378-437110.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.

Lattice Boltzmann model for high-order nonlinear partial differential equations

Science.gov (United States)

Chai, Zhenhua; He, Nanzhong; Guo, Zhaoli; Shi, Baochang

2018-01-01

In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂tϕ +∑k=1mαk∂xkΠk(ϕ ) =0 (1 ≤k ≤m ≤6 ), αk are constant coefficients, Πk(ϕ ) are some known differential functions of ϕ . As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K (n ,n ) -Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009), 10.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009), 10.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.

Numerical Simulation of Coupled Nonlinear Schrödinger Equations Using the Generalized Differential Quadrature Method

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)

Exploring inductive linearization for pharmacokinetic-pharmacodynamic systems of nonlinear ordinary differential equations.

Science.gov (United States)

Hasegawa, Chihiro; Duffull, Stephen B

2018-02-01

Pharmacokinetic-pharmacodynamic systems are often expressed with nonlinear ordinary differential equations (ODEs). While there are numerous methods to solve such ODEs these methods generally rely on time-stepping solutions (e.g. Runge-Kutta) which need to be matched to the characteristics of the problem at hand. The primary aim of this study was to explore the performance of an inductive approximation which iteratively converts nonlinear ODEs to linear time-varying systems which can then be solved algebraically or numerically. The inductive approximation is applied to three examples, a simple nonlinear pharmacokinetic model with Michaelis-Menten elimination (E1), an integrated glucose-insulin model and an HIV viral load model with recursive feedback systems (E2 and E3, respectively). The secondary aim of this study was to explore the potential advantages of analytically solving linearized ODEs with two examples, again E3 with stiff differential equations and a turnover model of luteinizing hormone with a surge function (E4). The inductive linearization coupled with a matrix exponential solution provided accurate predictions for all examples with comparable solution time to the matched time-stepping solutions for nonlinear ODEs. The time-stepping solutions however did not perform well for E4, particularly when the surge was approximated by a square wave. In circumstances when either a linear ODE is particularly desirable or the uncertainty in matching the integrator to the ODE system is of potential risk, then the inductive approximation method coupled with an analytical integration method would be an appropriate alternative.

Solving Differential Equations in R: Package deSolve

Directory of Open Access Journals (Sweden)

Karline Soetaert

2010-02-01

Full Text Available In this paper we present the R package deSolve to solve initial value problems (IVP written as ordinary differential equations (ODE, differential algebraic equations (DAE of index 0 or 1 and partial differential equations (PDE, the latter solved using the method of lines approach. The differential equations can be represented in R code or as compiled code. In the latter case, R is used as a tool to trigger the integration and post-process the results, which facilitates model development and application, whilst the compiled code significantly increases simulation speed. The methods implemented are efficient, robust, and well documented public-domain Fortran routines. They include four integrators from the ODEPACK package (LSODE, LSODES, LSODA, LSODAR, DVODE and DASPK2.0. In addition, a suite of Runge-Kutta integrators and special-purpose solvers to efficiently integrate 1-, 2- and 3-dimensional partial differential equations are available. The routines solve both stiff and non-stiff systems, and include many options, e.g., to deal in an efficient way with the sparsity of the Jacobian matrix, or finding the root of equations. In this article, our objectives are threefold: (1 to demonstrate the potential of using R for dynamic modeling, (2 to highlight typical uses of the different methods implemented and (3 to compare the performance of models specified in R code and in compiled code for a number of test cases. These comparisons demonstrate that, if the use of loops is avoided, R code can efficiently integrate problems comprising several thousands of state variables. Nevertheless, the same problem may be solved from 2 to more than 50 times faster by using compiled code compared to an implementation using only R code. Still, amongst the benefits of R are a more flexible and interactive implementation, better readability of the code, and access to R’s high-level procedures. deSolve is the successor of package odesolve which will be deprecated in

Nonlinear elliptic partial differential equations an introduction

CERN Document Server

Le Dret, Hervé

2018-01-01

This textbook presents the essential parts of the modern theory of nonlinear partial differential equations, including the calculus of variations. After a short review of results in real and functional analysis, the author introduces the main mathematical techniques for solving both semilinear and quasilinear elliptic PDEs, and the associated boundary value problems. Key topics include infinite dimensional fixed point methods, the Galerkin method, the maximum principle, elliptic regularity, and the calculus of variations. Aimed at graduate students and researchers, this textbook contains numerous examples and exercises and provides several comments and suggestions for further study.

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

International Nuclear Information System (INIS)

Levi, D.

1979-01-01

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

Solving Linear Differential Equations

NARCIS (Netherlands)

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

2010-01-01

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

Convergent Power Series of sech⁡(x and Solutions to Nonlinear Differential Equations

Directory of Open Access Journals (Sweden)

U. Al Khawaja

2018-01-01

Full Text Available It is known that power series expansion of certain functions such as sech⁡(x diverges beyond a finite radius of convergence. We present here an iterative power series expansion (IPS to obtain a power series representation of sech⁡(x that is convergent for all x. The convergent series is a sum of the Taylor series of sech⁡(x and a complementary series that cancels the divergence of the Taylor series for x≥π/2. The method is general and can be applied to other functions known to have finite radius of convergence, such as 1/(1+x2. A straightforward application of this method is to solve analytically nonlinear differential equations, which we also illustrate here. The method provides also a robust and very efficient numerical algorithm for solving nonlinear differential equations numerically. A detailed comparison with the fourth-order Runge-Kutta method and extensive analysis of the behavior of the error and CPU time are performed.

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.

Fault detection and diagnosis in nonlinear systems a differential and algebraic viewpoint

CERN Document Server

Martinez-Guerra, Rafael

2014-01-01

The high reliability required in industrial processes has created the necessity of detecting abnormal conditions, called faults, while processes are operating. The term fault generically refers to any type of process degradation, or degradation in equipment performance because of changes in the process's physical characteristics, process inputs or environmental conditions. This book is about the fundamentals of fault detection and diagnosis in a variety of nonlinear systems which are represented by ordinary differential equations. The fault detection problem is approached from a differential algebraic viewpoint, using residual generators based upon high-gain nonlinear auxiliary systems (‘observers’). A prominent role is played by the type of mathematical tools that will be used, requiring knowledge of differential algebra and differential equations. Specific theorems tailored to the needs of the problem-solving procedures are developed and proved. Applications to real-world problems, both with constant an...

An explicit MOT scheme for solving the TD-EFVIE on nonlinear and dispersive scatterers

KAUST Repository

Sayed, Sadeed Bin; Ulku, H. Arda; Bagci, Hakan

2017-01-01

An explicit marching-on-in-time (MOT) scheme for solving the time domain electric field volume integral equation (TD-EFVIE) on nonlinear and dispersive scatterers is described. The unknown electric field intensity, electric flux density, and polarization densities representing Kerr nonlinearity along with Lorentz dispersion relation, all of which are induced inside the scatterer upon excitation, are expanded using half and full Schaubert-Wilton-Glisson functions in space. The TD-EFVIE and the constitutive relations between polarization, field, and flux terms are cast in the form of a first-order ordinary differential equation. The resulting matrix system is integrated in time using a predictor-corrector scheme to obtain the time dependent unknown expansion coefficients. The resulting MOT scheme is explicit and accounts for nonlinearity by simple function evaluations.

An explicit MOT scheme for solving the TD-EFVIE on nonlinear and dispersive scatterers

KAUST Repository

Sayed, Sadeed Bin

2017-10-25

An explicit marching-on-in-time (MOT) scheme for solving the time domain electric field volume integral equation (TD-EFVIE) on nonlinear and dispersive scatterers is described. The unknown electric field intensity, electric flux density, and polarization densities representing Kerr nonlinearity along with Lorentz dispersion relation, all of which are induced inside the scatterer upon excitation, are expanded using half and full Schaubert-Wilton-Glisson functions in space. The TD-EFVIE and the constitutive relations between polarization, field, and flux terms are cast in the form of a first-order ordinary differential equation. The resulting matrix system is integrated in time using a predictor-corrector scheme to obtain the time dependent unknown expansion coefficients. The resulting MOT scheme is explicit and accounts for nonlinearity by simple function evaluations.

Neural network error correction for solving coupled ordinary differential equations

Science.gov (United States)

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

1992-01-01

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

A Numerical Algorithm for Solving a Four-Point Nonlinear Fractional Integro-Differential Equations

OpenAIRE

Gao, Er; Song, Songhe; Zhang, Xinjian

2012-01-01

We provide a new algorithm for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order q∈(1,2] based on reproducing kernel space method. According to our work, the analytical solution of the equations is represented in the reproducing kernel space which we construct and so the n-term approximation. At the same time, the n-term approximation is proved to converge to the analytical solution. An illustrative example is also presented, which sh...

Solving Large Scale Nonlinear Eigenvalue Problem in Next-Generation Accelerator Design

Energy Technology Data Exchange (ETDEWEB)

Liao, Ben-Shan; Bai, Zhaojun; /UC, Davis; Lee, Lie-Quan; Ko, Kwok; /SLAC

2006-09-28

A number of numerical methods, including inverse iteration, method of successive linear problem and nonlinear Arnoldi algorithm, are studied in this paper to solve a large scale nonlinear eigenvalue problem arising from finite element analysis of resonant frequencies and external Q{sub e} values of a waveguide loaded cavity in the next-generation accelerator design. They present a nonlinear Rayleigh-Ritz iterative projection algorithm, NRRIT in short and demonstrate that it is the most promising approach for a model scale cavity design. The NRRIT algorithm is an extension of the nonlinear Arnoldi algorithm due to Voss. Computational challenges of solving such a nonlinear eigenvalue problem for a full scale cavity design are outlined.

A Numerical Algorithm for Solving a Four-Point Nonlinear Fractional Integro-Differential Equations

Directory of Open Access Journals (Sweden)

Er Gao

2012-01-01

Full Text Available We provide a new algorithm for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order q∈(1,2] based on reproducing kernel space method. According to our work, the analytical solution of the equations is represented in the reproducing kernel space which we construct and so the n-term approximation. At the same time, the n-term approximation is proved to converge to the analytical solution. An illustrative example is also presented, which shows that the new algorithm is efficient and accurate.

Numerical approximations of nonlinear fractional differential difference equations by using modified He-Laplace method

Directory of Open Access Journals (Sweden)

J. Prakash

2016-03-01

Full Text Available In this paper, a numerical algorithm based on a modified He-Laplace method (MHLM is proposed to solve space and time nonlinear fractional differential-difference equations (NFDDEs arising in physical phenomena such as wave phenomena in fluids, coupled nonlinear optical waveguides and nanotechnology fields. The modified He-Laplace method is a combined form of the fractional homotopy perturbation method and Laplace transforms method. The nonlinear terms can be easily decomposed by the use of He’s polynomials. This algorithm has been tested against time-fractional differential-difference equations such as the modified Lotka Volterra and discrete (modified KdV equations. The proposed scheme grants the solution in the form of a rapidly convergent series. Three examples have been employed to illustrate the preciseness and effectiveness of the proposed method. The achieved results expose that the MHLM is very accurate, efficient, simple and can be applied to other nonlinear FDDEs.

Symbolic computation of analytic approximate solutions for nonlinear fractional differential equations

Science.gov (United States)

Lin, Yezhi; Liu, Yinping; Li, Zhibin

2013-01-01

The Adomian decomposition method (ADM) is one of the most effective methods to construct analytic approximate solutions for nonlinear differential equations. In this paper, based on the new definition of the Adomian polynomials, Rach (2008) [22], the Adomian decomposition method and the Padé approximants technique, a new algorithm is proposed to construct analytic approximate solutions for nonlinear fractional differential equations with initial or boundary conditions. Furthermore, a MAPLE software package is developed to implement this new algorithm, which is user-friendly and efficient. One only needs to input the system equation, initial or boundary conditions and several necessary parameters, then our package will automatically deliver the analytic approximate solutions within a few seconds. Several different types of examples are given to illustrate the scope and demonstrate the validity of our package, especially for non-smooth initial value problems. Our package provides a helpful and easy-to-use tool in science and engineering simulations. Program summaryProgram title: ADMP Catalogue identifier: AENE_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENE_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 12011 No. of bytes in distributed program, including test data, etc.: 575551 Distribution format: tar.gz Programming language: MAPLE R15. Computer: PCs. Operating system: Windows XP/7. RAM: 2 Gbytes Classification: 4.3. Nature of problem: Constructing analytic approximate solutions of nonlinear fractional differential equations with initial or boundary conditions. Non-smooth initial value problems can be solved by this program. Solution method: Based on the new definition of the Adomian polynomials [1], the Adomian decomposition method and the Pad

Unsteady Solution of Non-Linear Differential Equations Using Walsh Function Series

Science.gov (United States)

Gnoffo, Peter A.

2015-01-01

Walsh functions form an orthonormal basis set consisting of square waves. The discontinuous nature of square waves make the system well suited for representing functions with discontinuities. The product of any two Walsh functions is another Walsh function - a feature that can radically change an algorithm for solving non-linear partial differential equations (PDEs). The solution algorithm of non-linear differential equations using Walsh function series is unique in that integrals and derivatives may be computed using simple matrix multiplication of series representations of functions. Solutions to PDEs are derived as functions of wave component amplitude. Three sample problems are presented to illustrate the Walsh function series approach to solving unsteady PDEs. These include an advection equation, a Burgers equation, and a Riemann problem. The sample problems demonstrate the use of the Walsh function solution algorithms, exploiting Fast Walsh Transforms in multi-dimensions (O(Nlog(N))). Details of a Fast Walsh Reciprocal, defined here for the first time, enable inversion of aWalsh Symmetric Matrix in O(Nlog(N)) operations. Walsh functions have been derived using a fractal recursion algorithm and these fractal patterns are observed in the progression of pairs of wave number amplitudes in the solutions. These patterns are most easily observed in a remapping defined as a fractal fingerprint (FFP). A prolongation of existing solutions to the next highest order exploits these patterns. The algorithms presented here are considered a work in progress that provide new alternatives and new insights into the solution of non-linear PDEs.

Generalized solutions of nonlinear partial differential equations

CERN Document Server

Rosinger, EE

1987-01-01

During the last few years, several fairly systematic nonlinear theories of generalized solutions of rather arbitrary nonlinear partial differential equations have emerged. The aim of this volume is to offer the reader a sufficiently detailed introduction to two of these recent nonlinear theories which have so far contributed most to the study of generalized solutions of nonlinear partial differential equations, bringing the reader to the level of ongoing research.The essence of the two nonlinear theories presented in this volume is the observation that much of the mathematics concernin

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

Science.gov (United States)

Tripathi, Rajnee; Mishra, Hradyesh Kumar

2016-01-01

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

Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear partial differential evolution equations of dynamical systems

Institute of Scientific and Technical Information of China (English)

2008-01-01

Using functional derivative technique in quantum field theory, the algebraic dy-namics approach for solution of ordinary differential evolution equations was gen-eralized to treat partial differential evolution equations. The partial differential evo-lution equations were lifted to the corresponding functional partial differential equations in functional space by introducing the time translation operator. The functional partial differential evolution equations were solved by algebraic dynam-ics. The algebraic dynamics solutions are analytical in Taylor series in terms of both initial functions and time. Based on the exact analytical solutions, a new nu-merical algorithm—algebraic dynamics algorithm was proposed for partial differ-ential evolution equations. The difficulty of and the way out for the algorithm were discussed. The application of the approach to and computer numerical experi-ments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically.

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

International Nuclear Information System (INIS)

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

1989-01-01

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

On new classes of solutions of nonlinear partial differential equations in the form of convergent special series

Science.gov (United States)

Filimonov, M. Yu.

2017-12-01

The method of special series with recursively calculated coefficients is used to solve nonlinear partial differential equations. The recurrence of finding the coefficients of the series is achieved due to a special choice of functions, in powers of which the solution is expanded in a series. We obtain a sequence of linear partial differential equations to find the coefficients of the series constructed. In many cases, one can deal with a sequence of linear ordinary differential equations. We construct classes of solutions in the form of convergent series for a certain class of nonlinear evolution equations. A new class of solutions of generalized Boussinesque equation with an arbitrary function in the form of a convergent series is constructed.

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

International Nuclear Information System (INIS)

Huber, Alfred

2007-01-01

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

Quasi-exact solutions of nonlinear differential equations

OpenAIRE

Kudryashov, Nikolay A.; Kochanov, Mark B.

2014-01-01

The concept of quasi-exact solutions of nonlinear differential equations is introduced. Quasi-exact solution expands the idea of exact solution for additional values of parameters of differential equation. These solutions are approximate solutions of nonlinear differential equations but they are close to exact solutions. Quasi-exact solutions of the the Kuramoto--Sivashinsky, the Korteweg--de Vries--Burgers and the Kawahara equations are founded.

State-dependent differential Riccati equation to track control of time-varying systems with state and control nonlinearities.

Science.gov (United States)

Korayem, M H; Nekoo, S R

2015-07-01

This work studies an optimal control problem using the state-dependent Riccati equation (SDRE) in differential form to track for time-varying systems with state and control nonlinearities. The trajectory tracking structure provides two nonlinear differential equations: the state-dependent differential Riccati equation (SDDRE) and the feed-forward differential equation. The independence of the governing equations and stability of the controller are proven along the trajectory using the Lyapunov approach. Backward integration (BI) is capable of solving the equations as a numerical solution; however, the forward solution methods require the closed-form solution to fulfill the task. A closed-form solution is introduced for SDDRE, but the feed-forward differential equation has not yet been obtained. Different ways of solving the problem are expressed and analyzed. These include BI, closed-form solution with corrective assumption, approximate solution, and forward integration. Application of the tracking problem is investigated to control robotic manipulators possessing rigid or flexible joints. The intention is to release a general program for automatic implementation of an SDDRE controller for any manipulator that obeys the Denavit-Hartenberg (D-H) principle when only D-H parameters are received as input data. Copyright © 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Stochastic Computational Approach for Complex Nonlinear Ordinary Differential Equations

International Nuclear Information System (INIS)

Khan, Junaid Ali; Raja, Muhammad Asif Zahoor; Qureshi, Ijaz Mansoor

2011-01-01

We present an evolutionary computational approach for the solution of nonlinear ordinary differential equations (NLODEs). The mathematical modeling is performed by a feed-forward artificial neural network that defines an unsupervised error. The training of these networks is achieved by a hybrid intelligent algorithm, a combination of global search with genetic algorithm and local search by pattern search technique. The applicability of this approach ranges from single order NLODEs, to systems of coupled differential equations. We illustrate the method by solving a variety of model problems and present comparisons with solutions obtained by exact methods and classical numerical methods. The solution is provided on a continuous finite time interval unlike the other numerical techniques with comparable accuracy. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed. (general)

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

Energy Technology Data Exchange (ETDEWEB)

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

2016-12-15

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

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

International Conference on Differential Equations and Nonlinear Mechanics

CERN Document Server

2001-01-01

The International Conference on Differential Equations and Nonlinear Mechanics was hosted by the University of Central Florida in Orlando from March 17-19, 1999. One of the conference days was dedicated to Professor V. Lakshmikantham in th honor of his 75 birthday. 50 well established professionals (in differential equations, nonlinear analysis, numerical analysis, and nonlinear mechanics) attended the conference from 13 countries. Twelve of the attendees delivered hour long invited talks and remaining thirty-eight presented invited forty-five minute talks. In each of these talks, the focus was on the recent developments in differential equations and nonlinear mechanics and their applications. This book consists of 29 papers based on the invited lectures, and I believe that it provides a good selection of advanced topics of current interest in differential equations and nonlinear mechanics. I am indebted to the Department of Mathematics, College of Arts and Sciences, Department of Mechanical, Materials and Ae...

Application of power series to the solution of the boundary value problem for a second order nonlinear differential equation

International Nuclear Information System (INIS)

Semenova, V.N.

2016-01-01

A boundary value problem for a nonlinear second order differential equation has been considered. A numerical method has been proposed to solve this problem using power series. Results of numerical experiments have been presented in the paper [ru

Fuchs indices and the first integrals of nonlinear differential equations

International Nuclear Information System (INIS)

Kudryashov, Nikolai A.

2005-01-01

New method of finding the first integrals of nonlinear differential equations in polynomial form is presented. Basic idea of our approach is to use the scaling of solution of nonlinear differential equation and to find the dimensions of arbitrary constants in the Laurent expansion of the general solution. These dimensions allows us to obtain the scalings of members for the first integrals of nonlinear differential equations. Taking the polynomials with unknown coefficients into account we present the algorithm of finding the first integrals of nonlinear differential equations in the polynomial form. Our method is applied to look for the first integrals of eight nonlinear ordinary differential equations of the fourth order. The general solution of one of the fourth order ordinary differential equations is given

Parallel Algorithm Solves Coupled Differential Equations

Science.gov (United States)

Hayashi, A.

1987-01-01

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

Improved algorithm for solving nonlinear parabolized stability equations

International Nuclear Information System (INIS)

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

2016-01-01

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

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

International Nuclear Information System (INIS)

An Hengbin; Mo Zeyao; Xu Xiaowen; Liu Xu

2009-01-01

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

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

International Nuclear Information System (INIS)

Feng Qing-Hua

2014-01-01

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

Exact Solutions for Nonlinear Differential Difference Equations in Mathematical Physics

Directory of Open Access Journals (Sweden)

Khaled A. Gepreel

2013-01-01

Full Text Available We modified the truncated expansion method to construct the exact solutions for some nonlinear differential difference equations in mathematical physics via the general lattice equation, the discrete nonlinear Schrodinger with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system. Also, we put a rational solitary wave function method to find the rational solitary wave solutions for some nonlinear differential difference equations. The proposed methods are more effective and powerful to obtain the exact solutions for nonlinear difference differential equations.

Algorithms For Integrating Nonlinear Differential Equations

Science.gov (United States)

Freed, A. D.; Walker, K. P.

1994-01-01

Improved algorithms developed for use in numerical integration of systems of nonhomogenous, nonlinear, first-order, ordinary differential equations. In comparison with integration algorithms, these algorithms offer greater stability and accuracy. Several asymptotically correct, thereby enabling retention of stability and accuracy when large increments of independent variable used. Accuracies attainable demonstrated by applying them to systems of nonlinear, first-order, differential equations that arise in study of viscoplastic behavior, spread of acquired immune-deficiency syndrome (AIDS) virus and predator/prey populations.

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

Science.gov (United States)

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

2013-09-01

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

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

Science.gov (United States)

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

2016-11-01

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

Robustness of Operational Matrices of Differentiation for Solving State-Space Analysis and Optimal Control Problems

Directory of Open Access Journals (Sweden)

Emran Tohidi

2013-01-01

Full Text Available The idea of approximation by monomials together with the collocation technique over a uniform mesh for solving state-space analysis and optimal control problems (OCPs has been proposed in this paper. After imposing the Pontryagins maximum principle to the main OCPs, the problems reduce to a linear or nonlinear boundary value problem. In the linear case we propose a monomial collocation matrix approach, while in the nonlinear case, the general collocation method has been applied. We also show the efficiency of the operational matrices of differentiation with respect to the operational matrices of integration in our numerical examples. These matrices of integration are related to the Bessel, Walsh, Triangular, Laguerre, and Hermite functions.

Empirical Differential Balancing for Nonlinear Systems

NARCIS (Netherlands)

Kawano, Yu; Scherpen, Jacquelien M.A.; Dochain, Denis; Henrion, Didier; Peaucelle, Dimitri

In this paper, we consider empirical balancing of nonlinear systems by using its prolonged system, which consists of the original nonlinear system and its variational system. For the prolonged system, we define differential reachability and observability Gramians, which are matrix valued functions

Improved algorithm for solving nonlinear parabolized stability equations

Science.gov (United States)

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

2016-08-01

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

Spurious Solutions Of Nonlinear Differential Equations

Science.gov (United States)

Yee, H. C.; Sweby, P. K.; Griffiths, D. F.

1992-01-01

Report utilizes nonlinear-dynamics approach to investigate possible sources of errors and slow convergence and non-convergence of steady-state numerical solutions when using time-dependent approach for problems containing nonlinear source terms. Emphasizes implications for development of algorithms in CFD and computational sciences in general. Main fundamental conclusion of study is that qualitative features of nonlinear differential equations cannot be adequately represented by finite-difference method and vice versa.

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

Science.gov (United States)

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

2018-02-01

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

Numerical solution of two-dimensional non-linear partial differential ...

African Journals Online (AJOL)

linear partial differential equations using a hybrid method. The solution technique involves discritizing the non-linear system of partial differential equations (PDEs) to obtain a corresponding nonlinear system of algebraic difference equations to be ...

Analytical exact solution of the non-linear Schroedinger equation

International Nuclear Information System (INIS)

Martins, Alisson Xavier; Rocha Filho, Tarcisio Marciano da

2011-01-01

Full text: In this work we present how to classify and obtain analytical solutions of the Schroedinger equation with a generic non-linearity in 1+1 dimensions. Our approach is based on the determination of Lie symmetry transformation mapping solutions into solutions, and non-classical symmetry transformations, mapping a given solution into itself. From these symmetries it is then possible to reduce the equation to a system of ordinary differential equations which can then be solved using standard methods. The generic non-linearity is handled by considering it as an additional unknown in the determining equations for the symmetry transformations. This results in an over-determined system of non-linear partial differential equations. Its solution can then be determined in some cases by reducing it to the so called involutive (triangular) form, and then solved. This reduction is very tedious and can only performed using a computer algebra system. Once the determining system is solved, we obtain the explicit form for the non-linearity admitting a Lie or non-classical symmetry. The analytical solutions are then derived by solving the reduced ordinary differential equations. The non-linear determining system for the non-classical symmetry transformations and Lie symmetry generators are obtaining using the computer algebra package SADE (symmetry analysis of differential equations), developed at our group. (author)

Solution of Nonlinear Partial Differential Equations by New Laplace Variational Iteration Method

Directory of Open Access Journals (Sweden)

Eman M. A. Hilal

2014-01-01

Full Text Available The aim of this study is to give a good strategy for solving some linear and nonlinear partial differential equations in engineering and physics fields, by combining Laplace transform and the modified variational iteration method. This method is based on the variational iteration method, Laplace transforms, and convolution integral, introducing an alternative Laplace correction functional and expressing the integral as a convolution. Some examples in physical engineering are provided to illustrate the simplicity and reliability of this method. The solutions of these examples are contingent only on the initial conditions.

Solving Partial Differential Equations Using a New Differential Evolution Algorithm

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Natee Panagant

2014-01-01

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

Solving applied mathematical problems with Matlab

CERN Document Server

Xue, Dingyu

2008-01-01

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

Augmented nonlinear differentiator design and application to nonlinear uncertain systems.

Science.gov (United States)

Shao, Xingling; Liu, Jun; Li, Jie; Cao, Huiliang; Shen, Chong; Zhang, Xiaoming

2017-03-01

In this paper, an augmented nonlinear differentiator (AND) based on sigmoid function is developed to calculate the noise-less time derivative under noisy measurement condition. The essential philosophy of proposed AND in achieving high attenuation of noise effect is established by expanding the signal dynamics with extra state variable representing the integrated noisy measurement, then with the integral of measurement as input, the augmented differentiator is formulated to improve the estimation quality. The prominent advantages of the present differentiation technique are: (i) better noise suppression ability can be achieved without appreciable delay; (ii) the improved methodology can be readily extended to construct augmented high-order differentiator to obtain multiple derivatives. In addition, the convergence property and robustness performance against noises are investigated via singular perturbation theory and describing function method, respectively. Also, comparison with several classical differentiators is given to illustrate the superiority of AND in noise suppression. Finally, the robust control problems of nonlinear uncertain systems, including a numerical example and a mass spring system, are addressed to demonstrate the effectiveness of AND in precisely estimating the disturbance and providing the unavailable differential estimate to implement output feedback based controller. Copyright © 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Jacobi Elliptic Solutions for Nonlinear Differential Difference Equations in Mathematical Physics

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Khaled A. Gepreel

2012-01-01

Full Text Available We put a direct new method to construct the rational Jacobi elliptic solutions for nonlinear differential difference equations which may be called the rational Jacobi elliptic functions method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential difference equations in mathematical physics via the lattice equation and the discrete nonlinear Schrodinger equation with a saturable nonlinearity. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference equations.

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

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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.

Handbook of Nonlinear Partial Differential Equations

CERN Document Server

Polyanin, Andrei D

2011-01-01

New to the Second Edition More than 1,000 pages with over 1,500 new first-, second-, third-, fourth-, and higher-order nonlinear equations with solutions Parabolic, hyperbolic, elliptic, and other systems of equations with solutions Some exact methods and transformations Symbolic and numerical methods for solving nonlinear PDEs with Maple(t), Mathematica(R), and MATLAB(R) Many new illustrative examples and tables A large list of references consisting of over 1,300 sources To accommodate different mathematical backgrounds, the authors avoid wherever possible the use of special terminology. They

Entropy and convexity for nonlinear partial differential equations.

Science.gov (United States)

Ball, John M; Chen, Gui-Qiang G

2013-12-28

Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological and social processes. The concept of entropy originated in thermodynamics and statistical physics during the nineteenth century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear partial differential equations in recent decades. This opening article of the Theme Issue is intended to provide an introduction to entropy, convexity and related nonlinear methods for the analysis of nonlinear partial differential equations. We also provide a brief discussion about the content and contributions of the papers that make up this Theme Issue.

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

KAUST Repository

Bagci, Hakan

2015-01-07

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

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

KAUST Repository

Bagci, Hakan

2015-01-01

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

DISPL-1, 2. Order Nonlinear Partial Differential Equation System Solution for Kinetics Diffusion Problems

International Nuclear Information System (INIS)

Leaf, G.K.; Minkoff, M.

1982-01-01

1 - Description of problem or function: DISPL1 is a software package for solving second-order nonlinear systems of partial differential equations including parabolic, elliptic, hyperbolic, and some mixed types. The package is designed primarily for chemical kinetics- diffusion problems, although not limited to these problems. Fairly general nonlinear boundary conditions are allowed as well as inter- face conditions for problems in an inhomogeneous medium. The spatial domain is one- or two-dimensional with rectangular Cartesian, cylindrical, or spherical (in one dimension only) geometry. 2 - Method of solution: The numerical method is based on the use of Galerkin's procedure combined with the use of B-Splines (C.W.R. de-Boor's B-spline package) to generate a system of ordinary differential equations. These equations are solved by a sophisticated ODE software package which is a modified version of Hindmarsh's GEAR package, NESC Abstract 592. 3 - Restrictions on the complexity of the problem: The spatial domain must be rectangular with sides parallel to the coordinate geometry. Cross derivative terms are not permitted in the PDE. The order of the B-Splines is at most 12. Other parameters such as the number of mesh points in each coordinate direction, the number of PDE's etc. are set in a macro table used by the MORTRAn2 preprocessor in generating the object code

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

Science.gov (United States)

Khader, M M

2013-10-01

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

Neural network based online simultaneous policy update algorithm for solving the HJI equation in nonlinear H∞ control.

Science.gov (United States)

Wu, Huai-Ning; Luo, Biao

2012-12-01

It is well known that the nonlinear H∞ state feedback control problem relies on the solution of the Hamilton-Jacobi-Isaacs (HJI) equation, which is a nonlinear partial differential equation that has proven to be impossible to solve analytically. In this paper, a neural network (NN)-based online simultaneous policy update algorithm (SPUA) is developed to solve the HJI equation, in which knowledge of internal system dynamics is not required. First, we propose an online SPUA which can be viewed as a reinforcement learning technique for two players to learn their optimal actions in an unknown environment. The proposed online SPUA updates control and disturbance policies simultaneously; thus, only one iterative loop is needed. Second, the convergence of the online SPUA is established by proving that it is mathematically equivalent to Newton's method for finding a fixed point in a Banach space. Third, we develop an actor-critic structure for the implementation of the online SPUA, in which only one critic NN is needed for approximating the cost function, and a least-square method is given for estimating the NN weight parameters. Finally, simulation studies are provided to demonstrate the effectiveness of the proposed algorithm.

Global stability, periodic solutions, and optimal control in a nonlinear differential delay model

Directory of Open Access Journals (Sweden)

Anatoli F. Ivanov

2010-09-01

Full Text Available A nonlinear differential equation with delay serving as a mathematical model of several applied problems is considered. Sufficient conditions for the global asymptotic stability and for the existence of periodic solutions are given. Two particular applications are treated in detail. The first one is a blood cell production model by Mackey, for which new periodicity criteria are derived. The second application is a modified economic model with delay due to Ramsey. An optimization problem for a maximal consumption is stated and solved for the latter.

A Table Lookup Method for Exact Analytical Solutions of Nonlinear Fractional Partial Differential Equations

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

Science.gov (United States)

Chedjou, Jean Chamberlain; Kyamakya, Kyandoghere

2015-04-01

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

Nonlinear differential equations with exact solutions expressed via the Weierstrass function

NARCIS (Netherlands)

Kudryashov, NA

2004-01-01

A new problem is studied, that is to find nonlinear differential equations with special solutions expressed via the Weierstrass function. A method is discussed to construct nonlinear ordinary differential equations with exact solutions. The main step of our method is the assumption that nonlinear

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

International Nuclear Information System (INIS)

Kim, Ki Hong; Lee, Dong Hun

1999-01-01

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

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

Directory of Open Access Journals (Sweden)

San-Yang Liu

2014-01-01

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

Nonlinear dynamics of quadratically cubic systems

International Nuclear Information System (INIS)

Rudenko, O V

2013-01-01

We propose a modified form of the well-known nonlinear dynamic equations with quadratic relations used to model a cubic nonlinearity. We show that such quadratically cubic equations sometimes allow exact solutions and sometimes make the original problem easier to analyze qualitatively. Occasionally, exact solutions provide a useful tool for studying new phenomena. Examples considered include nonlinear ordinary differential equations and Hopf, Burgers, Korteweg–de Vries, and nonlinear Schrödinger partial differential equations. Some problems are solved exactly in the space–time and spectral representations. Unsolved problems potentially solvable by the proposed approach are listed. (methodological notes)

Analysis of an Nth-order nonlinear differential-delay equation

Science.gov (United States)

Vallée, Réal; Marriott, Christopher

1989-01-01

The problem of a nonlinear dynamical system with delay and an overall response time which is distributed among N individual components is analyzed. Such a system can generally be modeled by an Nth-order nonlinear differential delay equation. A linear-stability analysis as well as a numerical simulation of that equation are performed and a comparison is made with the experimental results. Finally, a parallel is established between the first-order differential equation with delay and the Nth-order differential equation without delay.

Nonlinear control and filtering using differential flatness approaches applications to electromechanical systems

CERN Document Server

Rigatos, Gerasimos G

2015-01-01

This monograph presents recent advances in differential flatness theory and analyzes its use for nonlinear control and estimation. It shows how differential flatness theory can provide solutions to complicated control problems, such as those appearing in highly nonlinear multivariable systems and distributed-parameter systems. Furthermore, it shows that differential flatness theory makes it possible to perform filtering and state estimation for a wide class of nonlinear dynamical systems and provides several descriptive test cases. The book focuses on the design of nonlinear adaptive controllers and nonlinear filters, using exact linearization based on differential flatness theory. The adaptive controllers obtained can be applied to a wide class of nonlinear systems with unknown dynamics, and assure reliable functioning of the control loop under uncertainty and varying operating conditions. The filters obtained outperform other nonlinear filters in terms of accuracy of estimation and computation speed. The bo...

[Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (2)].

Science.gov (United States)

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.

ADM For Solving Linear Second-Order Fredholm Integro-Differential Equations

Science.gov (United States)

Karim, Mohd F.; Mohamad, Mahathir; Saifullah Rusiman, Mohd; Che-Him, Norziha; Roslan, Rozaini; Khalid, Kamil

2018-04-01

In this paper, we apply Adomian Decomposition Method (ADM) as numerically analyse linear second-order Fredholm Integro-differential Equations. The approximate solutions of the problems are calculated by Maple package. Some numerical examples have been considered to illustrate the ADM for solving this equation. The results are compared with the existing exact solution. Thus, the Adomian decomposition method can be the best alternative method for solving linear second-order Fredholm Integro-Differential equation. It converges to the exact solution quickly and in the same time reduces computational work for solving the equation. The result obtained by ADM shows the ability and efficiency for solving these equations.

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.

Symbolic computation of analytic approximate solutions for nonlinear differential equations with initial conditions

Science.gov (United States)

Lin, Yezhi; Liu, Yinping; Li, Zhibin

2012-01-01

The Adomian decomposition method (ADM) is one of the most effective methods for constructing analytic approximate solutions of nonlinear differential equations. In this paper, based on the new definition of the Adomian polynomials, and the two-step Adomian decomposition method (TSADM) combined with the Padé technique, a new algorithm is proposed to construct accurate analytic approximations of nonlinear differential equations with initial conditions. Furthermore, a MAPLE package is developed, which is user-friendly and efficient. One only needs to input a system, initial conditions and several necessary parameters, then our package will automatically deliver analytic approximate solutions within a few seconds. Several different types of examples are given to illustrate the validity of the package. Our program provides a helpful and easy-to-use tool in science and engineering to deal with initial value problems. Program summaryProgram title: NAPA Catalogue identifier: AEJZ_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEJZ_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 4060 No. of bytes in distributed program, including test data, etc.: 113 498 Distribution format: tar.gz Programming language: MAPLE R13 Computer: PC Operating system: Windows XP/7 RAM: 2 Gbytes Classification: 4.3 Nature of problem: Solve nonlinear differential equations with initial conditions. Solution method: Adomian decomposition method and Padé technique. Running time: Seconds at most in routine uses of the program. Special tasks may take up to some minutes.

3D early embryogenesis image filtering by nonlinear partial differential equations.

Science.gov (United States)

Krivá, Z; Mikula, K; Peyriéras, N; Rizzi, B; Sarti, A; Stasová, O

2010-08-01

We present nonlinear diffusion equations, numerical schemes to solve them and their application for filtering 3D images obtained from laser scanning microscopy (LSM) of living zebrafish embryos, with a goal to identify the optimal filtering method and its parameters. In the large scale applications dealing with analysis of 3D+time embryogenesis images, an important objective is a correct detection of the number and position of cell nuclei yielding the spatio-temporal cell lineage tree of embryogenesis. The filtering is the first and necessary step of the image analysis chain and must lead to correct results, removing the noise, sharpening the nuclei edges and correcting the acquisition errors related to spuriously connected subregions. In this paper we study such properties for the regularized Perona-Malik model and for the generalized mean curvature flow equations in the level-set formulation. A comparison with other nonlinear diffusion filters, like tensor anisotropic diffusion and Beltrami flow, is also included. All numerical schemes are based on the same discretization principles, i.e. finite volume method in space and semi-implicit scheme in time, for solving nonlinear partial differential equations. These numerical schemes are unconditionally stable, fast and naturally parallelizable. The filtering results are evaluated and compared first using the Mean Hausdorff distance between a gold standard and different isosurfaces of original and filtered data. Then, the number of isosurface connected components in a region of interest (ROI) detected in original and after the filtering is compared with the corresponding correct number of nuclei in the gold standard. Such analysis proves the robustness and reliability of the edge preserving nonlinear diffusion filtering for this type of data and lead to finding the optimal filtering parameters for the studied models and numerical schemes. Further comparisons consist in ability of splitting the very close objects which

Exp-function method for solving fractional partial differential equations.

Science.gov (United States)

Zheng, Bin

2013-01-01

We extend the Exp-function method to fractional partial differential equations in the sense of modified Riemann-Liouville derivative based on nonlinear fractional complex transformation. For illustrating the validity of this method, we apply it to the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation. As a result, some new exact solutions for them are successfully established.

Evolutionary algorithm based heuristic scheme for nonlinear heat transfer equations.

Science.gov (United States)

Ullah, Azmat; Malik, Suheel Abdullah; Alimgeer, Khurram Saleem

2018-01-01

In this paper, a hybrid heuristic scheme based on two different basis functions i.e. Log Sigmoid and Bernstein Polynomial with unknown parameters is used for solving the nonlinear heat transfer equations efficiently. The proposed technique transforms the given nonlinear ordinary differential equation into an equivalent global error minimization problem. Trial solution for the given nonlinear differential equation is formulated using a fitness function with unknown parameters. The proposed hybrid scheme of Genetic Algorithm (GA) with Interior Point Algorithm (IPA) is opted to solve the minimization problem and to achieve the optimal values of unknown parameters. The effectiveness of the proposed scheme is validated by solving nonlinear heat transfer equations. The results obtained by the proposed scheme are compared and found in sharp agreement with both the exact solution and solution obtained by Haar Wavelet-Quasilinearization technique which witnesses the effectiveness and viability of the suggested scheme. Moreover, the statistical analysis is also conducted for investigating the stability and reliability of the presented scheme.

Evolutionary algorithm based heuristic scheme for nonlinear heat transfer equations.

Directory of Open Access Journals (Sweden)

Azmat Ullah

Full Text Available In this paper, a hybrid heuristic scheme based on two different basis functions i.e. Log Sigmoid and Bernstein Polynomial with unknown parameters is used for solving the nonlinear heat transfer equations efficiently. The proposed technique transforms the given nonlinear ordinary differential equation into an equivalent global error minimization problem. Trial solution for the given nonlinear differential equation is formulated using a fitness function with unknown parameters. The proposed hybrid scheme of Genetic Algorithm (GA with Interior Point Algorithm (IPA is opted to solve the minimization problem and to achieve the optimal values of unknown parameters. The effectiveness of the proposed scheme is validated by solving nonlinear heat transfer equations. The results obtained by the proposed scheme are compared and found in sharp agreement with both the exact solution and solution obtained by Haar Wavelet-Quasilinearization technique which witnesses the effectiveness and viability of the suggested scheme. Moreover, the statistical analysis is also conducted for investigating the stability and reliability of the presented scheme.

Regarding on the exact solutions for the nonlinear fractional differential equations

Directory of Open Access Journals (Sweden)

Kaplan Melike

2016-01-01

Full Text Available In this work, we have considered the modified simple equation (MSE method for obtaining exact solutions of nonlinear fractional-order differential equations. The space-time fractional equal width (EW and the modified equal width (mEW equation are considered for illustrating the effectiveness of the algorithm. It has been observed that all exact solutions obtained in this paper verify the nonlinear ordinary differential equations which was obtained from nonlinear fractional-order differential equations under the terms of wave transformation relationship. The obtained results are shown graphically.

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.

Modified Chebyshev Collocation Method for Solving Differential Equations

Directory of Open Access Journals (Sweden)

M Ziaul Arif

2015-05-01

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

Solving SAT Problem Based on Hybrid Differential Evolution Algorithm

Science.gov (United States)

Liu, Kunqi; Zhang, Jingmin; Liu, Gang; Kang, Lishan

Satisfiability (SAT) problem is an NP-complete problem. Based on the analysis about it, SAT problem is translated equally into an optimization problem on the minimum of objective function. A hybrid differential evolution algorithm is proposed to solve the Satisfiability problem. It makes full use of strong local search capacity of hill-climbing algorithm and strong global search capability of differential evolution algorithm, which makes up their disadvantages, improves the efficiency of algorithm and avoids the stagnation phenomenon. The experiment results show that the hybrid algorithm is efficient in solving SAT problem.

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

Science.gov (United States)

Murase, Kenya

2014-01-01

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

Nonlinear operators and nonlinear transformations studied via the differential form of the completeness relation in quantum mechanics

International Nuclear Information System (INIS)

Fan Hongyi; Yu Shenxi

1994-01-01

We show that the differential form of the fundamental completeness relation in quantum mechanics and the technique of differentiation within an ordered product (DWOP) of operators provide a new approach for calculating normal product expansions of some nonlinear operators and study some nonlinear transformations. Their usefulness in perturbative calculations is pointed out. (orig.)

On Newton-Kantorovich Method for Solving the Nonlinear Operator Equation

Directory of Open Access Journals (Sweden)

Hameed Husam Hameed

2015-01-01

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

Iterative Adaptive Dynamic Programming for Solving Unknown Nonlinear Zero-Sum Game Based on Online Data.

Science.gov (United States)

Zhu, Yuanheng; Zhao, Dongbin; Li, Xiangjun

2017-03-01

H ∞ control is a powerful method to solve the disturbance attenuation problems that occur in some control systems. The design of such controllers relies on solving the zero-sum game (ZSG). But in practical applications, the exact dynamics is mostly unknown. Identification of dynamics also produces errors that are detrimental to the control performance. To overcome this problem, an iterative adaptive dynamic programming algorithm is proposed in this paper to solve the continuous-time, unknown nonlinear ZSG with only online data. A model-free approach to the Hamilton-Jacobi-Isaacs equation is developed based on the policy iteration method. Control and disturbance policies and value are approximated by neural networks (NNs) under the critic-actor-disturber structure. The NN weights are solved by the least-squares method. According to the theoretical analysis, our algorithm is equivalent to a Gauss-Newton method solving an optimization problem, and it converges uniformly to the optimal solution. The online data can also be used repeatedly, which is highly efficient. Simulation results demonstrate its feasibility to solve the unknown nonlinear ZSG. When compared with other algorithms, it saves a significant amount of online measurement time.

Piecewise-linear and bilinear approaches to nonlinear differential equations approximation problem of computational structural mechanics

OpenAIRE

Leibov Roman

2017-01-01

This paper presents a bilinear approach to nonlinear differential equations system approximation problem. Sometimes the nonlinear differential equations right-hand sides linearization is extremely difficult or even impossible. Then piecewise-linear approximation of nonlinear differential equations can be used. The bilinear differential equations allow to improve piecewise-linear differential equations behavior and reduce errors on the border of different linear differential equations systems ...

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)

EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

Institute of Scientific and Technical Information of China (English)

无

2011-01-01

The initial value problem of a nonlinear fractional differential equation is discussed in this paper. Using the nonlinear alternative of Leray-Schauder type and the contraction mapping principle,we obtain the existence and uniqueness of solutions to the fractional differential equation,which extend some results of the previous papers.

Nonlinear problems in fluid dynamics and inverse scattering: Nonlinear waves and inverse scattering

Science.gov (United States)

Ablowitz, Mark J.

1994-12-01

Research investigations involving the fundamental understanding and applications of nonlinear wave motion and related studies of inverse scattering and numerical computation have been carried out and a number of significant results have been obtained. A class of nonlinear wave equations which can be solved by the inverse scattering transform (IST) have been studied, including the Kadaomtsev-Petviashvili (KP) equation, the Davey-Stewartson equation, and the 2+1 Toda system. The solutions obtained by IST correspond to the Cauchy initial value problem with decaying initial data. We have also solved two important systems via the IST method: a 'Volterra' system in 2+1 dimensions and a new one dimensional nonlinear equation which we refer to as the Toda differential-delay equation. Research in computational chaos in moderate to long time numerical simulations continues.

On implicit abstract neutral nonlinear differential equations

Energy Technology Data Exchange (ETDEWEB)

Hernández, Eduardo, E-mail: lalohm@ffclrp.usp.br [Universidade de São Paulo, Departamento de Computação e Matemática, Faculdade de Filosofia Ciências e Letras de Ribeirão Preto (Brazil); O’Regan, Donal, E-mail: donal.oregan@nuigalway.ie [National University of Ireland, School of Mathematics, Statistics and Applied Mathematics (Ireland)

2016-04-15

In this paper we continue our developments in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) on the existence of solutions for abstract neutral differential equations. In particular we extend the results in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) for the case of implicit nonlinear neutral equations and we focus on applications to partial “nonlinear” neutral differential equations. Some applications involving partial neutral differential equations are presented.

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

Energy Technology Data Exchange (ETDEWEB)

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

1996-12-31

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

Analysis of backward differentiation formula for nonlinear differential-algebraic equations with 2 delays.

Science.gov (United States)

Sun, Leping

2016-01-01

This paper is concerned with the backward differential formula or BDF methods for a class of nonlinear 2-delay differential algebraic equations. We obtain two sufficient conditions under which the methods are stable and asymptotically stable. At last, examples show that our methods are true.

Quantum hydrodynamics and nonlinear differential equations for degenerate Fermi gas

International Nuclear Information System (INIS)

Bettelheim, Eldad; Abanov, Alexander G; Wiegmann, Paul B

2008-01-01

We present new nonlinear differential equations for spacetime correlation functions of Fermi gas in one spatial dimension. The correlation functions we consider describe non-stationary processes out of equilibrium. The equations we obtain are integrable equations. They generalize known nonlinear differential equations for correlation functions at equilibrium [1-4] and provide vital tools for studying non-equilibrium dynamics of electronic systems. The method we developed is based only on Wick's theorem and the hydrodynamic description of the Fermi gas. Differential equations appear directly in bilinear form. (fast track communication)

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.

Pseudo-transient Continuation Based Variable Relaxation Solve in Nonlinear Magnetohydrodynamic Simulations

International Nuclear Information System (INIS)

Chen, Jin

2009-01-01

Efficient and robust Variable Relaxation Solver, based on pseudo-transient continuation, is developed to solve nonlinear anisotropic thermal conduction arising from fusion plasma simulations. By adding first and/or second order artificial time derivatives to the system, this type of method advances the resulting time-dependent nonlinear PDEs to steady state, which is the solution to be sought. In this process, only the stiffness matrix itself is involved so that the numerical complexity and errors can be greatly reduced. In fact, this work is an extension of integrating efficient linear elliptic solvers for fusion simulation on Cray XIE. Two schemes are derived in this work, first and second order Variable Relaxations. Four factors are observed to be critical for efficiency and preservation of solution's symmetric structure arising from periodic boundary condition: refining meshes in different coordinate directions, initializing nonlinear process, varying time steps in both temporal and spatial directions, and accurately generating nonlinear stiffness matrix. First finer mesh scale should be taken in strong transport direction; Next the system is carefully initialized by the solution with linear conductivity; Third, time step and relaxation factor are vertex-based varied and optimized at each time step; Finally, the nonlinear stiffness matrix is updated by just scaling corresponding linear one with the vector generated from nonlinear thermal conductivity.

On nonlinear differential equation with exact solutions having various pole orders

International Nuclear Information System (INIS)

Kudryashov, N.A.

2015-01-01

We consider a nonlinear ordinary differential equation having solutions with various movable pole order on the complex plane. We show that the pole order of exact solution is determined by values of parameters of the equation. Exact solutions in the form of the solitary waves for the second order nonlinear differential equation are found taking into account the method of the logistic function. Exact solutions of differential equations are discussed and analyzed

The improved fractional sub-equation method and its applications to the space–time fractional differential equations in fluid mechanics

International Nuclear Information System (INIS)

Guo, Shimin; Mei, Liquan; Li, Ying; Sun, Youfa

2012-01-01

By introducing a new general ansätz, the improved fractional sub-equation method is proposed to construct analytical solutions of nonlinear evolution equations involving Jumarie's modified Riemann–Liouville derivative. By means of this method, the space–time fractional Whitham–Broer–Kaup and generalized Hirota–Satsuma coupled KdV equations are successfully solved. The obtained results show that the proposed method is quite effective, promising and convenient for solving nonlinear fractional differential equations. -- Highlights: ► We propose a novel method for nonlinear fractional differential equations. ► Two important fractional differential equations in fluid mechanics are solved successfully. ► Some new exact solutions of the fractional differential equations are obtained. ► These solutions will advance the understanding of nonlinear physical phenomena.

Adams Predictor-Corrector Systems for Solving Fuzzy Differential Equations

Directory of Open Access Journals (Sweden)

Dequan Shang

2013-01-01

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

Oscillation criteria for third order delay nonlinear differential equations

Directory of Open Access Journals (Sweden)

E. M. Elabbasy

2012-01-01

via comparison with some first differential equations whose oscillatory characters are known. Our results generalize and improve some known results for oscillation of third order nonlinear differential equations. Some examples are given to illustrate the main results.

Reduced differential transform method for partial differential equations within local fractional derivative operators

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.

Differential constraints and exact solutions of nonlinear diffusion equations

International Nuclear Information System (INIS)

Kaptsov, Oleg V; Verevkin, Igor V

2003-01-01

The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining equations used in the search for classical Lie symmetries

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

Science.gov (United States)

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

2018-03-01

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

Pseudo-transient Continuation Based Variable Relaxation Solve in Nonlinear Magnetohydrodynamic Simulations

Energy Technology Data Exchange (ETDEWEB)

Jin Chen

2009-12-07

Efficient and robust Variable Relaxation Solver, based on pseudo-transient continuation, is developed to solve nonlinear anisotropic thermal conduction arising from fusion plasma simulations. By adding first and/or second order artificial time derivatives to the system, this type of method advances the resulting time-dependent nonlinear PDEs to steady state, which is the solution to be sought. In this process, only the stiffness matrix itself is involved so that the numerical complexity and errors can be greatly reduced. In fact, this work is an extension of integrating efficient linear elliptic solvers for fusion simulation on Cray XIE. Two schemes are derived in this work, first and second order Variable Relaxations. Four factors are observed to be critical for efficiency and preservation of solution's symmetric structure arising from periodic boundary condition: refining meshes in different coordinate directions, initializing nonlinear process, varying time steps in both temporal and spatial directions, and accurately generating nonlinear stiffness matrix. First finer mesh scale should be taken in strong transport direction; Next the system is carefully initialized by the solution with linear conductivity; Third, time step and relaxation factor are vertex-based varied and optimized at each time step; Finally, the nonlinear stiffness matrix is updated by just scaling corresponding linear one with the vector generated from nonlinear thermal conductivity.

Approximate analytical methods for solving ordinary differential equations

CERN Document Server

Radhika, TSL; Rani, T Raja

2015-01-01

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

A Generalized Halanay Inequality for Stability of Nonlinear Neutral Functional Differential Equations

Directory of Open Access Journals (Sweden)

Wansheng Wang

2010-01-01

Full Text Available This paper is devoted to generalize Halanay's inequality which plays an important rule in study of stability of differential equations. By applying the generalized Halanay inequality, the stability results of nonlinear neutral functional differential equations (NFDEs and nonlinear neutral delay integrodifferential equations (NDIDEs are obtained.

Universal formats for nonlinear ordinary differential systems

International Nuclear Information System (INIS)

Kerner, E.H.

1981-01-01

It is shown that very general nonlinear ordinary differential systems (embracing all that arise in practice) may, first, be brought down to polynomial systems (where the nonlinearities occur only as polynomials in the dependent variables) by introducing suitable new variables into the original system; second, that polynomial systems are reducible to ''Riccati systems,'' where the nonlinearities are quadratic at most; third, that Riccati systems may be brought to elemental universal formats containing purely quadratic terms with simple arrays of coefficients that are all zero or unity. The elemental systems have representations as novel types of matrix Riccati equations. Different starting systems and their associated Riccati systems differ from one another, at the final elemental level, in order and in initial data, but not in format

Explicit Solutions for Generalized (2+1)-Dimensional Nonlinear Zakharov-Kuznetsov Equation

International Nuclear Information System (INIS)

Sun Yuhuai; Ma Zhimin; Li Yan

2010-01-01

The exact solutions of the generalized (2+1)-dimensional nonlinear Zakharov-Kuznetsov (Z-K) equation are explored by the method of the improved generalized auxiliary differential equation. Many explicit analytic solutions of the Z-K equation are obtained. The methods used to solve the Z-K equation can be employed in further work to establish new solutions for other nonlinear partial differential equations. (general)

Analytic continuation of solutions of some nonlinear convolution partial differential equations

Directory of Open Access Journals (Sweden)

Hidetoshi Tahara

2015-01-01

Full Text Available The paper considers a problem of analytic continuation of solutions of some nonlinear convolution partial differential equations which naturally appear in the summability theory of formal solutions of nonlinear partial differential equations. Under a suitable assumption it is proved that any local holomorphic solution has an analytic extension to a certain sector and its extension has exponential growth when the variable goes to infinity in the sector.

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.

Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (3).

Science.gov (United States)

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.

New Exact Solutions for New Model Nonlinear Partial Differential Equation

OpenAIRE

Maher, A.; El-Hawary, H. M.; Al-Amry, M. S.

2013-01-01

In this paper we propose a new form of Padé-II equation, namely, a combined Padé-II and modified Padé-II equation. The mapping method is a promising method to solve nonlinear evaluation equations. Therefore, we apply it, to solve the combined Padé-II and modified Padé-II equation. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions, trigonometric functions, rational functions, and elliptic functions.

Stability of Nonlinear Neutral Stochastic Functional Differential Equations

Directory of Open Access Journals (Sweden)

Minggao Xue

2010-01-01

Full Text Available Neutral stochastic functional differential equations (NSFDEs have recently been studied intensively. The well-known conditions imposed for the existence and uniqueness and exponential stability of the global solution are the local Lipschitz condition and the linear growth condition. Therefore, the existing results cannot be applied to many important nonlinear NSFDEs. The main aim of this paper is to remove the linear growth condition and establish a Khasminskii-type test for nonlinear NSFDEs. New criteria not only cover a wide class of highly nonlinear NSFDEs but they can also be verified much more easily than the classical criteria. Finally, several examples are given to illustrate main results.

Stability analysis of solutions to nonlinear stiff Volterra functional differential equations in Banach spaces

Institute of Scientific and Technical Information of China (English)

LI Shoufu

2005-01-01

A series of stability, contractivity and asymptotic stability results of the solutions to nonlinear stiff Volterra functional differential equations (VFDEs) in Banach spaces is obtained, which provides the unified theoretical foundation for the stability analysis of solutions to nonlinear stiff problems in ordinary differential equations(ODEs), delay differential equations(DDEs), integro-differential equations(IDEs) and VFDEs of other type which appear in practice.

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.

Exact Solutions for Certain Nonlinear Autonomous Ordinary Differential Equations of the Second Order and Families of Two-Dimensional Autonomous Systems

Directory of Open Access Journals (Sweden)

M. P. Markakis

2010-01-01

Full Text Available Certain nonlinear autonomous ordinary differential equations of the second order are reduced to Abel equations of the first kind ((Ab-1 equations. Based on the results of a previous work, concerning a closed-form solution of a general (Ab-1 equation, and introducing an arbitrary function, exact one-parameter families of solutions are derived for the original autonomous equations, for the most of which only first integrals (in closed or parametric form have been obtained so far. Two-dimensional autonomous systems of differential equations of the first order, equivalent to the considered herein autonomous forms, are constructed and solved by means of the developed analysis.

Using Computer Symbolic Algebra to Solve Differential Equations.

Science.gov (United States)

Mathews, John H.

1989-01-01

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

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.

Nonlinear H-infinity control, Hamiltonian systems and Hamilton-Jacobi equations

CERN Document Server

Aliyu, MDS

2011-01-01

A comprehensive overview of nonlinear Haeu control theory for both continuous-time and discrete-time systems, Nonlinear Haeu-Control, Hamiltonian Systems and Hamilton-Jacobi Equations covers topics as diverse as singular nonlinear Haeu-control, nonlinear Haeu -filtering, mixed H2/ Haeu-nonlinear control and filtering, nonlinear Haeu-almost-disturbance-decoupling, and algorithms for solving the ubiquitous Hamilton-Jacobi-Isaacs equations. The link between the subject and analytical mechanics as well as the theory of partial differential equations is also elegantly summarized in a single chapter

On realization of nonlinear systems described by higher-order differential equations

NARCIS (Netherlands)

van der Schaft, Arjan

1987-01-01

We consider systems of smooth nonlinear differential and algebraic equations in which some of the variables are distinguished as “external variables.” The realization problem is to replace the higher-order implicit differential equations by first-order explicit differential equations and the

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

An effective method for finding special solutions of nonlinear differential equations with variable coefficients

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

Programmable Solution for Solving Non-linearity Characteristics of Smart Sensor Applications

Directory of Open Access Journals (Sweden)

S. Khan

2007-10-01

Full Text Available This paper presents a simple but programmable technique to solve the problem of non-linear characteristics of sensors used in more sensitive applications. The nonlinearity of the output response becomes a very sensitive issue in cases where a proportional increase in the physical quantity fails to bring about a proportional increase in the signal measured. The nonlinearity is addressed by using the interpolation method on the characteristics of a given sensor, approximating it to a set of tangent lines, the tangent points of which are recognized in the code of the processor by IF-THEN code. The method suggested here eliminates the use of external circuits for interfacing, and eases the programming burden on the processor at the cost of proportionally reduced memory requirements. The mathematically worked out results are compared with the simulation and experimental results for an IR sensor selected for the purpose and used for level measurement. This work will be of paramount importance and significance in applications where the controlled signal is required to follow the input signal precisely particularly in sensitive robotic applications.

Differential quadrature method of nonlinear bending of functionally graded beam

Science.gov (United States)

Gangnian, Xu; Liansheng, Ma; Wang, Youzhi; Quan, Yuan; Weijie, You

2018-02-01

Using the third-order shear deflection beam theory (TBT), nonlinear bending of functionally graded (FG) beams composed with various amounts of ceramic and metal is analyzed utilizing the differential quadrature method (DQM). The properties of beam material are supposed to accord with the power law index along to thickness. First, according to the principle of stationary potential energy, the partial differential control formulae of the FG beams subjected to a distributed lateral force are derived. To obtain numerical results of the nonlinear bending, non-dimensional boundary conditions and control formulae are dispersed by applying the DQM. To verify the present solution, several examples are analyzed for nonlinear bending of homogeneous beams with various edges. A minute parametric research is in progress about the effect of the law index, transverse shear deformation, distributed lateral force and boundary conditions.

Decomposition of a hierarchy of nonlinear evolution equations

International Nuclear Information System (INIS)

Geng Xianguo

2003-01-01

The generalized Hamiltonian structures for a hierarchy of nonlinear evolution equations are established with the aid of the trace identity. Using the nonlinearization approach, the hierarchy of nonlinear evolution equations is decomposed into a class of new finite-dimensional Hamiltonian systems. The generating function of integrals and their generator are presented, based on which the finite-dimensional Hamiltonian systems are proved to be completely integrable in the Liouville sense. As an application, solutions for the hierarchy of nonlinear evolution equations are reduced to solving the compatible Hamiltonian systems of ordinary differential equations

Nonlinear dynamics of two-phase flow

International Nuclear Information System (INIS)

Rizwan-uddin

1986-01-01

Unstable flow conditions can occur in a wide variety of laboratory and industry equipment that involve two-phase flow. Instabilities in industrial equipment, which include boiling water reactor (BWR) cores, steam generators, heated channels, cryogenic fluid heaters, heat exchangers, etc., are related to their nonlinear dynamics. These instabilities can be of static (Ledinegg instability) or dynamic (density wave oscillations) type. Determination of regions in parameters space where these instabilities can occur and knowledge of system dynamics in or near these regions is essential for the safe operation of such equipment. Many two-phase flow engineering components can be modeled as heated channels. The set of partial differential equations that describes the dynamics of single- and two-phase flow, for the special case of uniform heat flux along the length of the channel, can be reduced to a set of two coupled ordinary differential equations [in inlet velocity v/sub i/(t) and two-phase residence time tau(t)] involving history integrals: a nonlinear ordinary functional differential equation and an integral equation. Hence, to solve these equations, the dependent variables must be specified for -(nu + tau) ≤ t ≤ 0, where nu is the single-phase residence time. This system of nonlinear equations has been solved analytically using asymptotic expansion series for finite but small perturbations and numerically using finite difference techniques

New Solutions of Three Nonlinear Space- and Time-Fractional Partial Differential Equations in Mathematical Physics

International Nuclear Information System (INIS)

Yao Ruo-Xia; Wang Wei; Chen Ting-Hua

2014-01-01

Motivated by the widely used ansätz method and starting from the modified Riemann—Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper. (general)

Positive Solutions for Coupled Nonlinear Fractional Differential Equations

Directory of Open Access Journals (Sweden)

Wenning Liu

2014-01-01

Full Text Available We consider the existence of positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary values. Assume the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing two cones K1, K2 and computing the fixed point index in product cone K1×K2, we obtain that the system has a pair of positive solutions. It is remarkable that it is established on the Cartesian product of two cones, in which the feature of two equations can be opposite.

A numerical method for solving singular De`s

Energy Technology Data Exchange (ETDEWEB)

Mahaver, W.T.

1996-12-31

A numerical method is developed for solving singular differential equations using steepest descent based on weighted Sobolev gradients. The method is demonstrated on a variety of first and second order problems, including linear constrained, unconstrained, and partially constrained first order problems, a nonlinear first order problem with irregular singularity, and two second order variational problems.

Nonlinear analysis of composite thin-walled helicopter blades

Science.gov (United States)

Kalfon, J. P.; Rand, O.

Nonlinear theoretical modeling of laminated thin-walled composite helicopter rotor blades is presented. The derivation is based on nonlinear geometry with a detailed treatment of the body loads in the axial direction which are induced by the rotation. While the in-plane warping is neglected, a three-dimensional generic out-of-plane warping distribution is included. The formulation may also handle varying thicknesses and mass distribution along the cross-sectional walls. The problem is solved by successive iterations in which a system of equations is constructed and solved for each cross-section. In this method, the differential equations in the spanwise directions are formulated and solved using a finite-differences scheme which allows simple adaptation of the spanwise discretization mesh during iterations.

An analytical method for solving exact solutions of the nonlinear Bogoyavlenskii equation and the nonlinear diffusive predator–prey system

Directory of Open Access Journals (Sweden)

Md. Nur Alam

2016-06-01

Full Text Available In this article, we apply the exp(-Φ(ξ-expansion method to construct many families of exact solutions of nonlinear evolution equations (NLEEs via the nonlinear diffusive predator–prey system and the Bogoyavlenskii equations. These equations can be transformed to nonlinear ordinary differential equations. As a result, some new exact solutions are obtained through the hyperbolic function, the trigonometric function, the exponential functions and the rational forms. If the parameters take specific values, then the solitary waves are derived from the traveling waves. Also, we draw 2D and 3D graphics of exact solutions for the special diffusive predator–prey system and the Bogoyavlenskii equations by the help of programming language Maple.

Differential Polarization Nonlinear Optical Microscopy with Adaptive Optics Controlled Multiplexed Beams

Directory of Open Access Journals (Sweden)

Virginijus Barzda

2013-09-01

Full Text Available Differential polarization nonlinear optical microscopy has the potential to become an indispensable tool for structural investigations of ordered biological assemblies and microcrystalline aggregates. Their microscopic organization can be probed through fast and sensitive measurements of nonlinear optical signal anisotropy, which can be achieved with microscopic spatial resolution by using time-multiplexed pulsed laser beams with perpendicular polarization orientations and photon-counting detection electronics for signal demultiplexing. In addition, deformable membrane mirrors can be used to correct for optical aberrations in the microscope and simultaneously optimize beam overlap using a genetic algorithm. The beam overlap can be achieved with better accuracy than diffraction limited point-spread function, which allows to perform polarization-resolved measurements on the pixel-by-pixel basis. We describe a newly developed differential polarization microscope and present applications of the differential microscopy technique for structural studies of collagen and cellulose. Both, second harmonic generation, and fluorescence-detected nonlinear absorption anisotropy are used in these investigations. It is shown that the orientation and structural properties of the fibers in biological tissue can be deduced and that the orientation of fluorescent molecules (Congo Red, which label the fibers, can be determined. Differential polarization microscopy sidesteps common issues such as photobleaching and sample movement. Due to tens of megahertz alternating polarization of excitation pulses fast data acquisition can be conveniently applied to measure changes in the nonlinear signal anisotropy in dynamically changing in vivo structures.

Exact solutions of some nonlinear partial differential equations using ...

Indian Academy of Sciences (India)

Nonlinear partial differential equations (NPDEs) are encountered in various ... such as physics, mechanics, chemistry, biology, mathematics and engineering. ... In §3, this method is applied to the generalized forms of Klein–Gordon equation,.

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

Directory of Open Access Journals (Sweden)

Behzad Ghanbari

2014-01-01

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

Assessment of Two Analytical Methods in Solving the Linear and Nonlinear Elastic Beam Deformation Problems

DEFF Research Database (Denmark)

Barari, Amin; Ganjavi, B.; Jeloudar, M. Ghanbari

2010-01-01

and fluid mechanics. Design/methodology/approach – Two new but powerful analytical methods, namely, He's VIM and HPM, are introduced to solve some boundary value problems in structural engineering and fluid mechanics. Findings – Analytical solutions often fit under classical perturbation methods. However......, as with other analytical techniques, certain limitations restrict the wide application of perturbation methods, most important of which is the dependence of these methods on the existence of a small parameter in the equation. Disappointingly, the majority of nonlinear problems have no small parameter at all......Purpose – In the last two decades with the rapid development of nonlinear science, there has appeared ever-increasing interest of scientists and engineers in the analytical techniques for nonlinear problems. This paper considers linear and nonlinear systems that are not only regarded as general...

Solution of (3+1-Dimensional Nonlinear Cubic Schrodinger Equation by Differential Transform Method

Directory of Open Access Journals (Sweden)

Hassan A. Zedan

2012-01-01

Full Text Available Four-dimensional differential transform method has been introduced and fundamental theorems have been defined for the first time. Moreover, as an application of four-dimensional differential transform, exact solutions of nonlinear system of partial differential equations have been investigated. The results of the present method are compared very well with analytical solution of the system. Differential transform method can easily be applied to linear or nonlinear problems and reduces the size of computational work. With this method, exact solutions may be obtained without any need of cumbersome work, and it is a useful tool for analytical and numerical solutions.

Modified harmonic balance method for the solution of nonlinear jerk equations

Science.gov (United States)

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.

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

DEFF Research Database (Denmark)

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

2006-01-01

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

Entire solutions of nonlinear differential-difference equations.

Science.gov (United States)

Li, Cuiping; Lü, Feng; Xu, Junfeng

2016-01-01

In this paper, we describe the properties of entire solutions of a nonlinear differential-difference equation and a Fermat type equation, and improve several previous theorems greatly. In addition, we also deduce a uniqueness result for an entire function f(z) that shares a set with its shift [Formula: see text], which is a generalization of a result of Liu.

Contractivity and Exponential Stability of Solutions to Nonlinear Neutral Functional Differential Equations in Banach Spaces

Institute of Scientific and Technical Information of China (English)

Wan-sheng WANG; Shou-fu LI; Run-sheng YANG

2012-01-01

A series of contractivity and exponential stability results for the solutions to nonlinear neutral functional differential equations (NFDEs) in Banach spaces are obtained,which provide unified theoretical foundation for the contractivity analysis of solutions to nonlinear problems in functional differential equations (FDEs),neutral delay differential equations (NDDEs) and NFDEs of other types which appear in practice.

Nonlinear Dynamic Analysis and Optimization of Closed-Form Planetary Gear System

Directory of Open Access Journals (Sweden)

Qilin Huang

2013-01-01

Full Text Available A nonlinear purely rotational dynamic model of a multistage closed-form planetary gear set formed by two simple planetary stages is proposed in this study. The model includes time-varying mesh stiffness, excitation fluctuation and gear backlash nonlinearities. The nonlinear differential equations of motion are solved numerically using variable step-size Runge-Kutta. In order to obtain function expression of optimization objective, the nonlinear differential equations of motion are solved analytically using harmonic balance method (HBM. Based on the analytical solution of dynamic equations, the optimization mathematical model which aims at minimizing the vibration displacement of the low-speed carrier and the total mass of the gear transmission system is established. The optimization toolbox in MATLAB program is adopted to obtain the optimal solution. A case is studied to demonstrate the effectiveness of the dynamic model and the optimization method. The results show that the dynamic properties of the closed-form planetary gear transmission system have been improved and the total mass of the gear set has been decreased significantly.

Approximate Method for Solving the Linear Fuzzy Delay Differential Equations

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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.

Differential geometric methods in system theory.

Science.gov (United States)

Brockett, R. W.

1971-01-01

Discussion of certain problems in system theory which have been or might be solved using some basic concepts from differential geometry. The problems considered involve differential equations, controllability, optimal control, qualitative behavior, stochastic processes, and bilinear systems. The main goal is to extend the essentials of linear theory to some nonlinear classes of problems.

The lie-algebraic structures and integrability of differential and differential-difference nonlinear dynamical systems

International Nuclear Information System (INIS)

Prykarpatsky, A.K.; Blackmore, D.L.; Bogolubov, N.N. Jr.

2007-05-01

The infinite-dimensional operator Lie algebras of the related integrable nonlocal differential-difference dynamical systems are treated as their hidden symmetries. As a result of their dimerization the Lax type representations for both local differential-difference equations and nonlocal ones are obtained. An alternative approach to the Lie-algebraic interpretation of the integrable local differential-difference systems is also proposed. The Hamiltonian representation for a hierarchy of Lax type equations on a dual space to the centrally extended Lie algebra of integro-differential operators with matrix-valued coefficients coupled with suitable eigenfunctions and adjoint eigenfunctions evolutions of associated spectral problems is obtained by means of a specially constructed Baecklund transformation. The Hamiltonian description for the corresponding set of additional symmetry hierarchies is represented. The relation of these hierarchies with Lax type integrable (3+1)-dimensional nonlinear dynamical systems and their triple Lax type linearizations is analyzed. The Lie-algebraic structures, related with centrally extended current operator Lie algebras are discussed with respect to constructing new nonlinear integrable dynamical systems on functional manifolds and super-manifolds. Special Poisson structures and related with them factorized integrable operator dynamical systems having interesting applications in modern mathematical physics, quantum computing mathematics and other fields are constructed. The previous purely computational results are explained within the approach developed. (author)

A Generalized National Planning Approach for Admission Capacity in Higher Education: A Nonlinear Integer Goal Programming Model with a Novel Differential Evolution Algorithm.

Science.gov (United States)

El-Qulity, Said Ali; Mohamed, Ali Wagdy

2016-01-01

This paper proposes a nonlinear integer goal programming model (NIGPM) for solving the general problem of admission capacity planning in a country as a whole. The work aims to satisfy most of the required key objectives of a country related to the enrollment problem for higher education. The system general outlines are developed along with the solution methodology for application to the time horizon in a given plan. The up-to-date data for Saudi Arabia is used as a case study and a novel evolutionary algorithm based on modified differential evolution (DE) algorithm is used to solve the complexity of the NIGPM generated for different goal priorities. The experimental results presented in this paper show their effectiveness in solving the admission capacity for higher education in terms of final solution quality and robustness.

STABILITY OF NONLINEAR NEUTRAL DIFFERENTIAL EQUATION VIA FIXED POINT

Institute of Scientific and Technical Information of China (English)

无

2012-01-01

In this paper,a nonlinear neutral differential equation is considered.By a fixed point theory,we give some conditions to ensure that the zero solution to the equation is asymptotically stable.Some existing results are improved and generalized.

ON THE INSTABILITY OF SOLUTIONS TO A NONLINEAR VECTOR DIFFERENTIAL EQUATION OF FOURTH ORDER

Institute of Scientific and Technical Information of China (English)

无

2011-01-01

This paper presents a new result related to the instability of the zero solution to a nonlinear vector differential equation of fourth order.Our result includes and improves an instability result in the previous literature,which is related to the instability of the zero solution to a nonlinear scalar differential equation of fourth order.

A new sine-Gordon equation expansion algorithm to investigate some special nonlinear differential equations

International Nuclear Information System (INIS)

Yan Zhenya

2005-01-01

A new transformation method is developed using the general sine-Gordon travelling wave reduction equation and a generalized transformation. With the aid of symbolic computation, this method can be used to seek more types of solutions of nonlinear differential equations, which include not only the known solutions derived by some known methods but new solutions. Here we choose the double sine-Gordon equation, the Magma equation and the generalized Pochhammer-Chree (PC) equation to illustrate the method. As a result, many types of new doubly periodic solutions are obtained. Moreover when using the method to these special nonlinear differential equations, some transformations are firstly needed. The method can be also extended to other nonlinear differential equations

Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations

International Nuclear Information System (INIS)

Udaltsov, Vladimir S.; Goedgebuer, Jean-Pierre; Larger, Laurent; Cuenot, Jean-Baptiste; Levy, Pascal; Rhodes, William T.

2003-01-01

We report that signal encoding with high-dimensional chaos produced by delayed feedback systems with a strong nonlinearity can be broken. We describe the procedure and illustrate the method with chaotic waveforms obtained from a strongly nonlinear optical system that we used previously to demonstrate signal encryption/decryption with chaos in wavelength. The method can be extended to any systems ruled by nonlinear time-delayed differential equations

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

International Nuclear Information System (INIS)

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

2011-01-01

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

A Nonlinear Modal Aeroelastic Solver for FUN3D

Science.gov (United States)

Goldman, Benjamin D.; Bartels, Robert E.; Biedron, Robert T.; Scott, Robert C.

2016-01-01

A nonlinear structural solver has been implemented internally within the NASA FUN3D computational fluid dynamics code, allowing for some new aeroelastic capabilities. Using a modal representation of the structure, a set of differential or differential-algebraic equations are derived for general thin structures with geometric nonlinearities. ODEPACK and LAPACK routines are linked with FUN3D, and the nonlinear equations are solved at each CFD time step. The existing predictor-corrector method is retained, whereby the structural solution is updated after mesh deformation. The nonlinear solver is validated using a test case for a flexible aeroshell at transonic, supersonic, and hypersonic flow conditions. Agreement with linear theory is seen for the static aeroelastic solutions at relatively low dynamic pressures, but structural nonlinearities limit deformation amplitudes at high dynamic pressures. No flutter was found at any of the tested trajectory points, though LCO may be possible in the transonic regime.

Nonlinear Elliptic Differential Equations with Multivalued Nonlinearities

Indian Academy of Sciences (India)

In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all R R . Assuming the existence of an upper and of a lower ...

Modified Differential Transform Method for Two Singular Boundary Values Problems

Directory of Open Access Journals (Sweden)

Yinwei Lin

2014-01-01

Full Text Available This paper deals with the two singular boundary values problems of second order. Two singular points are both boundary values points of the differential equation. The numerical solutions are developed by modified differential transform method (DTM for expanded point. Linear and nonlinear models are solved by this method to get more reliable and efficient numerical results. It can also solve ordinary differential equations where the traditional one fails. Besides, we give the convergence of this new method.

From the hypergeometric differential equation to a non-linear Schrödinger one

International Nuclear Information System (INIS)

Plastino, A.; Rocca, M.C.

2015-01-01

We show that the q-exponential function is a hypergeometric function. Accordingly, it obeys the hypergeometric differential equation. We demonstrate that this differential equation can be transformed into a non-linear Schrödinger equation (NLSE). This NLSE exhibits both similarities and differences vis-a-vis the Nobre–Rego-Monteiro–Tsallis one. - Highlights: • We show that the q-exponential is a hypergeometric function. • It thus obeys the hypergeometric differential equation (HDE). • We show that the HDE can be cast as a non-linear Schrödinger equation. • This is different from the Nobre, Rego-Monteiro, Tsallis one.

Nonlinear partial differential equations for scientists and engineers

CERN Document Server

Debnath, Lokenath

1997-01-01

"An exceptionally complete overview. There are numerous examples and the emphasis is on applications to almost all areas of science and engineering. There is truly something for everyone here. This reviewer feels that it is a very hard act to follow, and recommends it strongly. [This book] is a jewel." ---Applied Mechanics Review (Review of First Edition) This expanded and revised second edition is a comprehensive and systematic treatment of linear and nonlinear partial differential equations and their varied applications. Building upon the successful material of the first book, this edition contains updated modern examples and applications from areas of fluid dynamics, gas dynamics, plasma physics, nonlinear dynamics, quantum mechanics, nonlinear optics, acoustics, and wave propagation. Methods and properties of solutions are presented, along with their physical significance, making the book more useful for a diverse readership. Topics and key features: * Thorough coverage of derivation and methods of soluti...

Superdiffusions and positive solutions of nonlinear partial differential equations

CERN Document Server

Dynkin, E B

2004-01-01

This book is devoted to the applications of probability theory to the theory of nonlinear partial differential equations. More precisely, it is shown that all positive solutions for a class of nonlinear elliptic equations in a domain are described in terms of their traces on the boundary of the domain. The main probabilistic tool is the theory of superdiffusions, which describes a random evolution of a cloud of particles. A substantial enhancement of this theory is presented that can be of interest for everybody who works on applications of probabilistic methods to mathematical analysis.

Nonlinear Dynamic Buckling of Damaged Composite Cylindrical Shells

Institute of Scientific and Technical Information of China (English)

WANG Tian-lin; TANG Wen-yong; ZHANG Sheng-kun

2007-01-01

Based on the first order shear deformation theory(FSDT), the nonlinear dynamic equations involving transverse shear deformation and initial geometric imperfections were obtained by Hamilton's philosophy. Geometric deformation of the composite cylindrical shell was treated as the initial geometric imperfection in the dynamic equations, which were solved by the semi-analytical method in this paper. Stiffness reduction was employed for the damaged sub-layer, and the equivalent stiffness matrix was obtained for the delaminated area. By circumferential Fourier series expansions for shell displacements and loads and by using Galerkin technique, the nonlinear partial differential equations were transformed to ordinary differential equations which were finally solved by the finite difference method. The buckling was judged from shell responses by B-R criteria, and critical loads were then determined. The effect of the initial geometric deformation on the dynamic response and buckling of composite cylindrical shell was also discussed, as well as the effects of concomitant delamination and sub-layer matrix damages.

Advances in nonlinear partial differential equations and stochastics

CERN Document Server

Kawashima, S

1998-01-01

In the past two decades, there has been great progress in the theory of nonlinear partial differential equations. This book describes the progress, focusing on interesting topics in gas dynamics, fluid dynamics, elastodynamics etc. It contains ten articles, each of which discusses a very recent result obtained by the author. Some of these articles review related results.

Estimation of delays and other parameters in nonlinear functional differential equations

Science.gov (United States)

Banks, H. T.; Lamm, P. K. D.

1983-01-01

A spline-based approximation scheme for nonlinear nonautonomous delay differential equations is discussed. Convergence results (using dissipative type estimates on the underlying nonlinear operators) are given in the context of parameter estimation problems which include estimation of multiple delays and initial data as well as the usual coefficient-type parameters. A brief summary of some of the related numerical findings is also given.

Non-linear mixed-effects pharmacokinetic/pharmacodynamic modelling in NLME using differential equations

DEFF Research Database (Denmark)

Tornøe, Christoffer Wenzel; Agersø, Henrik; Madsen, Henrik

2004-01-01

The standard software for non-linear mixed-effect analysis of pharmacokinetic/phar-macodynamic (PK/PD) data is NONMEM while the non-linear mixed-effects package NLME is an alternative as tong as the models are fairly simple. We present the nlmeODE package which combines the ordinary differential...... equation (ODE) solver package odesolve and the non-Linear mixed effects package NLME thereby enabling the analysis of complicated systems of ODEs by non-linear mixed-effects modelling. The pharmacokinetics of the anti-asthmatic drug theophylline is used to illustrate the applicability of the nlme...

Increase in speed of Wilkinson-type ADC and improvement of differential non-linearity

Energy Technology Data Exchange (ETDEWEB)

Kinbara, S [Japan Atomic Energy Research Inst., Tokai, Ibaraki. Tokai Research Establishment

1977-06-01

It is shown that the differential non-linearity of a Wilkinson-type analog-to-digital converter (ADC) is dominated by the unbalance of even-numbered periods caused by the action of interference resulting from operation of a channel scaler. To improve this situation, new methods were tested which allow such action of interference to be dispersed. Measurements show that a differential non-linearity value of +- 0.043% is attainable for a clock rate of 300 MHz.

A New Method to Solve Numeric Solution of Nonlinear Dynamic System

Directory of Open Access Journals (Sweden)

Min Hu

2016-01-01

Full Text Available It is well known that the cubic spline function has advantages of simple forms, good convergence, approximation, and second-order smoothness. A particular class of cubic spline function is constructed and an effective method to solve the numerical solution of nonlinear dynamic system is proposed based on the cubic spline function. Compared with existing methods, this method not only has high approximation precision, but also avoids the Runge phenomenon. The error analysis of several methods is given via two numeric examples, which turned out that the proposed method is a much more feasible tool applied to the engineering practice.

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

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Huiping Cao

2014-01-01

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

Intuitionistic Fuzzy Goal Programming Technique for Solving Non-Linear Multi-objective Structural Problem

Directory of Open Access Journals (Sweden)

Samir Dey

2015-07-01

Full Text Available This paper proposes a new multi-objective intuitionistic fuzzy goal programming approach to solve a multi-objective nonlinear programming problem in context of a structural design. Here we describe some basic properties of intuitionistic fuzzy optimization. We have considered a multi-objective structural optimization problem with several mutually conflicting objectives. The design objective is to minimize weight of the structure and minimize the vertical deflection at loading point of a statistically loaded three-bar planar truss subjected to stress constraints on each of the truss members. This approach is used to solve the above structural optimization model based on arithmetic mean and compare with the solution by intuitionistic fuzzy goal programming approach. A numerical solution is given to illustrate our approach.

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

Science.gov (United States)

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

2017-01-01

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

Multi-soliton management by the integrable nonautonomous nonlinear integro-differential Schrödinger equation

International Nuclear Information System (INIS)

Zhang, Yu-Juan; Zhao, Dun; Luo, Hong-Gang

2014-01-01

We consider a wide class of integrable nonautonomous nonlinear integro-differential Schrödinger equation which contains the models for the soliton management in Bose–Einstein condensates, nonlinear optics, and inhomogeneous Heisenberg spin chain. With the help of the nonisospectral AKNS hierarchy, we obtain the N-fold Darboux transformation and the N-fold soliton-like solutions for the equation. The soliton management, especially the synchronized dispersive and nonlinear management in optical fibers is discussed. It is found that in the situation without external potential, the synchronized dispersive and nonlinear management can keep the integrability of the nonlinear Schrödinger equation; this suggests that in optical fibers, the synchronized dispersive and nonlinear management can control and maintain the propagation of a multi-soliton. - Highlights: • We consider a unified model for soliton management by an integrable integro-differential Schrödinger equation. • Using Lax pair, the N-fold Darboux transformation for the equation is presented. • The multi-soliton management is considered. • The synchronized dispersive and nonlinear management is suggested

Exact solutions of some nonlinear partial differential equations using ...

Indian Academy of Sciences (India)

The functional variable method is a powerful solution method for obtaining exact solutions of some nonlinear partial differential equations. In this paper, the functional variable method is used to establish exact solutions of the generalized forms of Klein–Gordon equation, the (2 + 1)-dimensional Camassa–Holm ...

A nonlinear plate control without linearization

Directory of Open Access Journals (Sweden)

Yildirim Kenan

2017-03-01

Full Text Available In this paper, an optimal vibration control problem for a nonlinear plate is considered. In order to obtain the optimal control function, wellposedness and controllability of the nonlinear system is investigated. The performance index functional of the system, to be minimized by minimum level of control, is chosen as the sum of the quadratic 10 functional of the displacement. The velocity of the plate and quadratic functional of the control function is added to the performance index functional as a penalty term. By using a maximum principle, the nonlinear control problem is transformed to solving a system of partial differential equations including state and adjoint variables linked by initial-boundary-terminal conditions. Hence, it is shown that optimal control of the nonlinear systems can be obtained without linearization of the nonlinear term and optimal control function can be obtained analytically for nonlinear systems without linearization.

Conservation laws for certain time fractional nonlinear systems of partial differential equations

Science.gov (United States)

Singla, Komal; Gupta, R. K.

2017-12-01

In this study, an extension of the concept of nonlinear self-adjointness and Noether operators is proposed for calculating conserved vectors of the time fractional nonlinear systems of partial differential equations. In our recent work (J Math Phys 2016; 57: 101504), by proposing the symmetry approach for time fractional systems, the Lie symmetries for some fractional nonlinear systems have been derived. In this paper, the obtained infinitesimal generators are used to find conservation laws for the corresponding fractional systems.

Oscillation criteria for third order nonlinear delay differential equations with damping

Directory of Open Access Journals (Sweden)

Said R. Grace

2015-01-01

Full Text Available This note is concerned with the oscillation of third order nonlinear delay differential equations of the form \\[\\label{*} \\left( r_{2}(t\\left( r_{1}(ty^{\\prime}(t\\right^{\\prime}\\right^{\\prime}+p(ty^{\\prime}(t+q(tf(y(g(t=0.\\tag{\\(\\ast\\}\\] In the papers [A. Tiryaki, M. F. Aktas, Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl. 325 (2007, 54-68] and [M. F. Aktas, A. Tiryaki, A. Zafer, Oscillation criteria for third order nonlinear functional differential equations, Applied Math. Letters 23 (2010, 756-762], the authors established some sufficient conditions which insure that any solution of equation (\\(\\ast\\ oscillates or converges to zero, provided that the second order equation \\[\\left( r_{2}(tz^{\\prime }(t\\right^{\\prime}+\\left(p(t/r_{1}(t\\right z(t=0\\tag{\\(\\ast\\ast\\}\\] is nonoscillatory. Here, we shall improve and unify the results given in the above mentioned papers and present some new sufficient conditions which insure that any solution of equation (\\(\\ast\\ oscillates if equation (\\(\\ast\\ast\\ is nonoscillatory. We also establish results for the oscillation of equation (\\(\\ast\\ when equation (\\(\\ast\\ast\\ is oscillatory.

Multistage Spectral Relaxation Method for Solving the Hyperchaotic Complex Systems

Directory of Open Access Journals (Sweden)

Hassan Saberi Nik

2014-01-01

Full Text Available We present a pseudospectral method application for solving the hyperchaotic complex systems. The proposed method, called the multistage spectral relaxation method (MSRM is based on a technique of extending Gauss-Seidel type relaxation ideas to systems of nonlinear differential equations and using the Chebyshev pseudospectral methods to solve the resulting system on a sequence of multiple intervals. In this new application, the MSRM is used to solve famous hyperchaotic complex systems such as hyperchaotic complex Lorenz system and the complex permanent magnet synchronous motor. We compare this approach to the Runge-Kutta based ode45 solver to show that the MSRM gives accurate results.

Modified multiple time scale method for solving strongly nonlinear damped forced vibration systems

Science.gov (United States)

Razzak, M. A.; Alam, M. Z.; Sharif, M. N.

2018-03-01

In this paper, modified multiple time scale (MTS) method is employed to solve strongly nonlinear forced vibration systems. The first-order approximation is only considered in order to avoid complexicity. The formulations and the determination of the solution procedure are very easy and straightforward. The classical multiple time scale (MS) and multiple scales Lindstedt-Poincare method (MSLP) do not give desire result for the strongly damped forced vibration systems with strong damping effects. The main aim of this paper is to remove these limitations. Two examples are considered to illustrate the effectiveness and convenience of the present procedure. The approximate external frequencies and the corresponding approximate solutions are determined by the present method. The results give good coincidence with corresponding numerical solution (considered to be exact) and also provide better result than other existing results. For weak nonlinearities with weak damping effect, the absolute relative error measures (first-order approximate external frequency) in this paper is only 0.07% when amplitude A = 1.5 , while the relative error gives MSLP method is surprisingly 28.81%. Furthermore, for strong nonlinearities with strong damping effect, the absolute relative error found in this article is only 0.02%, whereas the relative error obtained by MSLP method is 24.18%. Therefore, the present method is not only valid for weakly nonlinear damped forced systems, but also gives better result for strongly nonlinear systems with both small and strong damping effect.

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

Directory of Open Access Journals (Sweden)

M. Bishehniasar

2017-01-01

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

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

Science.gov (United States)

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

2016-10-01

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

Analysis of Nonlinear Periodic and Aperiodic Media: Application to Optical Logic Gates

Science.gov (United States)

Yu, Yisheng

This dissertation is about the analysis of nonlinear periodic and aperiodic media and their application to the design of intensity controlled all optical logic gates: AND, OR, and NOT. A coupled nonlinear differential equation that characterizes the electromagnetic wave propagation in a nonlinear periodic (and aperiodic) medium has been derived from the first principle. The equations are general enough that it reflects the effect of transverse modal fields and can be used to analyze both co-propagating and counter propagating waves. A numerical technique based on the finite differences method and absorbing boundary condition has been developed to solve the coupled differential equations here. The numerical method is simple and accurate. Unlike the method based on characteristics that has been reported in the literature, this method does not involve integration and step sizes of time and space coordinates are decoupled. The decoupling provides independent choice for time and space step sizes. The concept of "gap soliton" has also been re-examined. The dissertation consists of four manuscripts. Manuscript I reports on the design of all optical logic gates: AND, OR, and NOT based on the bistability property of nonlinear periodic and aperiodic waveguiding structures. The functioning of the logic gates has been shown by analysis. The numerical technique that has been developed to solve the nonlinear differential equations are addressed in manuscript II. The effect of transverse modal fields on the bistable property of nonlinear periodic medium is reported in manuscript III. The concept of "gap soliton" that are generated in a nonlinear periodic medium has been re-examined. The details on the finding of the re-examination are discussed in manuscript IV.

Application of He’s Variational Iteration Method to Nonlinear Helmholtz Equation and Fifth-Order KDV Equation

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...

A Differential Geometric Approach to Nonlinear Filtering: The Projection Filter

NARCIS (Netherlands)

Brigo, D.; Hanzon, B.; LeGland, F.

1998-01-01

This paper presents a new and systematic method of approximating exact nonlinear filters with finite dimensional filters, using the differential geometric approach to statistics. The projection filter is defined rigorously in the case of exponential families. A convenient exponential family is

Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Volterra Delay Integro-Differential Equations

Directory of Open Access Journals (Sweden)

Haiyan Yuan

2013-01-01

Full Text Available This paper introduces the stability and convergence of two-step Runge-Kutta methods with compound quadrature formula for solving nonlinear Volterra delay integro-differential equations. First, the definitions of (k,l-algebraically stable and asymptotically stable are introduced; then the asymptotical stability of a (k,l-algebraically stable two-step Runge-Kutta method with 0

Numerical solution of distributed order fractional differential equations

Science.gov (United States)

Katsikadelis, John T.

2014-02-01

In this paper a method for the numerical solution of distributed order FDEs (fractional differential equations) of a general form is presented. The method applies to both linear and nonlinear equations. The Caputo type fractional derivative is employed. The distributed order FDE is approximated with a multi-term FDE, which is then solved by adjusting appropriately the numerical method developed for multi-term FDEs by Katsikadelis. Several example equations are solved and the response of mechanical systems described by such equations is studied. The convergence and the accuracy of the method for linear and nonlinear equations are demonstrated through well corroborated numerical results.

A MODIFIED DECOMPOSITION METHOD FOR SOLVING NONLINEAR PROBLEM OF FLOW IN CONVERGING- DIVERGING CHANNEL

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MOHAMED KEZZAR

2015-08-01

Full Text Available In this research, an efficient technique of computation considered as a modified decomposition method was proposed and then successfully applied for solving the nonlinear problem of the two dimensional flow of an incompressible viscous fluid between nonparallel plane walls. In fact this method gives the nonlinear term Nu and the solution of the studied problem as a power series. The proposed iterative procedure gives on the one hand a computationally efficient formulation with an acceleration of convergence rate and on the other hand finds the solution without any discretization, linearization or restrictive assumptions. The comparison of our results with those of numerical treatment and other earlier works shows clearly the higher accuracy and efficiency of the used Modified Decomposition Method.

Solving nonlinear nonstationary problem of heat-conductivity by finite element method

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Антон Янович Карвацький

2016-11-01

Full Text Available Methodology and effective solving algorithm of non-linear dynamic problems of thermal and electric conductivity with significant temperature dependence of thermal and physical properties are given on the basis of finite element method (FEM and Newton linearization method. Discrete equations system FEM was obtained with the use of Galerkin method, where the main function is the finite element form function. The methodology based on successive solving problems of thermal and electrical conductivity has been examined in the work in order to minimize the requirements for calculating resources (RAM. in particular. Having used Mathcad software original programming code was developed to solve the given problem. After investigation of the received results, comparative analyses of accurate solution data and results of numerical solutions, obtained with the use of Matlab programming products, was held. The geometry of one fourth part of the finite sized cylinder was used to test the given numerical model. The discretization of the calculation part was fulfilled using the open programming software for automated Gmsh nets with tetrahedral units, while ParaView, which is an open programming code as well, was used to visualize the calculation results. It was found out that the maximum value violation of potential and temperature determination doesn`t exceed 0,2-0,83% in the given work according to the problem conditions

System Entropy Measurement of Stochastic Partial Differential Systems

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Bor-Sen Chen

2016-03-01

Full Text Available System entropy describes the dispersal of a system’s energy and is an indication of the disorder of a physical system. Several system entropy measurement methods have been developed for dynamic systems. However, most real physical systems are always modeled using stochastic partial differential dynamic equations in the spatio-temporal domain. No efficient method currently exists that can calculate the system entropy of stochastic partial differential systems (SPDSs in consideration of the effects of intrinsic random fluctuation and compartment diffusion. In this study, a novel indirect measurement method is proposed for calculating of system entropy of SPDSs using a Hamilton–Jacobi integral inequality (HJII-constrained optimization method. In other words, we solve a nonlinear HJII-constrained optimization problem for measuring the system entropy of nonlinear stochastic partial differential systems (NSPDSs. To simplify the system entropy measurement of NSPDSs, the global linearization technique and finite difference scheme were employed to approximate the nonlinear stochastic spatial state space system. This allows the nonlinear HJII-constrained optimization problem for the system entropy measurement to be transformed to an equivalent linear matrix inequalities (LMIs-constrained optimization problem, which can be easily solved using the MATLAB LMI-toolbox (MATLAB R2014a, version 8.3. Finally, several examples are presented to illustrate the system entropy measurement of SPDSs.

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

Science.gov (United States)

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

2009-01-01

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

Intermittently chaotic oscillations for a differential-delay equation with Gaussian nonlinearity

Science.gov (United States)

Hamilton, Ian

1992-01-01

For a differential-delay equation the time dependence of the variable is a function of the variable at a previous time. We consider a differential-delay equation with Gaussian nonlinearity that displays intermittent chaos. Although not the first example of a differential-delay equation that displays such behavior, for this example the intermittency is classified as type III, and the origin of the intermittent chaos may be qualitatively understood from the limiting forms of the equation for large and small variable magnitudes.

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

Science.gov (United States)

Khataybeh, S. N.; Hashim, I.

2018-04-01

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

Algorithms of estimation for nonlinear systems a differential and algebraic viewpoint

CERN Document Server

Martínez-Guerra, Rafael

2017-01-01

This book acquaints readers with recent developments in dynamical systems theory and its applications, with a strong focus on the control and estimation of nonlinear systems. Several algorithms are proposed and worked out for a set of model systems, in particular so-called input-affine or bilinear systems, which can serve to approximate a wide class of nonlinear control systems. These can either take the form of state space models or be represented by an input-output equation. The approach taken here further highlights the role of modern mathematical and conceptual tools, including differential algebraic theory, observer design for nonlinear systems and generalized canonical forms.

Numerical solution for multi-term fractional (arbitrary) orders differential equations

OpenAIRE

El-Sayed, A. M. A.; El-Mesiry, A. E. M.; El-Saka, H. A. A.

2004-01-01

Our main concern here is to give a numerical scheme to solve a nonlinear multi-term fractional (arbitrary) orders differential equation. Some results concerning the existence and uniqueness have been also obtained.

Solution of stochastic nonlinear PDEs using Wiener-Hermite expansion of high orders

KAUST Repository

El Beltagy, Mohamed

2016-01-01

In this work, the Wiener-Hermite Expansion (WHE) is used to solve stochastic nonlinear PDEs excited with noise. The generation of the equivalent set of deterministic integro-differential equations is automated and hence allows for high order terms of WHE. The automation difficulties are discussed, solved and implemented to output the final system to be solved. A numerical Pikard-like algorithm is suggested to solve the resulting deterministic system. The automated WHE is applied to the 1D diffusion equation and to the heat equation. The results are compared with previous solutions obtained with WHEP (WHE with perturbation) technique. The solution obtained using the suggested WHE technique is shown to be the limit of the WHEP solutions with infinite number of corrections. The automation is extended easily to account for white-noise of higher dimension and for general nonlinear PDEs.

Solution of stochastic nonlinear PDEs using Wiener-Hermite expansion of high orders

KAUST Repository

El Beltagy, Mohamed

2016-01-06

In this work, the Wiener-Hermite Expansion (WHE) is used to solve stochastic nonlinear PDEs excited with noise. The generation of the equivalent set of deterministic integro-differential equations is automated and hence allows for high order terms of WHE. The automation difficulties are discussed, solved and implemented to output the final system to be solved. A numerical Pikard-like algorithm is suggested to solve the resulting deterministic system. The automated WHE is applied to the 1D diffusion equation and to the heat equation. The results are compared with previous solutions obtained with WHEP (WHE with perturbation) technique. The solution obtained using the suggested WHE technique is shown to be the limit of the WHEP solutions with infinite number of corrections. The automation is extended easily to account for white-noise of higher dimension and for general nonlinear PDEs.

Solving Second-Order Ordinary Differential Equations without Using Complex Numbers

Science.gov (United States)

Kougias, Ioannis E.

2009-01-01

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

Positive Solutions for System of Nonlinear Fractional Differential Equations in Two Dimensions with Delay

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Azizollah Babakhani

2010-01-01

Full Text Available We investigate the existence and uniqueness of positive solution for system of nonlinear fractional differential equations in two dimensions with delay. Our analysis relies on a nonlinear alternative of Leray-Schauder type and Krasnoselskii's fixed point theorem in a cone.

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

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Uswah Qasim

2016-03-01

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

Lectures on the theory of group properties of differential equations

CERN Document Server

Ovsyannikov, LV

2013-01-01

These lecturers provide a clear introduction to Lie group methods for determining and using symmetries of differential equations, a variety of their applications in gas dynamics and other nonlinear models as well as the author's remarkable contribution to this classical subject. It contains material that is useful for students and teachers but cannot be found in modern texts. For example, the theory of partially invariant solutions developed by Ovsyannikov provides a powerful tool for solving systems of nonlinear differential equations and investigating complicated mathematical models. Readers

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

Science.gov (United States)

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

2017-07-01

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

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

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Shahid Hasnain

2017-07-01

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

Reduced-order modelling of parameter-dependent, linear and nonlinear dynamic partial differential equation models.

Science.gov (United States)

Shah, A A; Xing, W W; Triantafyllidis, V

2017-04-01

In this paper, we develop reduced-order models for dynamic, parameter-dependent, linear and nonlinear partial differential equations using proper orthogonal decomposition (POD). The main challenges are to accurately and efficiently approximate the POD bases for new parameter values and, in the case of nonlinear problems, to efficiently handle the nonlinear terms. We use a Bayesian nonlinear regression approach to learn the snapshots of the solutions and the nonlinearities for new parameter values. Computational efficiency is ensured by using manifold learning to perform the emulation in a low-dimensional space. The accuracy of the method is demonstrated on a linear and a nonlinear example, with comparisons with a global basis approach.

Computation of Value Functions in Nonlinear Differential Games with State Constraints

KAUST Repository

Botkin, Nikolai; Hoffmann, Karl-Heinz; Mayer, Natalie; Turova, Varvara

2013-01-01

Finite-difference schemes for the computation of value functions of nonlinear differential games with non-terminal payoff functional and state constraints are proposed. The solution method is based on the fact that the value function is a

Existence and Uniqueness of Solutions for Coupled Systems of Higher-Order Nonlinear Fractional Differential Equations

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Ahmad Bashir

2010-01-01

Full Text Available We study an initial value problem for a coupled Caputo type nonlinear fractional differential system of higher order. As a first problem, the nonhomogeneous terms in the coupled fractional differential system depend on the fractional derivatives of lower orders only. Then the nonhomogeneous terms in the fractional differential system are allowed to depend on the unknown functions together with the fractional derivative of lower orders. Our method of analysis is based on the reduction of the given system to an equivalent system of integral equations. Applying the nonlinear alternative of Leray-Schauder, we prove the existence of solutions of the fractional differential system. The uniqueness of solutions of the fractional differential system is established by using the Banach contraction principle. An illustrative example is also presented.

Analytical approximate solutions for a general class of nonlinear delay differential equations.

Science.gov (United States)

Căruntu, Bogdan; Bota, Constantin

2014-01-01

We use the polynomial least squares method (PLSM), which allows us to compute analytical approximate polynomial solutions for a very general class of strongly nonlinear delay differential equations. The method is tested by computing approximate solutions for several applications including the pantograph equations and a nonlinear time-delay model from biology. The accuracy of the method is illustrated by a comparison with approximate solutions previously computed using other methods.

Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations

Energy Technology Data Exchange (ETDEWEB)

Mancas, Stefan C. [Department of Mathematics, Embry–Riddle Aeronautical University, Daytona Beach, FL 32114-3900 (United States); Rosu, Haret C., E-mail: hcr@ipicyt.edu.mx [IPICYT, Instituto Potosino de Investigacion Cientifica y Tecnologica, Apdo Postal 3-74 Tangamanga, 78231 San Luis Potosí, SLP (Mexico)

2013-09-02

We emphasize two connections, one well known and another less known, between the dissipative nonlinear second order differential equations and the Abel equations which in their first-kind form have only cubic and quadratic terms. Then, employing an old integrability criterion due to Chiellini, we introduce the corresponding integrable dissipative equations. For illustration, we present the cases of some integrable dissipative Fisher, nonlinear pendulum, and Burgers–Huxley type equations which are obtained in this way and can be of interest in applications. We also show how to obtain Abel solutions directly from the factorization of second order nonlinear equations.

Differential Neural Networks for Identification and Filtering in Nonlinear Dynamic Games

Directory of Open Access Journals (Sweden)

Emmanuel García

2014-01-01

Full Text Available This paper deals with the problem of identifying and filtering a class of continuous-time nonlinear dynamic games (nonlinear differential games subject to additive and undesired deterministic perturbations. Moreover, the mathematical model of this class is completely unknown with the exception of the control actions of each player, and even though the deterministic noises are known, their power (or their effect is not. Therefore, two differential neural networks are designed in order to obtain a feedback (perfect state information pattern for the mentioned class of games. In this way, the stability conditions for two state identification errors and for a filtering error are established, the upper bounds of these errors are obtained, and two new learning laws for each neural network are suggested. Finally, an illustrating example shows the applicability of this approach.

A preconditioner for the finite element computation of incompressible, nonlinear elastic deformations

Science.gov (United States)

Whiteley, J. P.

2017-10-01

Large, incompressible elastic deformations are governed by a system of nonlinear partial differential equations. The finite element discretisation of these partial differential equations yields a system of nonlinear algebraic equations that are usually solved using Newton's method. On each iteration of Newton's method, a linear system must be solved. We exploit the structure of the Jacobian matrix to propose a preconditioner, comprising two steps. The first step is the solution of a relatively small, symmetric, positive definite linear system using the preconditioned conjugate gradient method. This is followed by a small number of multigrid V-cycles for a larger linear system. Through the use of exemplar elastic deformations, the preconditioner is demonstrated to facilitate the iterative solution of the linear systems arising. The number of GMRES iterations required has only a very weak dependence on the number of degrees of freedom of the linear systems.

Waveform relaxation methods for implicit differential equations

NARCIS (Netherlands)

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

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

Science.gov (United States)

Goldston, J. W.

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

Improved differential evolution algorithms for handling economic dispatch optimization with generator constraints

International Nuclear Information System (INIS)

Coelho, Leandro dos Santos; Mariani, Viviana Cocco

2007-01-01

Global optimization based on evolutionary algorithms can be used as the important component for many engineering optimization problems. Evolutionary algorithms have yielded promising results for solving nonlinear, non-differentiable and multi-modal optimization problems in the power systems area. Differential evolution (DE) is a simple and efficient evolutionary algorithm for function optimization over continuous spaces. It has reportedly outperformed search heuristics when tested over both benchmark and real world problems. This paper proposes improved DE algorithms for solving economic load dispatch problems that take into account nonlinear generator features such as ramp rate limits and prohibited operating zones in the power system operation. The DE algorithms and its variants are validated for two test systems consisting of 6 and 15 thermal units. Various DE approaches outperforms other state of the art algorithms reported in the literature in solving load dispatch problems with generator constraints

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

Science.gov (United States)

Li, Shuai; Li, Yangming

2013-10-28

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

A new perspective for quintic B-spline based Crank-Nicolson-differential quadrature method algorithm for numerical solutions of the nonlinear Schrödinger equation

Science.gov (United States)

Başhan, Ali; Uçar, Yusuf; Murat Yağmurlu, N.; Esen, Alaattin

2018-01-01

In the present paper, a Crank-Nicolson-differential quadrature method (CN-DQM) based on utilizing quintic B-splines as a tool has been carried out to obtain the numerical solutions for the nonlinear Schrödinger (NLS) equation. For this purpose, first of all, the Schrödinger equation has been converted into coupled real value differential equations and then they have been discretized using both the forward difference formula and the Crank-Nicolson method. After that, Rubin and Graves linearization techniques have been utilized and the differential quadrature method has been applied to obtain an algebraic equation system. Next, in order to be able to test the efficiency of the newly applied method, the error norms, L2 and L_{∞}, as well as the two lowest invariants, I1 and I2, have been computed. Besides those, the relative changes in those invariants have been presented. Finally, the newly obtained numerical results have been compared with some of those available in the literature for similar parameters. This comparison clearly indicates that the currently utilized method, namely CN-DQM, is an effective and efficient numerical scheme and allows us to propose to solve a wide range of nonlinear equations.

Equivalence transformations and differential invariants of a generalized nonlinear Schroedinger equation

International Nuclear Information System (INIS)

Senthilvelan, M; Torrisi, M; Valenti, A

2006-01-01

By using Lie's invariance infinitesimal criterion, we obtain the continuous equivalence transformations of a class of nonlinear Schroedinger equations with variable coefficients. We construct the differential invariants of order 1 starting from a special equivalence subalgebra E χ o . We apply these latter ones to find the most general subclass of variable coefficient nonlinear Schr?dinger equations which can be mapped, by means of an equivalence transformation of E χ o , to the well-known cubic Schroedinger equation. We also provide the explicit form of the transformation

An Efficient Series Solution for Nonlinear Multiterm Fractional Differential Equations

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Moh’d Khier Al-Srihin

2017-01-01

Full Text Available In this paper, we introduce an efficient series solution for a class of nonlinear multiterm fractional differential equations of Caputo type. The approach is a generalization to our recent work for single fractional differential equations. We extend the idea of the Taylor series expansion method to multiterm fractional differential equations, where we overcome the difficulty of computing iterated fractional derivatives, which are difficult to be computed in general. The terms of the series are obtained sequentially using a closed formula, where only integer derivatives have to be computed. Several examples are presented to illustrate the efficiency of the new approach and comparison with the Adomian decomposition method is performed.

Solution of linear and nonlinear matrix systems. Application to a nonlinear diffusion equation

International Nuclear Information System (INIS)

Bonnet, M.; Meurant, G.

1978-01-01

The object of this study is to compare different methods of solving linear and nonlinear algebraic systems and to apply them to the nonlinear system obtained by discretizing a nonlinear diffusion equation. For linear systems the conventional methods of alternating direction type or Gauss Seidel's methods are compared to more recent ones of the type of generalized conjugate gradient; the superiority of the latter is shown by numerical examples. For nonlinear systems, a method of nonlinear conjugate gradient is studied together with Newton's method and some of its variants. It should be noted, however, that Newton's method is found to be more efficient when coupled with a good method for solving the linear system. As a conclusion, these methods are used to solve a nonlinear diffusion problem and the numerical results obtained are compared [fr

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

Science.gov (United States)

Murase, Kenya

2016-01-01

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

Nonlinear partial differential equations and their applications

CERN Document Server

Lions, Jacques Louis

2002-01-01

This book contains the written versions of lectures delivered since 1997 in the well-known weekly seminar on Applied Mathematics at the Collège de France in Paris, directed by Jacques-Louis Lions. It is the 14th and last of the series, due to the recent and untimely death of Professor Lions. The texts in this volume deal mostly with various aspects of the theory of nonlinear partial differential equations. They present both theoretical and applied results in many fields of growing importance such as Calculus of variations and optimal control, optimization, system theory and control, op

Differential morphology and image processing.

Science.gov (United States)

Maragos, P

1996-01-01

Image processing via mathematical morphology has traditionally used geometry to intuitively understand morphological signal operators and set or lattice algebra to analyze them in the space domain. We provide a unified view and analytic tools for morphological image processing that is based on ideas from differential calculus and dynamical systems. This includes ideas on using partial differential or difference equations (PDEs) to model distance propagation or nonlinear multiscale processes in images. We briefly review some nonlinear difference equations that implement discrete distance transforms and relate them to numerical solutions of the eikonal equation of optics. We also review some nonlinear PDEs that model the evolution of multiscale morphological operators and use morphological derivatives. Among the new ideas presented, we develop some general 2-D max/min-sum difference equations that model the space dynamics of 2-D morphological systems (including the distance computations) and some nonlinear signal transforms, called slope transforms, that can analyze these systems in a transform domain in ways conceptually similar to the application of Fourier transforms to linear systems. Thus, distance transforms are shown to be bandpass slope filters. We view the analysis of the multiscale morphological PDEs and of the eikonal PDE solved via weighted distance transforms as a unified area in nonlinear image processing, which we call differential morphology, and briefly discuss its potential applications to image processing and computer vision.

Analytical solutions of time-fractional models for homogeneous Gardner equation and non-homogeneous differential equations

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Olaniyi Samuel Iyiola

2014-09-01

Full Text Available In this paper, we obtain analytical solutions of homogeneous time-fractional Gardner equation and non-homogeneous time-fractional models (including Buck-master equation using q-Homotopy Analysis Method (q-HAM. Our work displays the elegant nature of the application of q-HAM not only to solve homogeneous non-linear fractional differential equations but also to solve the non-homogeneous fractional differential equations. The presence of the auxiliary parameter h helps in an effective way to obtain better approximation comparable to exact solutions. The fraction-factor in this method gives it an edge over other existing analytical methods for non-linear differential equations. Comparisons are made upon the existence of exact solutions to these models. The analysis shows that our analytical solutions converge very rapidly to the exact solutions.

Unique solvability of a non-linear non-local boundary-value problem for systems of non-linear functional differential equations

Czech Academy of Sciences Publication Activity Database

Dilna, N.; Rontó, András

2010-01-01

Roč. 60, č. 3 (2010), s. 327-338 ISSN 0139-9918 R&D Projects: GA ČR(CZ) GA201/06/0254 Institutional research plan: CEZ:AV0Z10190503 Keywords : non-linear boundary value-problem * functional differential equation * non-local condition * unique solvability * differential inequality Subject RIV: BA - General Mathematics Impact factor: 0.316, year: 2010 http://link.springer.com/article/10.2478%2Fs12175-010-0015-9

Algorithmic Verification of Linearizability for Ordinary Differential Equations

KAUST Repository

Lyakhov, Dmitry A.

2017-07-19

For a nonlinear ordinary differential equation solved with respect to the highest order derivative and rational in the other derivatives and in the independent variable, we devise two algorithms to check if the equation can be reduced to a linear one by a point transformation of the dependent and independent variables. The first algorithm is based on a construction of the Lie point symmetry algebra and on the computation of its derived algebra. The second algorithm exploits the differential Thomas decomposition and allows not only to test the linearizability, but also to generate a system of nonlinear partial differential equations that determines the point transformation and the coefficients of the linearized equation. The implementation of both algorithms is discussed and their application is illustrated using several examples.

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

Directory of Open Access Journals (Sweden)

Sufia Zulfa Ahmad

2016-01-01

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

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

Science.gov (United States)

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

2013-07-01

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

Methods of solving nonstandard problems

CERN Document Server

Grigorieva, Ellina

2015-01-01

This book, written by an accomplished female mathematician, is the second to explore nonstandard mathematical problems – those that are not directly solved by standard mathematical methods but instead rely on insight and the synthesis of a variety of mathematical ideas.   It promotes mental activity as well as greater mathematical skills, and is an ideal resource for successful preparation for the mathematics Olympiad. Numerous strategies and techniques are presented that can be used to solve intriguing and challenging problems of the type often found in competitions.  The author uses a friendly, non-intimidating approach to emphasize connections between different fields of mathematics and often proposes several different ways to attack the same problem.  Topics covered include functions and their properties, polynomials, trigonometric and transcendental equations and inequalities, optimization, differential equations, nonlinear systems, and word problems.   Over 360 problems are included with hints, ...

Right-invertibility for a class of nonlinear control systems: A geometric approach

NARCIS (Netherlands)

Nijmeijer, Henk

1986-01-01

In recent years it has become evident that various synthesis problems known from linear system theory can also be solved for nonlinear control systems by using differential geometric methods. The purpose of this paper is to use this mathematical framework for giving a preliminary account on the

Seven common errors in finding exact solutions of nonlinear differential equations

NARCIS (Netherlands)

Kudryashov, Nikolai A.

2009-01-01

We analyze the common errors of the recent papers in which the solitary wave solutions of nonlinear differential equations are presented. Seven common errors are formulated and classified. These errors are illustrated by using multiple examples of the common errors from the recent publications. We

New Look at Nonlinear Aerodynamics in Analysis of Hypersonic Panel Flutter

Directory of Open Access Journals (Sweden)

Dan Xie

2017-01-01

Full Text Available A simply supported plate fluttering in hypersonic flow is investigated considering both the airflow and structural nonlinearities. Third-order piston theory is used for nonlinear aerodynamic loading, and von Karman plate theory is used for modeling the nonlinear strain-displacement relation. The Galerkin method is applied to project the partial differential governing equations (PDEs into a set of ordinary differential equations (ODEs in time, which is then solved by numerical integration method. In observation of limit cycle oscillations (LCO and evolution of dynamic behaviors, nonlinear aerodynamic loading produces a smaller positive deflection peak and more complex bifurcation diagrams compared with linear aerodynamics. Moreover, a LCO obtained with the linear aerodynamics is mostly a nonsimple harmonic motion but when the aerodynamic nonlinearity is considered more complex motions are obtained, which is important in the evaluation of fatigue life. The parameters of Mach number, dynamic pressure, and in-plane thermal stresses all affect the aerodynamic nonlinearity. For a specific Mach number, there is a critical dynamic pressure beyond which the aerodynamic nonlinearity has to be considered. For a higher temperature, a lower critical dynamic pressure is required. Each nonlinear aerodynamic term in the full third-order piston theory is evaluated, based on which the nonlinear aerodynamic formulation has been simplified.

Nonclassical Symmetries for Nonlinear Partial Differential Equations via Compatibility

International Nuclear Information System (INIS)

El-Sabbagh, Mostafa F.; Ahmad, Ali T.

2011-01-01

The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the invariant surface conditions. The (2+1)-dimensional shallow water wave equation, Boussinesq equation, and the dispersive wave equations in shallow water serve as examples illustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries. (general)

Multiple Positive Solutions for Nonlinear Semipositone Fractional Differential Equations

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Wen-Xue Zhou

2012-01-01

Full Text Available We present some new multiplicity of positive solutions results for nonlinear semipositone fractional boundary value problem D0+αu(t=p(tf(t,u(t-q(t,0differentiation. One example is also given to illustrate the main result.

Viscous Flow over Nonlinearly Stretching Sheet with Effects of Viscous Dissipation

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Javad Alinejad

2012-01-01

Full Text Available The flow and heat transfer characteristics of incompressible viscous flow over a nonlinearly stretching sheet with the presence of viscous dissipation is investigated numerically. The similarity transformation reduces the time-independent boundary layer equations for momentum and thermal energy into a set of coupled ordinary differential equations. The obtained equations, including nonlinear equation for the velocity field and differential equation by variable coefficient for the temperature field , are solved numerically by using the fourth order of Runge-Kutta integration scheme accompanied by shooting technique with Newton-Raphson iteration method. The effect of various values of Prandtl number, Eckert number and nonlinear stretching parameter are studied. The results presented graphically show some behaviors such as decrease in dimensionless temperature due to increase in Pr number, and curve relocations are observed when heat dissipation is considered.

Numerical Integration of a Class of Singularly Perturbed Delay Differential Equations with Small Shift

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Gemechis File

2012-01-01

Full Text Available We have presented a numerical integration method to solve a class of singularly perturbed delay differential equations with small shift. First, we have replaced the second-order singularly perturbed delay differential equation by an asymptotically equivalent first-order delay differential equation. Then, Simpson’s rule and linear interpolation are employed to get the three-term recurrence relation which is solved easily by discrete invariant imbedding algorithm. The method is demonstrated by implementing it on several linear and nonlinear model examples by taking various values for the delay parameter and the perturbation parameter .

Nonlinear model predictive control of a wave energy converter based on differential flatness parameterisation

Science.gov (United States)

Li, Guang

2017-01-01

This paper presents a fast constrained optimization approach, which is tailored for nonlinear model predictive control of wave energy converters (WEC). The advantage of this approach relies on its exploitation of the differential flatness of the WEC model. This can reduce the dimension of the resulting nonlinear programming problem (NLP) derived from the continuous constrained optimal control of WEC using pseudospectral method. The alleviation of computational burden using this approach helps to promote an economic implementation of nonlinear model predictive control strategy for WEC control problems. The method is applicable to nonlinear WEC models, nonconvex objective functions and nonlinear constraints, which are commonly encountered in WEC control problems. Numerical simulations demonstrate the efficacy of this approach.

Applications of IBSOM and ETEM for solving the nonlinear chains of atoms with long-range interactions

Science.gov (United States)

Foroutan, Mohammadreza; Zamanpour, Isa; Manafian, Jalil

2017-10-01

This paper presents a number of new solutions obtained for solving a complex nonlinear equation describing dynamics of nonlinear chains of atoms via the improved Bernoulli sub-ODE method (IBSOM) and the extended trial equation method (ETEM). The proposed solutions are kink solitons, anti-kink solitons, soliton solutions, hyperbolic solutions, trigonometric solutions, and bellshaped soliton solutions. Then our new results are compared with the well-known results. The methods used here are very simple and succinct and can be also applied to other nonlinear models. The balance number of these methods is not constant contrary to other methods. The proposed methods also allow us to establish many new types of exact solutions. By utilizing the Maple software package, we show that all obtained solutions satisfy the conditions of the studied model. More importantly, the solutions found in this work can have significant applications in Hamilton's equations and generalized momentum where solitons are used for long-range interactions.

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

Measurable Strategies in Differential Games

Science.gov (United States)

Ivanov, R. P.

1990-02-01

Nonlinear approach-evasion differential games are considered in which the initial data depend on the time. These games are investigated in the class of strategies that are functions of three variables, namely, the time, the phase variable, and the current value of the other player's control, and are measurable jointly with respect to the time and the phase variable. The ideas of the Pontryagin methods in differential games and Krasovskiĭ's ideas on extremal aiming are developed, and it is shown that measurable strategies have broad applicability. It is proved that measurable strategies are compatible with differential equations with discontinuous right-hand side, and general theorems on the existence of solving measurable strategies in approach-evasion problems are proved, along with some auxiliary assertions. It is shown that the saddle point condition in the small game ensures the existence of solving measurable strategies. An example is given. Bibliography: 14 titles.

DOUBLE TRIALS METHOD FOR NONLINEAR PROBLEMS ARISING IN HEAT TRANSFER

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Chun-Hui He

2011-01-01

Full Text Available According to an ancient Chinese algorithm, the Ying Buzu Shu, in about second century BC, known as the rule of double false position in West after 1202 AD, two trial roots are assumed to solve algebraic equations. The solution procedure can be extended to solve nonlinear differential equations by constructing an approximate solution with an unknown parameter, and the unknown parameter can be easily determined using the Ying Buzu Shu. An example in heat transfer is given to elucidate the solution procedure.

Integrator Performance Analysis In Solving Stiff Differential Equation System

International Nuclear Information System (INIS)

B, Alhadi; Basaruddin, T.

2001-01-01

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

Thermal rectification and negative differential thermal conductance in harmonic chains with nonlinear system-bath coupling

Science.gov (United States)

Ming, Yi; Li, Hui-Min; Ding, Ze-Jun

2016-03-01

Thermal rectification and negative differential thermal conductance were realized in harmonic chains in this work. We used the generalized Caldeira-Leggett model to study the heat flow. In contrast to most previous studies considering only the linear system-bath coupling, we considered the nonlinear system-bath coupling based on recent experiment [Eichler et al., Nat. Nanotech. 6, 339 (2011), 10.1038/nnano.2011.71]. When the linear coupling constant is weak, the multiphonon processes induced by the nonlinear coupling allow more phonons transport across the system-bath interface and hence the heat current is enhanced. Consequently, thermal rectification and negative differential thermal conductance are achieved when the nonlinear couplings are asymmetric. However, when the linear coupling constant is strong, the umklapp processes dominate the multiphonon processes. Nonlinear coupling suppresses the heat current. Thermal rectification is also achieved. But the direction of rectification is reversed compared to the results of weak linear coupling constant.

AD Model Builder: using automatic differentiation for statistical inference of highly parameterized complex nonlinear models

DEFF Research Database (Denmark)

Fournier, David A.; Skaug, Hans J.; Ancheta, Johnoel

2011-01-01

Many criteria for statistical parameter estimation, such as maximum likelihood, are formulated as a nonlinear optimization problem.Automatic Differentiation Model Builder (ADMB) is a programming framework based on automatic differentiation, aimed at highly nonlinear models with a large number...... of such a feature is the generic implementation of Laplace approximation of high-dimensional integrals for use in latent variable models. We also review the literature in which ADMB has been used, and discuss future development of ADMB as an open source project. Overall, the main advantages ofADMB are flexibility...

Analytic solutions of a class of nonlinearly dynamic systems

International Nuclear Information System (INIS)

Wang, M-C; Zhao, X-S; Liu, X

2008-01-01

In this paper, the homotopy perturbation method (HPM) is applied to solve a coupled system of two nonlinear differential with first-order similar model of Lotka-Volterra and a Bratus equation with a source term. The analytic approximate solutions are derived. Furthermore, the analytic approximate solutions obtained by the HPM with the exact solutions reveals that the present method works efficiently

Derivation of a macroscale formulation for a class of nonlinear partial differential equations

International Nuclear Information System (INIS)

Pantelis, G.

1995-05-01

A macroscale formulation is constructed from a system of partial differential equations which govern the microscale dependent variables. The construction is based upon the requirement that the solutions of the macroscale partial differential equations satisfy, in some approximate sense, the system of partial differential equations associated with the microscale. These results are restricted to the class of nonlinear partial differential equations which can be expressed as polynomials of the dependent variables and their partial derivatives up to second order. A linear approximation of transformations of second order contact manifolds is employed. 6 refs

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

Science.gov (United States)

Khotimah, Rita Pramujiyanti; Masduki

2016-01-01

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

Stability and square integrability of solutions of nonlinear fourth order differential equations

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Moussadek Remili

2016-05-01

Full Text Available The aim of the present paper is to establish a new result, which guarantees the asymptotic stability of zero solution and square integrability of solutions and their derivatives to nonlinear differential equations of fourth order.

Nonlinear recurrent neural networks for finite-time solution of general time-varying linear matrix equations.

Science.gov (United States)

Xiao, Lin; Liao, Bolin; Li, Shuai; Chen, Ke

2018-02-01

In order to solve general time-varying linear matrix equations (LMEs) more efficiently, this paper proposes two nonlinear recurrent neural networks based on two nonlinear activation functions. According to Lyapunov theory, such two nonlinear recurrent neural networks are proved to be convergent within finite-time. Besides, by solving differential equation, the upper bounds of the finite convergence time are determined analytically. Compared with existing recurrent neural networks, the proposed two nonlinear recurrent neural networks have a better convergence property (i.e., the upper bound is lower), and thus the accurate solutions of general time-varying LMEs can be obtained with less time. At last, various different situations have been considered by setting different coefficient matrices of general time-varying LMEs and a great variety of computer simulations (including the application to robot manipulators) have been conducted to validate the better finite-time convergence of the proposed two nonlinear recurrent neural networks. Copyright © 2017 Elsevier Ltd. All rights reserved.

Nonlinear Maps and their Applications 2011 International Workshop

CERN Document Server

Fournier-Prunaret, Daniele; Ueta, Tetsushi; Nishio, Yoshifumi

2014-01-01

In the field of Dynamical Systems, nonlinear iterative processes play an important role. Nonlinear mappings can be found as immediate models for many systems from different scientific areas, such as engineering, economics, biology, or can also be obtained via numerical methods permitting to solve non-linear differential equations. In both cases, the understanding of specific dynamical behaviors and phenomena is of the greatest interest for scientists. This volume contains papers that were presented at the International Workshop on Nonlinear Maps and their Applications (NOMA 2011) held in Évora, Portugal, on September 15-16, 2011. This kind of collaborative effort is of paramount importance in promoting communication among the various groups that work in dynamical systems and networks in their research theoretical studies as well as for applications. This volume is suitable for graduate students as well as researchers in the field.

Adaptive Neural Control of Nonaffine Nonlinear Systems without Differential Condition for Nonaffine Function

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Chaojiao Sun

2016-01-01

Full Text Available An adaptive neural control scheme is proposed for nonaffine nonlinear system without using the implicit function theorem or mean value theorem. The differential conditions on nonaffine nonlinear functions are removed. The control-gain function is modeled with the nonaffine function probably being indifferentiable. Furthermore, only a semibounded condition for nonaffine nonlinear function is required in the proposed method, and the basic idea of invariant set theory is then constructively introduced to cope with the difficulty in the control design for nonaffine nonlinear systems. It is rigorously proved that all the closed-loop signals are bounded and the tracking error converges to a small residual set asymptotically. Finally, simulation examples are provided to demonstrate the effectiveness of the designed method.

Nonlinear behaviour of cantilevered carbon nanotube resonators based on a new nonlinear electrostatic load model

Science.gov (United States)

Farokhi, Hamed; Païdoussis, Michael P.; Misra, Arun K.

2018-04-01

The present study examines the nonlinear behaviour of a cantilevered carbon nanotube (CNT) resonator and its mass detection sensitivity, employing a new nonlinear electrostatic load model. More specifically, a 3D finite element model is developed in order to obtain the electrostatic load distribution on cantilevered CNT resonators. A new nonlinear electrostatic load model is then proposed accounting for the end effects due to finite length. Additionally, a new nonlinear size-dependent continuum model is developed for the cantilevered CNT resonator, employing the modified couple stress theory (to account for size-effects) together with the Kelvin-Voigt model (to account for nonlinear damping); the size-dependent model takes into account all sources of nonlinearity, i.e. geometrical and inertial nonlinearities as well as nonlinearities associated with damping, small-scale, and electrostatic load. The nonlinear equation of motion of the cantilevered CNT resonator is obtained based on the new models developed for the CNT resonator and the electrostatic load. The Galerkin method is then applied to the nonlinear equation of motion, resulting in a set of nonlinear ordinary differential equations, consisting of geometrical, inertial, electrical, damping, and size-dependent nonlinear terms. This high-dimensional nonlinear discretized model is solved numerically utilizing the pseudo-arclength continuation technique. The nonlinear static and dynamic responses of the system are examined for various cases, investigating the effect of DC and AC voltages, length-scale parameter, nonlinear damping, and electrostatic load. Moreover, the mass detection sensitivity of the system is examined for possible application of the CNT resonator as a nanosensor.

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

Time-domain Green's Function Method for three-dimensional nonlinear subsonic flows

Science.gov (United States)

Tseng, K.; Morino, L.

1978-01-01

The Green's Function Method for linearized 3D unsteady potential flow (embedded in the computer code SOUSSA P) is extended to include the time-domain analysis as well as the nonlinear term retained in the transonic small disturbance equation. The differential-delay equations in time, as obtained by applying the Green's Function Method (in a generalized sense) and the finite-element technique to the transonic equation, are solved directly in the time domain. Comparisons are made with both linearized frequency-domain calculations and existing nonlinear results.

The modified simplest equation method to look for exact solutions of nonlinear partial differential equations

OpenAIRE

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.

Darcy-Forchheimer flow of Maxwell nanofluid flow with nonlinear thermal radiation and activation energy

Directory of Open Access Journals (Sweden)

T. Sajid

2018-03-01

Full Text Available The present article is about the study of Darcy-Forchheimer flow of Maxwell nanofluid over a linear stretching surface. Effects like variable thermal conductivity, activation energy, nonlinear thermal radiation is also incorporated for the analysis of heat and mass transfer. The governing nonlinear partial differential equations (PDEs with convective boundary conditions are first converted into the nonlinear ordinary differential equations (ODEs with the help of similarity transformation, and then the resulting nonlinear ODEs are solved with the help of shooting method and MATLAB built-in bvp4c solver. The impact of different physical parameters like Brownian motion, thermophoresis parameter, Reynolds number, magnetic parameter, nonlinear radiative heat flux, Prandtl number, Lewis number, reaction rate constant, activation energy and Biot number on Nusselt number, velocity, temperature and concentration profile has been discussed. It is viewed that both thermophoresis parameter and activation energy parameter has ascending effect on the concentration profile.

A bivariate Chebyshev spectral collocation quasilinearization method for nonlinear evolution parabolic equations.

Science.gov (United States)

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.

A Bivariate Chebyshev Spectral Collocation Quasilinearization Method for Nonlinear Evolution Parabolic Equations

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.

A three operator split-step method covering a larger set of non-linear partial differential equations

Science.gov (United States)

Zia, Haider

2017-06-01

This paper describes an updated exponential Fourier based split-step method that can be applied to a greater class of partial differential equations than previous methods would allow. These equations arise in physics and engineering, a notable example being the generalized derivative non-linear Schrödinger equation that arises in non-linear optics with self-steepening terms. These differential equations feature terms that were previously inaccessible to model accurately with low computational resources. The new method maintains a 3rd order error even with these additional terms and models the equation in all three spatial dimensions and time. The class of non-linear differential equations that this method applies to is shown. The method is fully derived and implementation of the method in the split-step architecture is shown. This paper lays the mathematical ground work for an upcoming paper employing this method in white-light generation simulations in bulk material.

The generalized tanh method to obtain exact solutions of nonlinear partial differential equation

OpenAIRE

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.

EXISTENCE OF SOLUTION TO NONLINEAR SECOND ORDER NEUTRAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH DELAY

Institute of Scientific and Technical Information of China (English)

无

2010-01-01

This paper is concerned with nonlinear second order neutral stochastic differential equations with delay in a Hilbert space. Sufficient conditions for the existence of solution to the system are obtained by Picard iterations.

Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems

International Nuclear Information System (INIS)

Abdel-Halim Hassan, I.H.

2008-01-01

In this paper, we will compare the differential transformation method DTM and Adomian decomposition method ADM to solve partial differential equations (PDEs). The definition and operations of differential transform method was introduced by Zhou [Zhou JK. Differential transformation and its application for electrical circuits. Wuuhahn, China: Huarjung University Press; 1986 [in Chinese

Solving Boundary Value Problem for a Nonlinear Stationary Controllable System with Synthesizing Control

Directory of Open Access Journals (Sweden)

Alexander N. Kvitko

2017-01-01

Full Text Available An algorithm for constructing a control function that transfers a wide class of stationary nonlinear systems of ordinary differential equations from an initial state to a final state under certain control restrictions is proposed. The algorithm is designed to be convenient for numerical implementation. A constructive criterion of the desired transfer possibility is presented. The problem of an interorbital flight is considered as a test example and it is simulated numerically with the presented method.

Spectral methods for a nonlinear initial value problem involving pseudo differential operators

International Nuclear Information System (INIS)

Pasciak, J.E.

1982-01-01

Spectral methods (Fourier methods) for approximating the solution of a nonlinear initial value problem involving pseudo differential operators are defined and analyzed. A semidiscrete approximation to the nonlinear equation based on an L 2 projection is described. The semidiscrete L 2 approximation is shown to be a priori stable and convergent under sufficient decay and smoothness assumptions on the initial data. It is shown that the semidiscrete method converges with infinite order, that is, higher order decay and smoothness assumptions imply higher order error bounds. Spectral schemes based on spacial collocation are also discussed

Inexact Newton–Landweber iteration for solving nonlinear inverse problems in Banach spaces

International Nuclear Information System (INIS)

Jin, Qinian

2012-01-01

By making use of duality mappings, we formulate an inexact Newton–Landweber iteration method for solving nonlinear inverse problems in Banach spaces. The method consists of two components: an outer Newton iteration and an inner scheme providing the increments by applying the Landweber iteration in Banach spaces to the local linearized equations. It has the advantage of reducing computational work by computing more cheap steps in each inner scheme. We first prove a convergence result for the exact data case. When the data are given approximately, we terminate the method by a discrepancy principle and obtain a weak convergence result. Finally, we test the method by reporting some numerical simulations concerning the sparsity recovery and the noisy data containing outliers. (paper)

Composite spectral functions for solving Volterra's population model

International Nuclear Information System (INIS)

Ramezani, M.; Razzaghi, M.; Dehghan, M.

2007-01-01

An approximate method for solving Volterra's population model for population growth of a species in a closed system is proposed. Volterra's model is a nonlinear integro-differential equation, where the integral term represents the effect of toxin. The approach is based upon composite spectral functions approximations. The properties of composite spectral functions consisting of few terms of orthogonal functions are presented and are utilized to reduce the solution of the Volterra's model to the solution of a system of algebraic equations. The method is easy to implement and yields very accurate result

Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear ordinary differential equations

Institute of Scientific and Technical Information of China (English)

WANG; Shunjin; ZHANG; Hua

2006-01-01

The problem of preserving fidelity in numerical computation of nonlinear ordinary differential equations is studied in terms of preserving local differential structure and approximating global integration structure of the dynamical system.The ordinary differential equations are lifted to the corresponding partial differential equations in the framework of algebraic dynamics,and a new algorithm-algebraic dynamics algorithm is proposed based on the exact analytical solutions of the ordinary differential equations by the algebraic dynamics method.In the new algorithm,the time evolution of the ordinary differential system is described locally by the time translation operator and globally by the time evolution operator.The exact analytical piece-like solution of the ordinary differential equations is expressd in terms of Taylor series with a local convergent radius,and its finite order truncation leads to the new numerical algorithm with a controllable precision better than Runge Kutta Algorithm and Symplectic Geometric Algorithm.

Clawpack: Building an open source ecosystem for solving hyperbolic PDEs

Science.gov (United States)

Iverson, Richard M.; Mandli, K.T.; Ahmadia, Aron J.; Berger, M.J.; Calhoun, Donna; George, David L.; Hadjimichael, Y.; Ketcheson, David I.; Lemoine, Grady L.; LeVeque, Randall J.

2016-01-01

Clawpack is a software package designed to solve nonlinear hyperbolic partial differential equations using high-resolution finite volume methods based on Riemann solvers and limiters. The package includes a number of variants aimed at different applications and user communities. Clawpack has been actively developed as an open source project for over 20 years. The latest major release, Clawpack 5, introduces a number of new features and changes to the code base and a new development model based on GitHub and Git submodules. This article provides a summary of the most significant changes, the rationale behind some of these changes, and a description of our current development model. Clawpack: building an open source ecosystem for solving hyperbolic PDEs.

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.

"E pluribus unum" or How to Derive Single-equation Descriptions for Output-quantities in Nonlinear Circuits using Differential Algebra

OpenAIRE

Gerbracht, Eberhard H. -A.

2008-01-01

In this paper we describe by a number of examples how to deduce one single characterizing higher order differential equation for output quantities of an analog circuit. In the linear case, we apply basic "symbolic" methods from linear algebra to the system of differential equations which is used to model the analog circuit. For nonlinear circuits and their corresponding nonlinear differential equations, we show how to employ computer algebra tools implemented in Maple, which are based on diff...

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

Science.gov (United States)

Parand, K.; Nikarya, M.

2017-11-01

In this paper a novel method will be introduced to solve a nonlinear partial differential equation (PDE). In the proposed method, we use the spectral collocation method based on Bessel functions of the first kind and the Jacobian free Newton-generalized minimum residual (JFNGMRes) method with adaptive preconditioner. In this work a nonlinear PDE has been converted to a nonlinear system of algebraic equations using the collocation method based on Bessel functions without any linearization, discretization or getting the help of any other methods. Finally, by using JFNGMRes, the solution of the nonlinear algebraic system is achieved. To illustrate the reliability and efficiency of the proposed method, we solve some examples of the famous Fisher equation. We compare our results with other methods.

Solving a Local Boundary Value Problem for a Nonlinear Nonstationary System in the Class of Feedback Controls

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Kvitko, A. N.

2018-01-01

An algorithm convenient for numerical implementation is proposed for constructing differentiable control functions that transfer a wide class of nonlinear nonstationary systems of ordinary differential equations from an initial state to a given point of the phase space. Constructive sufficient conditions imposed on the right-hand side of the controlled system are obtained under which this transfer is possible. The control of a robotic manipulator is considered, and its numerical simulation is performed.

An iterative kernel based method for fourth order nonlinear equation with nonlinear boundary condition

Science.gov (United States)

Azarnavid, Babak; Parand, Kourosh; Abbasbandy, Saeid

2018-06-01

This article discusses an iterative reproducing kernel method with respect to its effectiveness and capability of solving a fourth-order boundary value problem with nonlinear boundary conditions modeling beams on elastic foundations. Since there is no method of obtaining reproducing kernel which satisfies nonlinear boundary conditions, the standard reproducing kernel methods cannot be used directly to solve boundary value problems with nonlinear boundary conditions as there is no knowledge about the existence and uniqueness of the solution. The aim of this paper is, therefore, to construct an iterative method by the use of a combination of reproducing kernel Hilbert space method and a shooting-like technique to solve the mentioned problems. Error estimation for reproducing kernel Hilbert space methods for nonlinear boundary value problems have yet to be discussed in the literature. In this paper, we present error estimation for the reproducing kernel method to solve nonlinear boundary value problems probably for the first time. Some numerical results are given out to demonstrate the applicability of the method.

Maillet type theorem for singular first order nonlinear partial differential equations of totally characteristic type. Part II

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Akira Shirai

2015-01-01

Full Text Available In this paper, we study the following nonlinear first order partial differential equation: \\[f(t,x,u,\\partial_t u,\\partial_x u=0\\quad\\text{with}\\quad u(0,x\\equiv 0.\\] The purpose of this paper is to determine the estimate of Gevrey order under the condition that the equation is singular of a totally characteristic type. The Gevrey order is indicated by the rate of divergence of a formal power series. This paper is a continuation of the previous papers [Convergence of formal solutions of singular first order nonlinear partial differential equations of totally characteristic type, Funkcial. Ekvac. 45 (2002, 187-208] and [Maillet type theorem for singular first order nonlinear partial differential equations of totally characteristic type, Surikaiseki Kenkyujo Kokyuroku, Kyoto University 1431 (2005, 94-106]. Especially the last-mentioned paper is regarded as part I of this paper.

Solution of Fractional Partial Differential Equations in Fluid Mechanics by Extension of Some Iterative Method

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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.

Periodic solutions of certain third order nonlinear differential systems with delay

International Nuclear Information System (INIS)

Tejumola, H.O.; Afuwape, A.U.

1990-12-01

This paper investigates the existence of 2π-periodic solutions of systems of third-order nonlinear differential equations, with delay, under varied assumptions. The results obtained extend earlier works of Tejumola and generalize to third order systems those of Conti, Iannacci and Nkashama as well as DePascale and Iannacci and Iannacci and Nkashama. 16 refs

Third-order nonlinear differential operators preserving invariant subspaces of maximal dimension

International Nuclear Information System (INIS)

Qu Gai-Zhu; Zhang Shun-Li; Li Yao-Long

2014-01-01

In this paper, third-order nonlinear differential operators are studied. It is shown that they are quadratic forms when they preserve invariant subspaces of maximal dimension. A complete description of third-order quadratic operators with constant coefficients is obtained. One example is given to derive special solutions for evolution equations with third-order quadratic operators. (general)

A practical course in differential equations and mathematical modeling

CERN Document Server

Ibragimov , Nail H

2009-01-01

A Practical Course in Differential Equations and Mathematical Modelling is a unique blend of the traditional methods of ordinary and partial differential equations with Lie group analysis enriched by the author's own theoretical developments. The book which aims to present new mathematical curricula based on symmetry and invariance principles is tailored to develop analytic skills and working knowledge in both classical and Lie's methods for solving linear and nonlinear equations. This approach helps to make courses in differential equations, mathematical modelling, distributions and fundame

Lie Group Classifications and Non-differentiable Solutions for Time-Fractional Burgers Equation

International Nuclear Information System (INIS)

Wu Guocheng

2011-01-01

Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests Lie group method for fractional partial differential equations. A time-fractional Burgers equation is used as an example to illustrate the effectiveness of the Lie group method and some classes of exact solutions are obtained. (electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics)

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

Science.gov (United States)

Parker, Adam E.

2013-01-01

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

Nonlinear Waves In A Stenosed Elastic Tube Filled With Viscous Fluid: Forced Perturbed Korteweg-De Vries Equation

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Gaik*, Tay Kim; Demiray, Hilmi; Tiong, Ong Chee

In the present work, treating the artery as a prestressed thin-walled and long circularly cylindrical elastic tube with a mild symmetrical stenosis and the blood as an incompressible Newtonian fluid, we have studied the pro pagation of weakly nonlinear waves in such a composite medium, in the long wave approximation, by use of the reductive perturbation method. By intro ducing a set of stretched coordinates suitable for the boundary value type of problems and expanding the field variables into asymptotic series of the small-ness parameter of nonlinearity and dispersion, we obtained a set of nonlinear differential equations governing the terms at various order. By solving these nonlinear differential equations, we obtained the forced perturbed Korteweg-de Vries equation with variable coefficient as the nonlinear evolution equation. By use of the coordinate transformation, it is shown that this type of nonlinear evolution equation admits a progressive wave solution with variable wave speed.

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

Science.gov (United States)

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

2014-01-01

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

A New Spectral Local Linearization Method for Nonlinear Boundary Layer Flow Problems

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S. S. Motsa

2013-01-01

Full Text Available We propose a simple and efficient method for solving highly nonlinear systems of boundary layer flow problems with exponentially decaying profiles. The algorithm of the proposed method is based on an innovative idea of linearizing and decoupling the governing systems of equations and reducing them into a sequence of subsystems of differential equations which are solved using spectral collocation methods. The applicability of the proposed method, hereinafter referred to as the spectral local linearization method (SLLM, is tested on some well-known boundary layer flow equations. The numerical results presented in this investigation indicate that the proposed method, despite being easy to develop and numerically implement, is very robust in that it converges rapidly to yield accurate results and is more efficient in solving very large systems of nonlinear boundary value problems of the similarity variable boundary layer type. The accuracy and numerical stability of the SLLM can further be improved by using successive overrelaxation techniques.

High-accuracy power series solutions with arbitrarily large radius of convergence for the fractional nonlinear Schrödinger-type equations

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Khawaja, U. Al; Al-Refai, M.; Shchedrin, Gavriil; Carr, Lincoln D.

2018-06-01

Fractional nonlinear differential equations present an interplay between two common and important effective descriptions used to simplify high dimensional or more complicated theories: nonlinearity and fractional derivatives. These effective descriptions thus appear commonly in physical and mathematical modeling. We present a new series method providing systematic controlled accuracy for solutions of fractional nonlinear differential equations, including the fractional nonlinear Schrödinger equation and the fractional nonlinear diffusion equation. The method relies on spatially iterative use of power series expansions. Our approach permits an arbitrarily large radius of convergence and thus solves the typical divergence problem endemic to power series approaches. In the specific case of the fractional nonlinear Schrödinger equation we find fractional generalizations of cnoidal waves of Jacobi elliptic functions as well as a fractional bright soliton. For the fractional nonlinear diffusion equation we find the combination of fractional and nonlinear effects results in a more strongly localized solution which nevertheless still exhibits power law tails, albeit at a much lower density.

Customized Steady-State Constraints for Parameter Estimation in Non-Linear Ordinary Differential Equation Models.

Science.gov (United States)

Rosenblatt, Marcus; Timmer, Jens; Kaschek, Daniel

2016-01-01

Ordinary differential equation models have become a wide-spread approach to analyze dynamical systems and understand underlying mechanisms. Model parameters are often unknown and have to be estimated from experimental data, e.g., by maximum-likelihood estimation. In particular, models of biological systems contain a large number of parameters. To reduce the dimensionality of the parameter space, steady-state information is incorporated in the parameter estimation process. For non-linear models, analytical steady-state calculation typically leads to higher-order polynomial equations for which no closed-form solutions can be obtained. This can be circumvented by solving the steady-state equations for kinetic parameters, which results in a linear equation system with comparatively simple solutions. At the same time multiplicity of steady-state solutions is avoided, which otherwise is problematic for optimization. When solved for kinetic parameters, however, steady-state constraints tend to become negative for particular model specifications, thus, generating new types of optimization problems. Here, we present an algorithm based on graph theory that derives non-negative, analytical steady-state expressions by stepwise removal of cyclic dependencies between dynamical variables. The algorithm avoids multiple steady-state solutions by construction. We show that our method is applicable to most common classes of biochemical reaction networks containing inhibition terms, mass-action and Hill-type kinetic equations. Comparing the performance of parameter estimation for different analytical and numerical methods of incorporating steady-state information, we show that our approach is especially well-tailored to guarantee a high success rate of optimization.

Robust fast controller design via nonlinear fractional differential equations.

Science.gov (United States)

Zhou, Xi; Wei, Yiheng; Liang, Shu; Wang, Yong

2017-07-01

A new method for linear system controller design is proposed whereby the closed-loop system achieves both robustness and fast response. The robustness performance considered here means the damping ratio of closed-loop system can keep its desired value under system parameter perturbation, while the fast response, represented by rise time of system output, can be improved by tuning the controller parameter. We exploit techniques from both the nonlinear systems control and the fractional order systems control to derive a novel nonlinear fractional order controller. For theoretical analysis of the closed-loop system performance, two comparison theorems are developed for a class of fractional differential equations. Moreover, the rise time of the closed-loop system can be estimated, which facilitates our controller design to satisfy the fast response performance and maintain the robustness. Finally, numerical examples are given to illustrate the effectiveness of our methods. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Numerical study of nonlinear singular fractional differential equations arising in biology by operational matrix of shifted Legendre polynomials

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D. Jabari Sabeg

2016-10-01

Full Text Available In this paper, we present a new computational method for solving nonlinear singular boundary value problems of fractional order arising in biology. To this end, we apply the operational matrices of derivatives of shifted Legendre polynomials to reduce such problems to a system of nonlinear algebraic equations. To demonstrate the validity and applicability of the presented method, we present some numerical examples.

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.)

Nonlinear differential equations

CERN Document Server

Struble, Raimond A

2017-01-01

Detailed treatment covers existence and uniqueness of a solution of the initial value problem, properties of solutions, properties of linear systems, stability of nonlinear systems, and two-dimensional systems. 1962 edition.

Nonlinear perturbations of systems of partial differential equations with constant coefficients

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Carmen J. Vanegas

2000-01-01

Full Text Available In this article, we show the existence of solutions to boundary-value problems, consisting of nonlinear systems of partial differential equations with constant coefficients. For this purpose, we use the right inverse of an associated operator and a fix point argument. As illustrations, we apply this method to Helmholtz equations and to second order systems of elliptic equations.

Solving differential equations for Feynman integrals by expansions near singular points

Science.gov (United States)

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

2018-03-01

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

Monitoring inter-channel nonlinearity based on differential pilot

Science.gov (United States)

Wang, Wanli; Yang, Aiying; Guo, Peng; Lu, Yueming; Qiao, Yaojun

2018-06-01

We modify and simplify the inter-channel nonlinearity (NL) estimation method by using differential pilot. Compared to previous works, the inter-channel NL estimation method we propose has much lower complexity and does not need modification of the transmitter. The performance of inter-channel NL monitoring with different launch power is tested. For both QPSK and 16QAM systems with 9 channels, the estimation error of inter-channel NL is lower than 1 dB when the total launch power is bigger than 12 dBm after 1000 km optical transmission. At last, we compare our inter-channel NL estimation method with other methods.

Synthesis of robust nonlinear autopilots using differential game theory

Science.gov (United States)

Menon, P. K. A.

1991-01-01

A synthesis technique for handling unmodeled disturbances in nonlinear control law synthesis was advanced using differential game theory. Two types of modeling inaccuracies can be included in the formulation. The first is a bias-type error, while the second is the scale-factor-type error in the control variables. The disturbances were assumed to satisfy an integral inequality constraint. Additionally, it was assumed that they act in such a way as to maximize a quadratic performance index. Expressions for optimal control and worst-case disturbance were then obtained using optimal control theory.

Multiple positive solutions to a coupled systems of nonlinear fractional differential equations.

Science.gov (United States)

Shah, Kamal; Khan, Rahmat Ali

2016-01-01

In this article, we study existence, uniqueness and nonexistence of positive solution to a highly nonlinear coupled system of fractional order differential equations. Necessary and sufficient conditions for the existence and uniqueness of positive solution are developed by using Perov's fixed point theorem for the considered problem. Further, we also established sufficient conditions for existence of multiplicity results for positive solutions. Also, we developed some conditions under which the considered coupled system of fractional order differential equations has no positive solution. Appropriate examples are also provided which demonstrate our results.

A method for exponential propagation of large systems of stiff nonlinear differential equations

Science.gov (United States)

Friesner, Richard A.; Tuckerman, Laurette S.; Dornblaser, Bright C.; Russo, Thomas V.

1989-01-01

A new time integrator for large, stiff systems of linear and nonlinear coupled differential equations is described. For linear systems, the method consists of forming a small (5-15-term) Krylov space using the Jacobian of the system and carrying out exact exponential propagation within this space. Nonlinear corrections are incorporated via a convolution integral formalism; the integral is evaluated via approximate Krylov methods as well. Gains in efficiency ranging from factors of 2 to 30 are demonstrated for several test problems as compared to a forward Euler scheme and to the integration package LSODE.

Symmetric and arbitrarily high-order Birkhoff-Hermite time integrators and their long-time behaviour for solving nonlinear Klein-Gordon equations

Science.gov (United States)

Liu, Changying; Iserles, Arieh; Wu, Xinyuan

2018-03-01

The Klein-Gordon equation with nonlinear potential occurs in a wide range of application areas in science and engineering. Its computation represents a major challenge. The main theme of this paper is the construction of symmetric and arbitrarily high-order time integrators for the nonlinear Klein-Gordon equation by integrating Birkhoff-Hermite interpolation polynomials. To this end, under the assumption of periodic boundary conditions, we begin with the formulation of the nonlinear Klein-Gordon equation as an abstract second-order ordinary differential equation (ODE) and its operator-variation-of-constants formula. We then derive a symmetric and arbitrarily high-order Birkhoff-Hermite time integration formula for the nonlinear abstract ODE. Accordingly, the stability, convergence and long-time behaviour are rigorously analysed once the spatial differential operator is approximated by an appropriate positive semi-definite matrix, subject to suitable temporal and spatial smoothness. A remarkable characteristic of this new approach is that the requirement of temporal smoothness is reduced compared with the traditional numerical methods for PDEs in the literature. Numerical results demonstrate the advantage and efficiency of our time integrators in comparison with the existing numerical approaches.

A new differential equations-based model for nonlinear history-dependent magnetic behaviour

International Nuclear Information System (INIS)

Aktaa, J.; Weth, A. von der

2000-01-01

The paper presents a new kind of numerical model describing nonlinear magnetic behaviour. The model is formulated as a set of differential equations taking into account history dependence phenomena like the magnetisation hysteresis as well as saturation effects. The capability of the model is demonstrated carrying out comparisons between measurements and calculations

Darboux transformations and linear parabolic partial differential equations

International Nuclear Information System (INIS)

Arrigo, Daniel J.; Hickling, Fred

2002-01-01

Solutions for a class of linear parabolic partial differential equation are provided. These solutions are obtained by first solving a system of (n+1) nonlinear partial differential equations. This system arises as the coefficients of a Darboux transformation and is equivalent to a matrix Burgers' equation. This matrix equation is solved using a generalized Hopf-Cole transformation. The solutions for the original equation are given in terms of solutions of the heat equation. These results are applied to the (1+1)-dimensional Schroedinger equation where all bound state solutions are obtained for a 2n-parameter family of potentials. As a special case, the solutions for integral members of the regular and modified Poeschl-Teller potentials are recovered. (author). Letter-to-the-editor

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

International Nuclear Information System (INIS)

Ka-Lin, Su; Yuan-Xi, Xie

2010-01-01

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

A toolbox to solve coupled systems of differential and difference equations

International Nuclear Information System (INIS)

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

2016-01-01

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

A toolbox to solve coupled systems of differential and difference equations

Energy Technology Data Exchange (ETDEWEB)

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

2016-01-15

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

Maglev Train Signal Processing Architecture Based on Nonlinear Discrete Tracking Differentiator.

Science.gov (United States)

Wang, Zhiqiang; Li, Xiaolong; Xie, Yunde; Long, Zhiqiang

2018-05-24

In a maglev train levitation system, signal processing plays an important role for the reason that some sensor signals are prone to be corrupted by noise due to the harsh installation and operation environment of sensors and some signals cannot be acquired directly via sensors. Based on these concerns, an architecture based on a new type of nonlinear second-order discrete tracking differentiator is proposed. The function of this signal processing architecture includes filtering signal noise and acquiring needed signals for levitation purposes. The proposed tracking differentiator possesses the advantages of quick convergence, no fluttering, and simple calculation. Tracking differentiator's frequency characteristics at different parameter values are studied in this paper. The performance of this new type of tracking differentiator is tested in a MATLAB simulation and this tracking-differentiator is implemented in Very-High-Speed Integrated Circuit Hardware Description Language (VHDL). In the end, experiments are conducted separately on a test board and a maglev train model. Simulation and experiment results show that the performance of this novel signal processing architecture can fulfill the real system requirement.

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

Science.gov (United States)

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

1988-01-01

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

Non-linear analysis of skew thin plate by finite difference method

International Nuclear Information System (INIS)

Kim, Chi Kyung; Hwang, Myung Hwan

2012-01-01

This paper deals with a discrete analysis capability for predicting the geometrically nonlinear behavior of skew thin plate subjected to uniform pressure. The differential equations are discretized by means of the finite difference method which are used to determine the deflections and the in-plane stress functions of plates and reduced to several sets of linear algebraic simultaneous equations. For the geometrically non-linear, large deflection behavior of the plate, the non-linear plate theory is used for the analysis. An iterative scheme is employed to solve these quasi-linear algebraic equations. Several problems are solved which illustrate the potential of the method for predicting the finite deflection and stress. For increasing lateral pressures, the maximum principal tensile stress occurs at the center of the plate and migrates toward the corners as the load increases. It was deemed important to describe the locations of the maximum principal tensile stress as it occurs. The load-deflection relations and the maximum bending and membrane stresses for each case are presented and discussed

A Novel Partial Differential Algebraic Equation (PDAE) Solver

DEFF Research Database (Denmark)

Lim, Young-il; Chang, Sin-Chung; Jørgensen, Sten Bay

2004-01-01

For solving partial differential algebraic equations (PDAEs), the space-time conservation element/solution element (CE/SE) method is addressed in this study. The method of lines (MOL) using an implicit time integrator is compared with the CE/SE method in terms of computational efficiency, solution...... or nonlinear adsorption isotherm are solved by the two methods. The CE/SE method enforces both local and global flux conservation in space and time, and uses a simple stencil structure (two points at the previous time level and one point at the present time level). Thus, accurate and computationally...

Nonlinear vibration of a traveling belt with non-homogeneous boundaries

Science.gov (United States)

Ding, Hu; Lim, C. W.; Chen, Li-Qun

2018-06-01

Free and forced nonlinear vibrations of a traveling belt with non-homogeneous boundary conditions are studied. The axially moving materials in operation are always externally excited and produce strong vibrations. The moving materials with the homogeneous boundary condition are usually considered. In this paper, the non-homogeneous boundaries are introduced by the support wheels. Equilibrium deformation of the belt is produced by the non-homogeneous boundaries. In order to solve the equilibrium deformation, the differential and integral quadrature methods (DIQMs) are utilized to develop an iterative scheme. The influence of the equilibrium deformation on free and forced nonlinear vibrations of the belt is explored. The DIQMs are applied to solve the natural frequencies and forced resonance responses of transverse vibration around the equilibrium deformation. The Galerkin truncation method (GTM) is utilized to confirm the DIQMs' results. The numerical results demonstrate that the non-homogeneous boundary conditions cause the transverse vibration to deviate from the straight equilibrium, increase the natural frequencies, and lead to coexistence of square nonlinear terms and cubic nonlinear terms. Moreover, the influence of non-homogeneous boundaries can be exacerbated by the axial speed. Therefore, non-homogeneous boundary conditions of axially moving materials especially should be taken into account.

A direct algebraic method applied to obtain complex solutions of some nonlinear partial differential equations

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.

Oscillation and asymptotic stability of a delay differential equation with Richard's nonlinearity

Directory of Open Access Journals (Sweden)

Leonid Berezansky

2005-04-01

Full Text Available We obtain sufficient conditions for oscillation of solutions, and for asymptotical stability of the positive equilibrium, of the scalar nonlinear delay differential equation $$ frac{dN}{dt} = r(tN(tBig[a-Big(sum_{k=1}^m b_k N(g_k(tBig^{gamma}Big], $$ where $ g_k(tleq t$.

Numerical treatments for solving nonlinear mixed integral equation

Directory of Open Access Journals (Sweden)

M.A. Abdou

2016-12-01

Full Text Available We consider a mixed type of nonlinear integral equation (MNLIE of the second kind in the space C[0,T]×L2(Ω,T<1. The Volterra integral terms (VITs are considered in time with continuous kernels, while the Fredholm integral term (FIT is considered in position with singular general kernel. Using the quadratic method and separation of variables method, we obtain a nonlinear system of Fredholm integral equations (NLSFIEs with singular kernel. A Toeplitz matrix method, in each case, is then used to obtain a nonlinear algebraic system. Numerical results are calculated when the kernels take a logarithmic form or Carleman function. Moreover, the error estimates, in each case, are then computed.

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

Science.gov (United States)

Alam Khan, Najeeb; Razzaq, Oyoon Abdul

2016-03-01

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

An Efficient Higher-Order Quasilinearization Method for Solving Nonlinear BVPs

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Eman S. Alaidarous

2013-01-01

Full Text Available In this research paper, we present higher-order quasilinearization methods for the boundary value problems as well as coupled boundary value problems. The construction of higher-order convergent methods depends on a decomposition method which is different from Adomain decomposition method (Motsa and Sibanda, 2013. The reported method is very general and can be extended to desired order of convergence for highly nonlinear differential equations and also computationally superior to proposed iterative method based on Adomain decomposition because our proposed iterative scheme avoids the calculations of Adomain polynomials and achieves the same computational order of convergence as authors have claimed in Motsa and Sibanda, 2013. In order to check the validity and computational performance, the constructed iterative schemes are also successfully applied to bifurcation problems to calculate the values of critical parameters. The numerical performance is also tested for one-dimension Bratu and Frank-Kamenetzkii equations.

Solving differential equations with unknown constitutive relations as recurrent neural networks

Energy Technology Data Exchange (ETDEWEB)

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

2017-12-08

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

Nonlocal symmetries of a class of scalar and coupled nonlinear ordinary differential equations of any order

International Nuclear Information System (INIS)

Pradeep, R Gladwin; Chandrasekar, V K; Senthilvelan, M; Lakshmanan, M

2011-01-01

In this paper, we devise a systematic procedure to obtain nonlocal symmetries of a class of scalar nonlinear ordinary differential equations (ODEs) of arbitrary order related to linear ODEs through nonlocal relations. The procedure makes use of the Lie point symmetries of the linear ODEs and the nonlocal connection to deduce the nonlocal symmetries of the corresponding nonlinear ODEs. Using these nonlocal symmetries, we obtain reduction transformations and reduced equations to specific examples. We find that the reduced equations can be explicitly integrated to deduce the general solutions for these cases. We also extend this procedure to coupled higher order nonlinear ODEs with specific reference to second-order nonlinear ODEs. (paper)

ON THE BOUNDEDNESS AND THE STABILITY OF SOLUTION TO THIRD ORDER NON-LINEAR DIFFERENTIAL EQUATIONS

Institute of Scientific and Technical Information of China (English)

无

2008-01-01

In this paper we investigate the global asymptotic stability,boundedness as well as the ultimate boundedness of solutions to a general third order nonlinear differential equation,using complete Lyapunov function.

Biorthogonal Systems Approximating the Solution of the Nonlinear Volterra Integro-Differential Equation

Directory of Open Access Journals (Sweden)

Berenguer MI

2010-01-01

Full Text Available This paper deals with obtaining a numerical method in order to approximate the solution of the nonlinear Volterra integro-differential equation. We define, following a fixed-point approach, a sequence of functions which approximate the solution of this type of equation, due to some properties of certain biorthogonal systems for the Banach spaces and .

Oscillation of Nonlinear Delay Differential Equation with Non-Monotone Arguments

Directory of Open Access Journals (Sweden)

Özkan Öcalan

2017-07-01

Full Text Available Consider the first-order nonlinear retarded differential equation $$ x^{\\prime }(t+p(tf\\left( x\\left( \\tau (t\\right \\right =0, t\\geq t_{0} $$ where $p(t$ and $\\tau (t$ are function of positive real numbers such that $%\\tau (t\\leq t$ for$\\ t\\geq t_{0},\\ $and$\\ \\lim_{t\\rightarrow \\infty }\\tau(t=\\infty $. Under the assumption that the retarded argument is non-monotone, new oscillation results are given. An example illustrating the result is also given.

Nonlinear reaction-diffusion equations with delay: some theorems, test problems, exact and numerical solutions

Science.gov (United States)

Polyanin, A. D.; Sorokin, V. G.

2017-12-01

The paper deals with nonlinear reaction-diffusion equations with one or several delays. We formulate theorems that allow constructing exact solutions for some classes of these equations, which depend on several arbitrary functions. Examples of application of these theorems for obtaining new exact solutions in elementary functions are provided. We state basic principles of construction, selection, and use of test problems for nonlinear partial differential equations with delay. Some test problems which can be suitable for estimating accuracy of approximate analytical and numerical methods of solving reaction-diffusion equations with delay are presented. Some examples of numerical solutions of nonlinear test problems with delay are considered.

Identification of time-varying nonlinear systems using differential evolution algorithm

DEFF Research Database (Denmark)

Perisic, Nevena; Green, Peter L; Worden, Keith

2013-01-01

(DE) algorithm for the identification of time-varying systems. DE is an evolutionary optimisation method developed to perform direct search in a continuous space without requiring any derivative estimation. DE is modified so that the objective function changes with time to account for the continuing......, thus identification of time-varying systems with nonlinearities can be a very challenging task. In order to avoid conventional least squares and gradient identification methods which require uni-modal and double differentiable objective functions, this work proposes a modified differential evolution...... inclusion of new data within an error metric. This paper presents results of identification of a time-varying SDOF system with Coulomb friction using simulated noise-free and noisy data for the case of time-varying friction coefficient, stiffness and damping. The obtained results are promising and the focus...

Computation of Value Functions in Nonlinear Differential Games with State Constraints

KAUST Repository

Botkin, Nikolai

2013-01-01

Finite-difference schemes for the computation of value functions of nonlinear differential games with non-terminal payoff functional and state constraints are proposed. The solution method is based on the fact that the value function is a generalized viscosity solution of the corresponding Hamilton-Jacobi-Bellman-Isaacs equation. Such a viscosity solution is defined as a function satisfying differential inequalities introduced by M. G. Crandall and P. L. Lions. The difference with the classical case is that these inequalities hold on an unknown in advance subset of the state space. The convergence rate of the numerical schemes is given. Numerical solution to a non-trivial three-dimensional example is presented. © 2013 IFIP International Federation for Information Processing.

One-dimensional nonlinear theory for rectangular helix traveling-wave tube

Energy Technology Data Exchange (ETDEWEB)

Fu, Chengfang, E-mail: fchffchf@126.com; Zhao, Bo; Yang, Yudong; Ju, Yongfeng [Faculty of Electronic Information Engineering, Huaiyin Institute of Technology, Huai' an 223003 (China); Wei, Yanyu [School of Physical Electronics, University of Electronic and Technology of China, Chengdu 610054 (China)

2016-08-15

A 1-D nonlinear theory of a rectangular helix traveling-wave tube (TWT) interacting with a ribbon beam is presented in this paper. The RF field is modeled by a transmission line equivalent circuit, the ribbon beam is divided into a sequence of thin rectangular electron discs with the same cross section as the beam, and the charges are assumed to be uniformly distributed over these discs. Then a method of computing the space-charge field by solving Green's Function in the Cartesian Coordinate-system is fully described. Nonlinear partial differential equations for field amplitudes and Lorentz force equations for particles are solved numerically using the fourth-order Runge-Kutta technique. The tube's gain, output power, and efficiency of the above TWT are computed. The results show that increasing the cross section of the ribbon beam will improve a rectangular helix TWT's efficiency and reduce the saturated length.

The Semianalytical Solutions for Stiff Systems of Ordinary Differential Equations by Using Variational Iteration Method and Modified Variational Iteration Method with Comparison to Exact Solutions

Directory of Open Access Journals (Sweden)

Mehmet Tarik Atay

2013-01-01

Full Text Available The Variational Iteration Method (VIM and Modified Variational Iteration Method (MVIM are used to find solutions of systems of stiff ordinary differential equations for both linear and nonlinear problems. Some examples are given to illustrate the accuracy and effectiveness of these methods. We compare our results with exact results. In some studies related to stiff ordinary differential equations, problems were solved by Adomian Decomposition Method and VIM and Homotopy Perturbation Method. Comparisons with exact solutions reveal that the Variational Iteration Method (VIM and the Modified Variational Iteration Method (MVIM are easier to implement. In fact, these methods are promising methods for various systems of linear and nonlinear stiff ordinary differential equations. Furthermore, VIM, or in some cases MVIM, is giving exact solutions in linear cases and very satisfactory solutions when compared to exact solutions for nonlinear cases depending on the stiffness ratio of the stiff system to be solved.

Existence Results for Differential Inclusions with Nonlinear Growth Conditions in Banach Spaces

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Messaoud Bounkhel

2013-01-01

Full Text Available In the Banach space setting, the existence of viable solutions for differential inclusions with nonlinear growth; that is, ẋ(t∈F(t,x(t a.e. on I, x(t∈S, ∀t∈I, x(0=x0∈S, (*, where S is a closed subset in a Banach space , I=[0,T], (T>0, F:I×S→, is an upper semicontinuous set-valued mapping with convex values satisfying F(t,x⊂c(tx+xp, ∀(t,x∈I×S, where p∈ℝ, with p≠1, and c∈C([0,T],ℝ+. The existence of solutions for nonconvex sweeping processes with perturbations with nonlinear growth is also proved in separable Hilbert spaces.

Application of Reproducing Kernel Method for Solving Nonlinear Fredholm-Volterra Integrodifferential Equations

Directory of Open Access Journals (Sweden)

Omar Abu Arqub

2012-01-01

Full Text Available This paper investigates the numerical solution of nonlinear Fredholm-Volterra integro-differential equations using reproducing kernel Hilbert space method. The solution ( is represented in the form of series in the reproducing kernel space. In the mean time, the n-term approximate solution ( is obtained and it is proved to converge to the exact solution (. Furthermore, the proposed method has an advantage that it is possible to pick any point in the interval of integration and as well the approximate solution and its derivative will be applicable. Numerical examples are included to demonstrate the accuracy and applicability of the presented technique. The results reveal that the method is very effective and simple.

A direct method for numerical solution of a class of nonlinear Volterra integro-differential equations and its application to the nonlinear fission and fusion reactor kinetics

International Nuclear Information System (INIS)

Nakahara, Yasuaki; Ise, Takeharu; Kobayashi, Kensuke; Itoh, Yasuyuki

1975-12-01

A new method has been developed for numerical solution of a class of nonlinear Volterra integro-differential equations with quadratic nonlinearity. After dividing the domain of the variable into subintervals, piecewise approximations are applied in the subintervals. The equation is first integrated over a subinterval to obtain the piecewise equation, to which six approximate treatments are applied, i.e. fully explicit, fully implicit, Crank-Nicolson, linear interpolation, quadratic and cubic spline. The numerical solution at each time step is obtained directly as a positive root of the resulting algebraic quadratic equation. The point reactor kinetics with a ramp reactivity insertion, linear temperature feedback and delayed neutrons can be described by one of this type of nonlinear Volterra integro-differential equations. The algorithm is applied to the Argonne benchmark problem and a model problem for a fast reactor without delayed neutrons. The fully implicit method has been found to be unconditionally stable in the sense that it always gives the positive real roots. The cubic spline method is divergent, and the other four methods are intermediate in between. From the estimation of the stability, convergency, accuracy and CPU time, it is concluded that the Crank-Nicolson method is best, then the linear interpolation method comes closely next to it. Discussions are also made on the possibility of applying the algorithm to the fusion reactor kinetics in the form of a nonlinear partial differential equation. (auth.)

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

Science.gov (United States)

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

2017-04-01

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

NONLINEAR MODEL PREDICTIVE CONTROL OF CHEMICAL PROCESSES

Directory of Open Access Journals (Sweden)

SILVA R. G.

1999-01-01

Full Text Available A new algorithm for model predictive control is presented. The algorithm utilizes a simultaneous solution and optimization strategy to solve the model's differential equations. The equations are discretized by equidistant collocation, and along with the algebraic model equations are included as constraints in a nonlinear programming (NLP problem. This algorithm is compared with the algorithm that uses orthogonal collocation on finite elements. The equidistant collocation algorithm results in simpler equations, providing a decrease in computation time for the control moves. Simulation results are presented and show a satisfactory performance of this algorithm.

Maglev Train Signal Processing Architecture Based on Nonlinear Discrete Tracking Differentiator

Directory of Open Access Journals (Sweden)

Zhiqiang Wang

2018-05-01

Full Text Available In a maglev train levitation system, signal processing plays an important role for the reason that some sensor signals are prone to be corrupted by noise due to the harsh installation and operation environment of sensors and some signals cannot be acquired directly via sensors. Based on these concerns, an architecture based on a new type of nonlinear second-order discrete tracking differentiator is proposed. The function of this signal processing architecture includes filtering signal noise and acquiring needed signals for levitation purposes. The proposed tracking differentiator possesses the advantages of quick convergence, no fluttering, and simple calculation. Tracking differentiator’s frequency characteristics at different parameter values are studied in this paper. The performance of this new type of tracking differentiator is tested in a MATLAB simulation and this tracking-differentiator is implemented in Very-High-Speed Integrated Circuit Hardware Description Language (VHDL. In the end, experiments are conducted separately on a test board and a maglev train model. Simulation and experiment results show that the performance of this novel signal processing architecture can fulfill the real system requirement.

The Extended Fractional Subequation Method for Nonlinear Fractional Differential Equations

OpenAIRE

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...

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

Science.gov (United States)

Bilyeu, David

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

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

OpenAIRE

Waleed K. Ahmed

2013-01-01

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

Nonlinear dynamics in integrated coupled DFB lasers with ultra-short delay.

Science.gov (United States)

Liu, Dong; Sun, Changzheng; Xiong, Bing; Luo, Yi

2014-03-10

We report rich nonlinear dynamics in integrated coupled lasers with ultra-short coupling delay. Mutually stable locking, period-1 oscillation, frequency locking, quasi-periodicity and chaos are observed experimentally. The dynamic behaviors are reproduced numerically by solving coupled delay differential equations that take the variation of both frequency detuning and coupling phase into account. Moreover, it is pointed out that the round-trip frequency is not involved in the above nonlinear dynamical behaviors. Instead, the relationship between the frequency detuning Δν and the relaxation oscillation frequency νr under mutual injection are found to be critical for the various observed dynamics in mutually coupled lasers with very short delay.

Free vibration of geometrically nonlinear micro-switches under electrostatic and Casimir forces

International Nuclear Information System (INIS)

Jia, X L; Kitipornchai, S; Lim, C W; Yang, J

2010-01-01

This paper investigates the free vibration characteristics of micro-switches under combined electrostatic, intermolecular forces and axial residual stress, with an emphasis on the effect of geometric nonlinear deformation due to mid-plane stretching and the influence of Casimir force. The micro-switch considered in this study is made of either homogeneous material or non-homogeneous functionally graded material with two material phases. The Euler–Bernoulli beam theory with von Karman type nonlinear kinematics is applied in the theoretical formulation. The principle of virtual work is used to derive the nonlinear governing differential equation. The eigenvalue problem which describes free vibration of the micro-beam at its statically deflected state is then solved using the differential quadrature method. The natural frequencies and mode shapes of micro-switches for four different boundary conditions (i.e. clamped–clamped, clamped–simply supported, simply supported and clamped–free) are obtained. The solutions are validated through direct comparisons with experimental and other existing results reported in previous studies. A parametric study is conducted to show the significant effects of geometric nonlinearity, Casimir force, axial residual stress and material composition for the natural frequencies

Nonlinear evolution equations

CERN Document Server

Uraltseva, N N

1995-01-01

This collection focuses on nonlinear problems in partial differential equations. Most of the papers are based on lectures presented at the seminar on partial differential equations and mathematical physics at St. Petersburg University. Among the topics explored are the existence and properties of solutions of various classes of nonlinear evolution equations, nonlinear imbedding theorems, bifurcations of solutions, and equations of mathematical physics (Navier-Stokes type equations and the nonlinear Schrödinger equation). The book will be useful to researchers and graduate students working in p

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).

Development of a nonlinear unsteady transonic flow theory

Science.gov (United States)

Stahara, S. S.; Spreiter, J. R.

1973-01-01

A nonlinear, unsteady, small-disturbance theory capable of predicting inviscid transonic flows about aerodynamic configurations undergoing both rigid body and elastic oscillations was developed. The theory is based on the concept of dividing the flow into steady and unsteady components and then solving, by method of local linearization, the coupled differential equation for unsteady surface pressure distribution. The equations, valid at all frequencies, were derived for two-dimensional flows, numerical results, were obtained for two classses of airfoils and two types of oscillatory motions.

Bifurcations of solitary wave solutions for (two and three)-dimensional nonlinear partial differential equation in quantum and magnetized plasma by using two different methods

Science.gov (United States)

Khater, Mostafa M. A.; Seadawy, Aly R.; Lu, Dianchen

2018-06-01

In this research, we study new two techniques that called the extended simple equation method and the novel (G‧/G) -expansion method. The extended simple equation method depend on the auxiliary equation (dϕ/dξ = α + λϕ + μϕ2) which has three ways for solving depends on the specific condition on the parameters as follow: When (λ = 0) this auxiliary equation reduces to Riccati equation, when (α = 0) this auxiliary equation reduces to Bernoulli equation and when (α ≠ 0, λ ≠ 0, μ ≠ 0) we the general solutions of this auxiliary equation while the novel (G‧/G) -expansion method depends also on similar auxiliary equation (G‧/G)‧ = μ + λ(G‧/G) + (v - 1)(G‧/G) 2 which depend also on the value of (λ2 - 4 (v - 1) μ) and the specific condition on the parameters as follow: When (λ = 0) this auxiliary equation reduces to Riccati equation, when (μ = 0) this auxiliary equation reduces to Bernoulli equation and when (λ2 ≠ 4 (v - 1) μ) we the general solutions of this auxiliary equation. This show how both of these auxiliary equation are special cases of Riccati equation. We apply these methods on two dimensional nonlinear Kadomtsev-Petviashvili Burgers equation in quantum plasma and three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma. We obtain the exact traveling wave solutions of these important models and under special condition on the parameters, we get solitary traveling wave solutions. All calculations in this study have been established and verified back with the aid of the Maple package program. The executed method is powerful, effective and straightforward for solving nonlinear partial differential equations to obtain more and new solutions.

ESTIMATION OF CONSTANT AND TIME-VARYING DYNAMIC PARAMETERS OF HIV INFECTION IN A NONLINEAR DIFFERENTIAL EQUATION MODEL.

Science.gov (United States)

Liang, Hua; Miao, Hongyu; Wu, Hulin

2010-03-01

Modeling viral dynamics in HIV/AIDS studies has resulted in deep understanding of pathogenesis of HIV infection from which novel antiviral treatment guidance and strategies have been derived. Viral dynamics models based on nonlinear differential equations have been proposed and well developed over the past few decades. However, it is quite challenging to use experimental or clinical data to estimate the unknown parameters (both constant and time-varying parameters) in complex nonlinear differential equation models. Therefore, investigators usually fix some parameter values, from the literature or by experience, to obtain only parameter estimates of interest from clinical or experimental data. However, when such prior information is not available, it is desirable to determine all the parameter estimates from data. In this paper, we intend to combine the newly developed approaches, a multi-stage smoothing-based (MSSB) method and the spline-enhanced nonlinear least squares (SNLS) approach, to estimate all HIV viral dynamic parameters in a nonlinear differential equation model. In particular, to the best of our knowledge, this is the first attempt to propose a comparatively thorough procedure, accounting for both efficiency and accuracy, to rigorously estimate all key kinetic parameters in a nonlinear differential equation model of HIV dynamics from clinical data. These parameters include the proliferation rate and death rate of uninfected HIV-targeted cells, the average number of virions produced by an infected cell, and the infection rate which is related to the antiviral treatment effect and is time-varying. To validate the estimation methods, we verified the identifiability of the HIV viral dynamic model and performed simulation studies. We applied the proposed techniques to estimate the key HIV viral dynamic parameters for two individual AIDS patients treated with antiretroviral therapies. We demonstrate that HIV viral dynamics can be well characterized and

Consideration of Transient Stream/Aquifer Interaction with the Nonlinear Boussinesq Equation using HPM

DEFF Research Database (Denmark)

Ganji, S. S.; Barari, Amin; Sfahani, M. G.

2011-01-01

of time. The differential equations were solved using the method of Homotopy Perturbation. The simplicity and accuracy of the approximation are compared with “exact” solution and illustrated numerically and graphically. The results reveal that the HPM is very effective and simple and provides highly...... accurate solutions for nonlinear differential equations.......The phenomenon of stream–aquifer interaction was investigated via mathematical modeling using the Boussinesq equation. A new approximate solution of the one-dimensional Boussinesq equation is presented for a semi-infinite aquifer when the hydraulic head at the source is an arbitrary function...

Riccati-parameter solutions of nonlinear second-order ODEs

International Nuclear Information System (INIS)

Reyes, M A; Rosu, H C

2008-01-01

It has been proven by Rosu and Cornejo-Perez (Rosu and Cornejo-Perez 2005 Phys. Rev. E 71 046607, Cornejo-Perez and Rosu 2005 Prog. Theor. Phys. 114 533) that for some nonlinear second-order ODEs it is a very simple task to find one particular solution once the nonlinear equation is factorized with the use of two first-order differential operators. Here, it is shown that an interesting class of parametric solutions is easy to obtain if the proposed factorization has a particular form, which happily turns out to be the case in many problems of physical interest. The method that we exemplify with a few explicitly solved cases consists in using the general solution of the Riccati equation, which contributes with one parameter to this class of parametric solutions. For these nonlinear cases, the Riccati parameter serves as a 'growth' parameter from the trivial null solution up to the particular solution found through the factorization procedure

The verification of the Taylor-expansion moment method in solving aerosol breakage

Directory of Open Access Journals (Sweden)

Yu Ming-Zhou

2012-01-01

Full Text Available The combination of the method of moment, characterizing the particle population balance, and the computational fluid dynamics has been an emerging research issue in the studies on the aerosol science and on the multiphase flow science. The difficulty of solving the moment equation arises mainly from the closure of some fractal moment variables which appears in the transform from the non-linear integral-differential population balance equation to the moment equations. Within the Taylor-expansion moment method, the breakage-dominated Taylor-expansion moment equation is first derived here when the symmetric fragmentation mechanism is involved. Due to the high efficiency and the high precision, this proposed moment model is expected to become an important tool for solving population balance equations.

Nonlinear static analysis of single layer annular/circular graphene sheets embedded in Winkler–Pasternak elastic matrix based on non-local theory of Eringen

Directory of Open Access Journals (Sweden)

Shahriar Dastjerdi

2016-06-01

Full Text Available Nonlinear bending analysis of orthotropic annular/circular graphene sheets has been studied based on the non-local elasticity theory. The first order shear deformation theory (FSDT is applied in combination with the nonlinear Von-Karman strain field. The obtained differential equations are solved by using two methods, first the differential quadrature method (DQM and a new semi-analytical polynomial method (SAPM which is innovated by the authors. Applying the DQM or SAPM, the differential equations are transformed to nonlinear algebraic equations system. Then the Newton–Raphson iterative scheme is used. First, the obtained results from DQM and SAPM are compared and it is concluded that although the SAPM’s formulation is considerably simpler than DQM, however, the SAPM’s results are so close to DQM. The results are validated with available papers. Finally, the effects of small scale parameter on the results, the comparison between local and non-local theories, and linear to nonlinear analyses are investigated.

Operator overloading as an enabling technology for automatic differentiation

International Nuclear Information System (INIS)

Corliss, G.F.; Griewank, A.

1993-01-01

We present an example of the science that is enabled by object-oriented programming techniques. Scientific computation often needs derivatives for solving nonlinear systems such as those arising in many PDE algorithms, optimization, parameter identification, stiff ordinary differential equations, or sensitivity analysis. Automatic differentiation computes derivatives accurately and efficiently by applying the chain rule to each arithmetic operation or elementary function. Operator overloading enables the techniques of either the forward or the reverse mode of automatic differentiation to be applied to real-world scientific problems. We illustrate automatic differentiation with an example drawn from a model of unsaturated flow in a porous medium. The problem arises from planning for the long-term storage of radioactive waste

Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations

Science.gov (United States)

Indekeu, Joseph O.; Smets, Ruben

2017-08-01

Physically motivated modified Fisher equations are studied in which nonlinear convection and nonlinear diffusion is allowed for besides the usual growth and spread of a population. It is pointed out that in a large variety of cases separable functions in the form of exponentially decaying sharp wavefronts solve the differential equation exactly provided a co-moving point source or sink is active at the wavefront. The velocity dispersion and front steepness may differ from those of some previously studied exact smooth traveling wave solutions. For an extension of the reaction-diffusion-convection equation, featuring a memory effect in the form of a maturity delay for growth and spread, also smooth exact wavefront solutions are obtained. The stability of the solutions is verified analytically and numerically.

Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations

International Nuclear Information System (INIS)

Indekeu, Joseph O; Smets, Ruben

2017-01-01

Physically motivated modified Fisher equations are studied in which nonlinear convection and nonlinear diffusion is allowed for besides the usual growth and spread of a population. It is pointed out that in a large variety of cases separable functions in the form of exponentially decaying sharp wavefronts solve the differential equation exactly provided a co-moving point source or sink is active at the wavefront. The velocity dispersion and front steepness may differ from those of some previously studied exact smooth traveling wave solutions. For an extension of the reaction-diffusion-convection equation, featuring a memory effect in the form of a maturity delay for growth and spread, also smooth exact wavefront solutions are obtained. The stability of the solutions is verified analytically and numerically. (paper)

Nonlinear $q$-fractional differential equations with nonlocal and sub-strip type boundary conditions

Directory of Open Access Journals (Sweden)

Bashir Ahmad

2014-06-01

Full Text Available This paper is concerned with new boundary value problems of nonlinear $q$-fractional differential equations with nonlocal and sub-strip type boundary conditions. Our results are new in the present setting and rely on the contraction mapping principle and a fixed point theorem due to O'Regan. Some illustrative examples are also presented.

Nonlinear elliptic equations and nonassociative algebras

CERN Document Server

Nadirashvili, Nikolai; Vlăduţ, Serge

2014-01-01

This book presents applications of noncommutative and nonassociative algebras to constructing unusual (nonclassical and singular) solutions to fully nonlinear elliptic partial differential equations of second order. The methods described in the book are used to solve a longstanding problem of the existence of truly weak, nonsmooth viscosity solutions. Moreover, the authors provide an almost complete description of homogeneous solutions to fully nonlinear elliptic equations. It is shown that even in the very restricted setting of "Hessian equations", depending only on the eigenvalues of the Hessian, these equations admit homogeneous solutions of all orders compatible with known regularity for viscosity solutions provided the space dimension is five or larger. To the contrary, in dimension four or less the situation is completely different, and our results suggest strongly that there are no nonclassical homogeneous solutions at all in dimensions three and four. Thus this book gives a complete list of dimensions...

Solving Ordinary Differential Equations

Science.gov (United States)

Krogh, F. T.

1987-01-01

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

The existence of periodic solutions for nonlinear beam equations on Td by a para-differential method

Science.gov (United States)

Chen, Bochao; Li, Yong; Gao, Yixian

2018-05-01

This paper focuses on the construction of periodic solutions of nonlinear beam equations on the $d$-dimensional tori. For a large set of frequencies, we demonstrate that an equivalent form of the nonlinear equations can be obtained by a para-differential conjugation. Given the non-resonant conditions on each finite dimensional subspaces, it is shown that the periodic solutions can be constructed for the block diagonal equation by a classical iteration scheme.

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

Science.gov (United States)

Biala, T A; Jator, S N

2015-01-01

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

Nonlinear dynamics analysis of a low-temperature-differential kinematic Stirling heat engine

Science.gov (United States)

Izumida, Yuki

2018-03-01

The low-temperature-differential (LTD) Stirling heat engine technology constitutes one of the important sustainable energy technologies. The basic question of how the rotational motion of the LTD Stirling heat engine is maintained or lost based on the temperature difference is thus a practically and physically important problem that needs to be clearly understood. Here, we approach this problem by proposing and investigating a minimal nonlinear dynamic model of an LTD kinematic Stirling heat engine. Our model is described as a driven nonlinear pendulum where the motive force is the temperature difference. The rotational state and the stationary state of the engine are described as a stable limit cycle and a stable fixed point of the dynamical equations, respectively. These two states coexist under a sufficient temperature difference, whereas the stable limit cycle does not exist under a temperature difference that is too small. Using a nonlinear bifurcation analysis, we show that the disappearance of the stable limit cycle occurs via a homoclinic bifurcation, with the temperature difference being the bifurcation parameter.

on the properties of solutions and some applications on the TOV differential equation with a model of nuclear equation of state

International Nuclear Information System (INIS)

Esmail, S.F.H.

2006-01-01

the mathematical formulation of numerous physical problems results in differential equations actually non-linear differential equations . in our study we are interested in solutions of differential equations which describe the structure of neutron star in non-relativistic and relativistic cases. the aim of this work is to determine the mass and the radius of a neutron star, by solving the tolmann-oppenheimer-volkoff (TOV) differential equation using different models of the nuclear equation of state (EOS). analytically solutions are obtained for a simple form of the nuclear equation of state of Clayton model and poly trope model. for a more realistic equation of state the TOV differential equation is solved numerically using rung -Kutta method

A lattice Boltzmann model with an amending function for simulating nonlinear partial differential equations

International Nuclear Information System (INIS)

Lin-Jie, Chen; Chang-Feng, Ma

2010-01-01

This paper proposes a lattice Boltzmann model with an amending function for one-dimensional nonlinear partial differential equations (NPDEs) in the form u t + αuu x + βu n u x + γu xx + δu xxx + ζu xxxx = 0. This model is different from existing models because it lets the time step be equivalent to the square of the space step and derives higher accuracy and nonlinear terms in NPDEs. With the Chapman–Enskog expansion, the governing evolution equation is recovered correctly from the continuous Boltzmann equation. The numerical results agree well with the analytical solutions. (general)

An introduction to differential equations

CERN Document Server

Ladde, Anil G

2012-01-01

This is a twenty-first century book designed to meet the challenges of understanding and solving interdisciplinary problems. The book creatively incorporates "cutting-edge" research ideas and techniques at the undergraduate level. The book also is a unique research resource for undergraduate/graduate students and interdisciplinary researchers. It emphasizes and exhibits the importance of conceptual understandings and its symbiotic relationship in the problem solving process. The book is proactive in preparing for the modeling of dynamic processes in various disciplines. It introduces a "break-down-the problem" type of approach in a way that creates "fun" and "excitement". The book presents many learning tools like "step-by-step procedures (critical thinking)", the concept of "math" being a language, applied examples from diverse fields, frequent recaps, flowcharts and exercises. Uniquely, this book introduces an innovative and unified method of solving nonlinear scalar differential equations. This is called ...

Study of coupled nonlinear partial differential equations for finding exact analytical solutions.

Science.gov (United States)

Khan, Kamruzzaman; Akbar, M Ali; Koppelaar, H

2015-07-01

Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (G'/G)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd-Sokolov-Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics.

Differential Transform Method for Mathematical Modeling of Jamming Transition Problem in Traffic Congestion Flow

DEFF Research Database (Denmark)

Ganji, S. S.; Barari, Amin; Ibsen, Lars Bo

2010-01-01

. In current research the authors utilized the Differential Transformation Method (DTM) for solving the nonlinear problem and compared the analytical results with those ones obtained by the 4th order Runge-Kutta Method (RK4) as a numerical method. Further illustration embedded in this paper shows the ability...

Differential Transform Method for Mathematical Modeling of Jamming Transition Problem in Traffic Congestion Flow

DEFF Research Database (Denmark)

Ganji, S.; Barari, Amin; Ibsen, Lars Bo

2012-01-01

. In current research the authors utilized the Differential Transformation Method (DTM) for solving the nonlinear problem and compared the analytical results with those ones obtained by the 4th order Runge-Kutta Method (RK4) as a numerical method. Further illustration embedded in this paper shows the ability...

A Novel Operational Matrix of Caputo Fractional Derivatives of Fibonacci Polynomials: Spectral Solutions of Fractional Differential Equations

Directory of Open Access Journals (Sweden)

Waleed M. Abd-Elhameed

2016-09-01

Full Text Available Herein, two numerical algorithms for solving some linear and nonlinear fractional-order differential equations are presented and analyzed. For this purpose, a novel operational matrix of fractional-order derivatives of Fibonacci polynomials was constructed and employed along with the application of the tau and collocation spectral methods. The convergence and error analysis of the suggested Fibonacci expansion were carefully investigated. Some numerical examples with comparisons are presented to ensure the efficiency, applicability and high accuracy of the proposed algorithms. Two accurate semi-analytic polynomial solutions for linear and nonlinear fractional differential equations are the result.

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

International Nuclear Information System (INIS)

Provost, J-P; Bracco, C

2009-01-01

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

Dynamics of excited instantons in the system of forced Gursey nonlinear differential equations

Energy Technology Data Exchange (ETDEWEB)

Aydogmus, F., E-mail: fatma.aydogmus@gmail.com [Istanbul University, Department of Physics, Faculty of Science (Turkey)

2015-02-15

The Gursey model is a 4D conformally invariant pure fermionic model with a nonlinear spinor self-coupled term. Gursey proposed his model as a possible basis for a unitary description of elementary particles following the “Heisenberg dream.” In this paper, we consider the system of Gursey nonlinear differential equations (GNDEs) formed by using the Heisenberg ansatz. We use it to understand how the behavior of spinor-type Gursey instantons can be affected by excitations. For this, the regular and chaotic numerical solutions of forced GNDEs are investigated by constructing their Poincaré sections in phase space. A hierarchical cluster analysis method for investigating the forced GNDEs is also presented.

Nonlinear dynamics of resistive electrostatic drift waves

DEFF Research Database (Denmark)

Korsholm, Søren Bang; Michelsen, Poul; Pécseli, H.L.

1999-01-01

The evolution of weakly nonlinear electrostatic drift waves in an externally imposed strong homogeneous magnetic field is investigated numerically in three spatial dimensions. The analysis is based on a set of coupled, nonlinear equations, which are solved for an initial condition which is pertur......The evolution of weakly nonlinear electrostatic drift waves in an externally imposed strong homogeneous magnetic field is investigated numerically in three spatial dimensions. The analysis is based on a set of coupled, nonlinear equations, which are solved for an initial condition which...... polarity, i.e. a pair of electrostatic convective cells....

Penalized Nonlinear Least Squares Estimation of Time-Varying Parameters in Ordinary Differential Equations

KAUST Repository

Cao, Jiguo; Huang, Jianhua Z.; Wu, Hulin

2012-01-01

Ordinary differential equations (ODEs) are widely used in biomedical research and other scientific areas to model complex dynamic systems. It is an important statistical problem to estimate parameters in ODEs from noisy observations. In this article we propose a method for estimating the time-varying coefficients in an ODE. Our method is a variation of the nonlinear least squares where penalized splines are used to model the functional parameters and the ODE solutions are approximated also using splines. We resort to the implicit function theorem to deal with the nonlinear least squares objective function that is only defined implicitly. The proposed penalized nonlinear least squares method is applied to estimate a HIV dynamic model from a real dataset. Monte Carlo simulations show that the new method can provide much more accurate estimates of functional parameters than the existing two-step local polynomial method which relies on estimation of the derivatives of the state function. Supplemental materials for the article are available online.

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

International Nuclear Information System (INIS)

Ablinger, Jakob; Schneider, Carsten

2016-08-01

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

Simulation of creep effects in framework of a geometrically nonlinear endochronic theory of inelasticity

Science.gov (United States)

Zabavnikova, T. A.; Kadashevich, Yu. I.; Pomytkin, S. P.

2018-05-01

A geometric non-linear endochronic theory of inelasticity in tensor parametric form is considered. In the framework of this theory, the creep strains are modelled. The effect of various schemes of applying stresses and changing of material properties on the development of creep strains is studied. The constitutive equations of the model are represented by non-linear systems of ordinary differential equations which are solved in MATLAB environment by implicit difference method. Presented results demonstrate a good qualitative agreement of theoretical data and experimental observations including the description of the tertiary creep and pre-fracture of materials.

MINPACK-1, Subroutine Library for Nonlinear Equation System

International Nuclear Information System (INIS)

Garbow, Burton S.

1984-01-01

1 - Description of problem or function: MINPACK1 is a package of FORTRAN subprograms for the numerical solution of systems of non- linear equations and nonlinear least-squares problems. The individual programs are: Identification/Description: - CHKDER: Check gradients for consistency with functions, - DOGLEG: Determine combination of Gauss-Newton and gradient directions, - DPMPAR: Provide double precision machine parameters, - ENORM: Calculate Euclidean norm of vector, - FDJAC1: Calculate difference approximation to Jacobian (nonlinear equations), - FDJAC2: Calculate difference approximation to Jacobian (least squares), - HYBRD: Solve system of nonlinear equations (approximate Jacobian), - HYBRD1: Easy-to-use driver for HYBRD, - HYBRJ: Solve system of nonlinear equations (analytic Jacobian), - HYBRJ1: Easy-to-use driver for HYBRJ, - LMDER: Solve nonlinear least squares problem (analytic Jacobian), - LMDER1: Easy-to-use driver for LMDER, - LMDIF: Solve nonlinear least squares problem (approximate Jacobian), - LMDIF1: Easy-to-use driver for LMDIF, - LMPAR: Determine Levenberg-Marquardt parameter - LMSTR: Solve nonlinear least squares problem (analytic Jacobian, storage conserving), - LMSTR1: Easy-to-use driver for LMSTR, - QFORM: Accumulate orthogonal matrix from QR factorization QRFAC Compute QR factorization of rectangular matrix, - QRSOLV: Complete solution of least squares problem, - RWUPDT: Update QR factorization after row addition, - R1MPYQ: Apply orthogonal transformations from QR factorization, - R1UPDT: Update QR factorization after rank-1 addition, - SPMPAR: Provide single precision machine parameters. 4. Method of solution - MINPACK1 uses the modified Powell hybrid method and the Levenberg-Marquardt algorithm

Sturm-Picone type theorems for second-order nonlinear differential equations

Directory of Open Access Journals (Sweden)

Aydin Tiryaki

2014-06-01

Full Text Available The aim of this article is to give Sturm-Picone type theorems for the pair of second-order nonlinear differential equations $$\\displaylines{ (p_1(t|x'|^{\\alpha-1}x''+q_1(tf_1(x=0 \\cr (p_2(t|y'|^{\\alpha-1}y''+q_2(tf_2(y=0,\\quad t_1

Uniqueness of global quasi-classical solutions of the Cauchy problems for first-order nonlinear partial differential equations

International Nuclear Information System (INIS)

Tran Duc Van

1994-01-01

The notion of global quasi-classical solutions of the Cauchy problems for first-order nonlinear partial differential equations is presented, some uniqueness theorems and a stability result are established by the method based on the theory of differential inclusions. In particular, the answer to an open problem of S.N. Kruzhkov is given. (author). 10 refs, 1 fig

Parameter Estimation for Partial Differential Equations by Collage-Based Numerical Approximation

Directory of Open Access Journals (Sweden)

Xiaoyan Deng

2009-01-01

into a minimization problem of a function of several variables after the partial differential equation is approximated by a differential dynamical system. Then numerical schemes for solving this minimization problem are proposed, including grid approximation and ant colony optimization. The proposed schemes are applied to a parameter estimation problem for the Belousov-Zhabotinskii equation, and the results show that the proposed approximation method is efficient for both linear and nonlinear partial differential equations with respect to unknown parameters. At worst, the presented method provides an excellent starting point for traditional inversion methods that must first select a good starting point.

Stability of abstract nonlinear nonautonomous differential-delay equations with unbounded history-responsive operators

Science.gov (United States)

Gil', M. I.

2005-08-01

We consider a class of nonautonomous functional-differential equations in a Banach space with unbounded nonlinear history-responsive operators, which have the local Lipshitz property. Conditions for the boundedness of solutions, Lyapunov stability, absolute stability and input-output one are established. Our approach is based on a combined usage of properties of sectorial operators and spectral properties of commuting operators.

On periodic bounded and unbounded solutions of second order nonlinear ordinary differential equations

Czech Academy of Sciences Publication Activity Database

Lomtatidze, Alexander

2017-01-01

Roč. 24, č. 2 (2017), s. 241-263 ISSN 1072-947X Institutional support: RVO:67985840 Keywords : nonlinear ordinary differential equations * periodic boundary value problem * solvability Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics Impact factor: 0.290, year: 2016 https://www.degruyter.com/view/j/gmj.2017.24.issue-2/gmj-2017-0009/gmj-2017-0009. xml

On periodic bounded and unbounded solutions of second order nonlinear ordinary differential equations

Czech Academy of Sciences Publication Activity Database

Lomtatidze, Alexander

2017-01-01

Roč. 24, č. 2 (2017), s. 241-263 ISSN 1072-947X Institutional support: RVO:67985840 Keywords : nonlinear ordinary differential equations * periodic boundary value problem * solvability Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics Impact factor: 0.290, year: 2016 https://www.degruyter.com/view/j/gmj.2017.24.issue-2/gmj-2017-0009/gmj-2017-0009.xml

Advanced differential quadrature methods

CERN Document Server

Zong, Zhi

2009-01-01

Modern Tools to Perform Numerical DifferentiationThe original direct differential quadrature (DQ) method has been known to fail for problems with strong nonlinearity and material discontinuity as well as for problems involving singularity, irregularity, and multiple scales. But now researchers in applied mathematics, computational mechanics, and engineering have developed a range of innovative DQ-based methods to overcome these shortcomings. Advanced Differential Quadrature Methods explores new DQ methods and uses these methods to solve problems beyond the capabilities of the direct DQ method.After a basic introduction to the direct DQ method, the book presents a number of DQ methods, including complex DQ, triangular DQ, multi-scale DQ, variable order DQ, multi-domain DQ, and localized DQ. It also provides a mathematical compendium that summarizes Gauss elimination, the Runge-Kutta method, complex analysis, and more. The final chapter contains three codes written in the FORTRAN language, enabling readers to q...

Analysis and classification of nonlinear dispersive evolution equations in the potential representation

International Nuclear Information System (INIS)

Eichmann, U.A.; Draayer, J.P.; Ludu, A.

2002-01-01

A potential representation for the subset of travelling solutions of nonlinear dispersive evolution equations is introduced. The procedure involves reduction of a third-order partial differential equation to a first-order ordinary differential equation. The potential representation allows us to deduce certain properties of the solutions without the actual need to solve the underlying evolution equation. In particular, the paper deals with the so-called K(n, m) equations. Starting from their respective potential representations it is shown that these equations can be classified according to a simple point transformation. As a result, e.g., all equations with linear dispersion join the same equivalence class with the Korteweg-deVries equation being its representative, and all soliton solutions of higher order nonlinear equations are thus equivalent to the KdV soliton. Certain equations with both linear and quadratic dispersions can also be treated within this equivalence class. (author)

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

KAUST Repository

Piret, Cécile

2012-05-01

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

Study of coupled nonlinear partial differential equations for finding exact analytical solutions

Science.gov (United States)

Khan, Kamruzzaman; Akbar, M. Ali; Koppelaar, H.

2015-01-01

Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (G′/G)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd–Sokolov–Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics. PMID:26587256

Differential equations methods and applications

CERN Document Server

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. .

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

Nonlinear vibration of rectangular atomic force microscope cantilevers by considering the Hertzian contact theory

Energy Technology Data Exchange (ETDEWEB)

Sadeghi, A., E-mail: a_sadeghi@srbiau.ac.ir [Islamic Azad Univ., Dept. of Mechanical and Aerospace Engineering, Science and Research Branch, Tehran (Iran, Islamic Republic of); Zohoor, H. [Sharif Univ. of Technology, Center of Excellence in Design, Robotics and Automation, Tehran (Iran, Islamic Republic of); The Academy of Sciences if I.R. Iran (Iran, Islamic Republic of)

2010-05-15

The nonlinear flexural vibration for a rectangular atomic force microscope cantilever is investigated by using Timoshenko beam theory. In this paper, the normal and tangential tip-sample interaction forces are found from a Hertzian contact model and the effects of the contact position, normal and lateral contact stiffness, tip height, thickness of the beam, and the angle between the cantilever and the sample surface on the nonlinear frequency to linear frequency ratio are studied. The differential quadrature method is employed to solve the nonlinear differential equations of motion. The results show that softening behavior is seen for most cases and by increasing the normal contact stiffness, the frequency ratio increases for the first mode, but for the second mode, the situation is reversed. The nonlinear-frequency to linear-frequency ratio increases by increasing the Timoshenko beam parameter, but decreases by increasing the contact position for constant amplitude for the first and second modes. For the first mode, the frequency ratio decreases by increasing both of the lateral contact stiffness and the tip height, but increases by increasing the angle α between the cantilever and sample surface. (author)

On magnetohydrodynamic flow of second grade nanofluid over a nonlinear stretching sheet

Energy Technology Data Exchange (ETDEWEB)

Hayat, Tasawar [Department of Mathematics, Quaid-I-Azam University, Islamabad 44000 (Pakistan); Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589 (Saudi Arabia); Aziz, Arsalan [Department of Mathematics, Quaid-I-Azam University, Islamabad 44000 (Pakistan); Muhammad, Taseer, E-mail: taseer_qau@yahoo.com [Department of Mathematics, Quaid-I-Azam University, Islamabad 44000 (Pakistan); Ahmad, Bashir [Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589 (Saudi Arabia)

2016-06-15

This research article addresses the magnetohydrodynamic (MHD) flow of second grade nanofluid over a nonlinear stretching sheet. Heat and mass transfer aspects are investigated through the thermophoresis and Brownian motion effects. Second grade fluid is assumed electrically conducting through a non-uniform applied magnetic field. Mathematical formulation is developed subject to small magnetic Reynolds number and boundary layer assumptions. Newly constructed condition having zero mass flux of nanoparticles at the boundary is incorporated. Transformations have been invoked for the reduction of partial differential systems into the set of nonlinear ordinary differential systems. The governing nonlinear systems have been solved for local behavior. Graphical results of different influential parameters are studied and discussed in detail. Computations for skin friction coefficient and local Nusselt number have been carried out. It is observed that the effects of thermophoresis parameter on the temperature and nanoparticles concentration distributions are qualitatively similar. The temperature and nanoparticles concentration distributions are enhanced for the larger magnetic parameter. - Highlights: • Constitutive relation for second grade fluid is employed. • Flow is caused by a nonlinear stretching surface. • Magnetic field applied is in transverse direction. • Nanofluid model consists of Brownian motion and thermophoresis. • Magnetic Reynolds number is assumed small.

On magnetohydrodynamic flow of second grade nanofluid over a nonlinear stretching sheet

International Nuclear Information System (INIS)

Hayat, Tasawar; Aziz, Arsalan; Muhammad, Taseer; Ahmad, Bashir

2016-01-01

This research article addresses the magnetohydrodynamic (MHD) flow of second grade nanofluid over a nonlinear stretching sheet. Heat and mass transfer aspects are investigated through the thermophoresis and Brownian motion effects. Second grade fluid is assumed electrically conducting through a non-uniform applied magnetic field. Mathematical formulation is developed subject to small magnetic Reynolds number and boundary layer assumptions. Newly constructed condition having zero mass flux of nanoparticles at the boundary is incorporated. Transformations have been invoked for the reduction of partial differential systems into the set of nonlinear ordinary differential systems. The governing nonlinear systems have been solved for local behavior. Graphical results of different influential parameters are studied and discussed in detail. Computations for skin friction coefficient and local Nusselt number have been carried out. It is observed that the effects of thermophoresis parameter on the temperature and nanoparticles concentration distributions are qualitatively similar. The temperature and nanoparticles concentration distributions are enhanced for the larger magnetic parameter. - Highlights: • Constitutive relation for second grade fluid is employed. • Flow is caused by a nonlinear stretching surface. • Magnetic field applied is in transverse direction. • Nanofluid model consists of Brownian motion and thermophoresis. • Magnetic Reynolds number is assumed small.

Solving eigenvalue response matrix equations with nonlinear techniques

International Nuclear Information System (INIS)

Roberts, Jeremy A.; Forget, Benoit

2014-01-01

Highlights: • High performance solvers were applied within ERMM for the first time. • Accelerated fixed-point methods were developed that reduce computational times by 2–3. • A nonlinear, Newton-based ERMM led to similar improvement and more robustness. • A 3-D, SN-based ERMM shows how ERMM can apply fine-mesh methods to full-core analysis. - Abstract: This paper presents new algorithms for use in the eigenvalue response matrix method (ERMM) for reactor eigenvalue problems. ERMM spatially decomposes a domain into independent nodes linked via boundary conditions approximated as truncated orthogonal expansions, the coefficients of which are response functions. In its simplest form, ERMM consists of a two-level eigenproblem: an outer Picard iteration updates the k-eigenvalue via balance, while the inner λ-eigenproblem imposes neutron balance between nodes. Efficient methods are developed for solving the inner λ-eigenvalue problem within the outer Picard iteration. Based on results from several diffusion and transport benchmark models, it was found that the Krylov–Schur method applied to the λ-eigenvalue problem reduces Picard solver times (excluding response generation) by a factor of 2–5. Furthermore, alternative methods, including Picard acceleration schemes, Steffensen’s method, and Newton’s method, are developed in this paper. These approaches often yield faster k-convergence and a need for fewer k-dependent response function evaluations, which is important because response generation is often the primary cost for problems using responses computed online (i.e., not from a precomputed database). Accelerated Picard iteration was found to reduce total computational times by 2–3 compared to the unaccelerated case for problems dominated by response generation. In addition, Newton’s method was found to provide nearly the same performance with improved robustness

Analytical approaches for the approximate solution of a nonlinear fractional ordinary differential equation

International Nuclear Information System (INIS)

Basak, K C; Ray, P C; Bera, R K

2009-01-01

The aim of the present analysis is to apply the Adomian decomposition method and He's variational method for the approximate analytical solution of a nonlinear ordinary fractional differential equation. The solutions obtained by the above two methods have been numerically evaluated and presented in the form of tables and also compared with the exact solution. It was found that the results obtained by the above two methods are in excellent agreement with the exact solution. Finally, a surface plot of the approximate solutions of the fractional differential equation by the above two methods is drawn for 0≤t≤2 and 1<α≤2.

Initial-value problems for first-order differential recurrence equations with auto-convolution

Directory of Open Access Journals (Sweden)

Mircea Cirnu

2011-01-01

Full Text Available A differential recurrence equation consists of a sequence of differential equations, from which must be determined by recurrence a sequence of unknown functions. In this article, we solve two initial-value problems for some new types of nonlinear (quadratic first order homogeneous differential recurrence equations, namely with discrete auto-convolution and with combinatorial auto-convolution of the unknown functions. In both problems, all initial values form a geometric progression, but in the second problem the first initial value is exempted and has a prescribed form. Some preliminary results showing the importance of the initial conditions are obtained by reducing the differential recurrence equations to algebraic type. Final results about solving the considered initial value problems, are shown by mathematical induction. However, they can also be shown by changing the unknown functions, or by the generating function method. So in a remark, we give a proof of the first theorem by the generating function method.

The focal boundary value problem for strongly singular higher-order nonlinear functional-differential equations

Czech Academy of Sciences Publication Activity Database

Mukhigulashvili, Sulkhan; Půža, B.

2015-01-01

Roč. 2015, January (2015), s. 17 ISSN 1687-2770 Institutional support: RVO:67985840 Keywords : higher order nonlinear functional-differential equations * two-point right-focal boundary value problem * strong singularity Subject RIV: BA - General Mathematics Impact factor: 0.642, year: 2015 http://link.springer.com/article/10.1186%2Fs13661-014-0277-1

Three-Field Modelling of Nonlinear Nonsmooth Boundary Value Problems and Stability of Differential Mixed Variational Inequalities

Directory of Open Access Journals (Sweden)

J. Gwinner

2013-01-01

Full Text Available The purpose of this paper is twofold. Firstly we consider nonlinear nonsmooth elliptic boundary value problems, and also related parabolic initial boundary value problems that model in a simplified way steady-state unilateral contact with Tresca friction in solid mechanics, respectively, stem from nonlinear transient heat conduction with unilateral boundary conditions. Here a recent duality approach, that augments the classical Babuška-Brezzi saddle point formulation for mixed variational problems to twofold saddle point formulations, is extended to the nonsmooth problems under consideration. This approach leads to variational inequalities of mixed form for three coupled fields as unknowns and to related differential mixed variational inequalities in the time-dependent case. Secondly we are concerned with the stability of the solution set of a general class of differential mixed variational inequalities. Here we present a novel upper set convergence result with respect to perturbations in the data, including perturbations of the associated nonlinear maps, the nonsmooth convex functionals, and the convex constraint set. We employ epiconvergence for the convergence of the functionals and Mosco convergence for set convergence. We impose weak convergence assumptions on the perturbed maps using the monotonicity method of Browder and Minty.

Finite-horizon differential games for missile-target interception system using adaptive dynamic programming with input constraints

Science.gov (United States)

Sun, Jingliang; Liu, Chunsheng

2018-01-01

In this paper, the problem of intercepting a manoeuvring target within a fixed final time is posed in a non-linear constrained zero-sum differential game framework. The Nash equilibrium solution is found by solving the finite-horizon constrained differential game problem via adaptive dynamic programming technique. Besides, a suitable non-quadratic functional is utilised to encode the control constraints into a differential game problem. The single critic network with constant weights and time-varying activation functions is constructed to approximate the solution of associated time-varying Hamilton-Jacobi-Isaacs equation online. To properly satisfy the terminal constraint, an additional error term is incorporated in a novel weight-updating law such that the terminal constraint error is also minimised over time. By utilising Lyapunov's direct method, the closed-loop differential game system and the estimation weight error of the critic network are proved to be uniformly ultimately bounded. Finally, the effectiveness of the proposed method is demonstrated by using a simple non-linear system and a non-linear missile-target interception system, assuming first-order dynamics for the interceptor and target.

A semi-analytical iterative technique for solving chemistry problems

Directory of Open Access Journals (Sweden)

Majeed Ahmed AL-Jawary

2017-07-01

Full Text Available The main aim and contribution of the current paper is to implement a semi-analytical iterative method suggested by Temimi and Ansari in 2011 namely (TAM to solve two chemical problems. An approximate solution obtained by the TAM provides fast convergence. The current chemical problems are the absorption of carbon dioxide into phenyl glycidyl ether and the other system is a chemical kinetics problem. These problems are represented by systems of nonlinear ordinary differential equations that contain boundary conditions and initial conditions. Error analysis of the approximate solutions is studied using the error remainder and the maximal error remainder. Exponential rate for the convergence is observed. For both problems the results of the TAM are compared with other results obtained by previous methods available in the literature. The results demonstrate that the method has many merits such as being derivative-free, and overcoming the difficulty arising in calculating Adomian polynomials to handle the non-linear terms in Adomian Decomposition Method (ADM. It does not require to calculate Lagrange multiplier in Variational Iteration Method (VIM in which the terms of the sequence become complex after several iterations, thus, analytical evaluation of terms becomes very difficult or impossible in VIM. No need to construct a homotopy in Homotopy Perturbation Method (HPM and solve the corresponding algebraic equations. The MATHEMATICA® 9 software was used to evaluate terms in the iterative process.

A linearizing transformation for the Korteweg-de Vries equation; generalizations to higher-dimensional nonlinear partial differential equations

NARCIS (Netherlands)

Dorren, H.J.S.

1998-01-01

It is shown that the Korteweg–de Vries (KdV) equation can be transformed into an ordinary linear partial differential equation in the wave number domain. Explicit solutions of the KdV equation can be obtained by subsequently solving this linear differential equation and by applying a cascade of

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

Science.gov (United States)

Kiryakova, Virginia S.

2012-11-01

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

Optimization of Thermal Object Nonlinear Control Systems by Energy Efficiency Criterion.

Science.gov (United States)

Velichkin, Vladimir A.; Zavyalov, Vladimir A.

2018-03-01

This article presents the results of thermal object functioning control analysis (heat exchanger, dryer, heat treatment chamber, etc.). The results were used to determine a mathematical model of the generalized thermal control object. The appropriate optimality criterion was chosen to make the control more energy-efficient. The mathematical programming task was formulated based on the chosen optimality criterion, control object mathematical model and technological constraints. The “maximum energy efficiency” criterion helped avoid solving a system of nonlinear differential equations and solve the formulated problem of mathematical programming in an analytical way. It should be noted that in the case under review the search for optimal control and optimal trajectory reduces to solving an algebraic system of equations. In addition, it is shown that the optimal trajectory does not depend on the dynamic characteristics of the control object.

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

Directory of Open Access Journals (Sweden)

Jianping Liu

2016-01-01

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

Hidden physics models: Machine learning of nonlinear partial differential equations

Science.gov (United States)

Raissi, Maziar; Karniadakis, George Em

2018-03-01

While there is currently a lot of enthusiasm about "big data", useful data is usually "small" and expensive to acquire. In this paper, we present a new paradigm of learning partial differential equations from small data. In particular, we introduce hidden physics models, which are essentially data-efficient learning machines capable of leveraging the underlying laws of physics, expressed by time dependent and nonlinear partial differential equations, to extract patterns from high-dimensional data generated from experiments. The proposed methodology may be applied to the problem of learning, system identification, or data-driven discovery of partial differential equations. Our framework relies on Gaussian processes, a powerful tool for probabilistic inference over functions, that enables us to strike a balance between model complexity and data fitting. The effectiveness of the proposed approach is demonstrated through a variety of canonical problems, spanning a number of scientific domains, including the Navier-Stokes, Schrödinger, Kuramoto-Sivashinsky, and time dependent linear fractional equations. The methodology provides a promising new direction for harnessing the long-standing developments of classical methods in applied mathematics and mathematical physics to design learning machines with the ability to operate in complex domains without requiring large quantities of data.

Influence of nonlinear thermal radiation and viscous dissipation on three-dimensional flow of Jeffrey nano fluid over a stretching sheet in the presence of Joule heating

Science.gov (United States)

Ganesh Kumar, K.; Rudraswamy, N. G.; Gireesha, B. J.; Krishnamurthy, M. R.

2017-09-01

Present exploration discusses the combined effect of viscous dissipation and Joule heating on three dimensional flow and heat transfer of a Jeffrey nanofluid in the presence of nonlinear thermal radiation. Here the flow is generated over bidirectional stretching sheet in the presence of applied magnetic field by accounting thermophoresis and Brownian motion of nanoparticles. Suitable similarity transformations are employed to reduce the governing partial differential equations into coupled nonlinear ordinary differential equations. These nonlinear ordinary differential equations are solved numerically by using the Runge-Kutta-Fehlberg fourth-fifth order method with shooting technique. Graphically results are presented and discussed for various parameters. Validation of the current method is proved by comparing our results with the existing results under limiting situations. It can be concluded that combined effect of Joule and viscous heating increases the temperature profile and thermal boundary layer thickness.

Existence and Multiplicity Results for Nonlinear Differential Equations Depending on a Parameter in Semipositone Case

Directory of Open Access Journals (Sweden)

Hailong Zhu

2012-01-01

Full Text Available The existence and multiplicity of solutions for second-order differential equations with a parameter are discussed in this paper. We are mainly concerned with the semipositone case. The analysis relies on the nonlinear alternative principle of Leray-Schauder and Krasnosel'skii's fixed point theorem in cones.

Clawpack: building an open source ecosystem for solving hyperbolic PDEs

KAUST Repository

Mandli, Kyle T.; Ahmadia, Aron J.; Berger, Marsha; Calhoun, Donna; George, David L.; Hadjimichael, Yiannis; Ketcheson, David I.; Lemoine, Grady I.; LeVeque, Randall J.

2016-01-01

Clawpack is a software package designed to solve nonlinear hyperbolic partial differential equations using high-resolution finite volume methods based on Riemann solvers and limiters. The package includes a number of variants aimed at different applications and user communities. Clawpack has been actively developed as an open source project for over 20 years. The latest major release, Clawpack 5, introduces a number of new features and changes to the code base and a new development model based on GitHub and Git submodules. This article provides a summary of the most significant changes, the rationale behind some of these changes, and a description of our current development model.

Clawpack: building an open source ecosystem for solving hyperbolic PDEs

KAUST Repository

Mandli, Kyle T.

2016-08-08

Clawpack is a software package designed to solve nonlinear hyperbolic partial differential equations using high-resolution finite volume methods based on Riemann solvers and limiters. The package includes a number of variants aimed at different applications and user communities. Clawpack has been actively developed as an open source project for over 20 years. The latest major release, Clawpack 5, introduces a number of new features and changes to the code base and a new development model based on GitHub and Git submodules. This article provides a summary of the most significant changes, the rationale behind some of these changes, and a description of our current development model.

Nonlinear tension-bending deformation of a shape memory alloy rod

International Nuclear Information System (INIS)

Shang, Zejin; Wang, Zhongmin

2012-01-01

Based on the measured shape memory alloy (SMA) stress–strain curve and the nonlinear large deformation theory of extensible beams (or rods), the first-order nonlinear governing equations of a SMA cantilever straight rod are established. They consist of a boundary-value problem of ordinary differential equations with a strong nonlinearity, in which seven unknown functions are contained and the arc length of the deformed axis is considered as one of the basic unknown functions. The shooting method combining with the Newton–Raphson iteration method is applied to solve the equations numerically. For a SMA cantilever rod subjected to a transverse uniformly distributed force, the deformation characteristics curves, the maximum strain and the maximum stress distribution curves along the longitudinal direction of rod, and the relation curves between deformation characteristic parameters and transverse uniformly force under different slenderness ratios are obtained. The effects of material nonlinearity, geometrical nonlinearity and slenderness ratio on the tension-bending deformation of the SMA cantilever rod are investigated. The numerical simulation results are in good agreement with the experimental data from the literature, verifying the soundness of the entire numerical simulation scheme. (paper)

Unified algorithm for partial differential equations and examples of numerical computation

International Nuclear Information System (INIS)

Watanabe, Tsuguhiro

1999-01-01

A new unified algorithm is proposed to solve partial differential equations which describe nonlinear boundary value problems, eigenvalue problems and time developing boundary value problems. The algorithm is composed of implicit difference scheme and multiple shooting scheme and is named as HIDM (Higher order Implicit Difference Method). A new prototype computer programs for 2-dimensional partial differential equations is constructed and tested successfully to several problems. Extension of the computer programs to 3 or more higher order dimension problems will be easy due to the direct product type difference scheme. (author)

Quantifying non-linear dynamics of mass-springs in series oscillators via asymptotic approach

Science.gov (United States)

Starosta, Roman; Sypniewska-Kamińska, Grażyna; Awrejcewicz, Jan

2017-05-01

Dynamical regular response of an oscillator with two serially connected springs with nonlinear characteristics of cubic type and governed by a set of differential-algebraic equations (DAEs) is studied. The classical approach of the multiple scales method (MSM) in time domain has been employed and appropriately modified to solve the governing DAEs of two systems, i.e. with one- and two degrees-of-freedom. The approximate analytical solutions have been verified by numerical simulations.

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

KAUST Repository

Piret, Cé cile

2012-01-01

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

Numerical Solution of Stochastic Nonlinear Fractional Differential Equations

KAUST Repository

El-Beltagy, Mohamed A.

2015-01-07

Using Wiener-Hermite expansion (WHE) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. WHE is the only known expansion that handles the white/colored noise exactly. This work introduces a numerical estimation of the stochastic response of the Duffing oscillator with fractional or variable order damping and driven by white noise. The WHE technique is integrated with the Grunwald-Letnikov approximation in case of fractional order and with Coimbra approximation in case of variable-order damping. The numerical solver was tested with the analytic solution and with Monte-Carlo simulations. The developed mixed technique was shown to be efficient in simulating SPDEs.

Numerical Solution of Stochastic Nonlinear Fractional Differential Equations

KAUST Repository

El-Beltagy, Mohamed A.; Al-Juhani, Amnah

2015-01-01

Using Wiener-Hermite expansion (WHE) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. WHE is the only known expansion that handles the white/colored noise exactly. This work introduces a numerical estimation of the stochastic response of the Duffing oscillator with fractional or variable order damping and driven by white noise. The WHE technique is integrated with the Grunwald-Letnikov approximation in case of fractional order and with Coimbra approximation in case of variable-order damping. The numerical solver was tested with the analytic solution and with Monte-Carlo simulations. The developed mixed technique was shown to be efficient in simulating SPDEs.

Memory functioning and negative symptoms as differential predictors of social problem solving skills in schizophrenia.

Science.gov (United States)

Ventura, Joseph; Tom, Shelley R; Jetton, Chris; Kern, Robert S

2013-02-01

Neurocognition in general, and memory functioning in particular, as well as symptoms have all been shown to be related to social problem solving (SPS) in schizophrenia. However, few studies have directly compared the relative contribution of neurocognition vs. psychiatric symptoms to the components of SPS. Sixty outpatients (aged 21-65) who met DSM-IV criteria for schizophrenia or schizoaffective disorder were administered a broad battery of memory tests and assessed for severity of positive and negative symptoms as part of a baseline assessment of a study of psychiatric rehabilitation. Multiple regression analyses were used to examine the contribution of memory functioning vs. symptoms on receiving, processing, and sending skill areas of social problem solving ability. An index of verbal learning was the strongest predictor of processing skills whereas negative symptoms were the strongest predictor of sending skills. Positive symptoms were not related to any of the three skill areas of social problem solving. Memory functioning and psychiatric symptoms differentially predict selected areas of social problem solving ability in persons with schizophrenia. Consistent with other reports, positive symptoms were not related to social problem solving. Consideration of both neurocognition and negative symptoms may be important to the development of rehabilitation interventions in this area of functioning. Copyright © 2012 Elsevier B.V. All rights reserved.

Thermophoretic diffusion and nonlinear radiative heat transfer due to a contracting cylinder in a nanofluid with generalized slip condition

Directory of Open Access Journals (Sweden)

Z. Abbas

Full Text Available An analysis is carried out to study the generalized slip condition and MHD flow of a nanofluid due to a contracting cylinder in the presence of non-linear radiative heat transfer using Buongiorno’s model. The Navier-Stokes along with energy and nanoparticle concentration equations is transformed to highly nonlinear ordinary differential equations using similarity transformations. These similar differential equations are then solved numerically by employing a shooting technique with Runge–Kutta–Fehlberg method. Dual solutions exist for a particular range of the unsteadiness parameter. The physical influence of the several important fluid parameters on the flow velocity, temperature and nanoparticle volume fraction is discussed and shown through graphs and table in detail. The present study indicates that as increase of Brownian motion parameter and slip velocity is to decrease the nanoparticle volume fraction. Keywords: Nanofluid, Contracting cylinder, Nonlinear thermal radiation, Generalized slip condition, Numerical solution

Nonlinear analysis of 0-3 polarized PLZT microplate based on the new modified couple stress theory

Science.gov (United States)

Wang, Liming; Zheng, Shijie

2018-02-01

In this study, based on the new modified couple stress theory, the size- dependent model for nonlinear bending analysis of a pure 0-3 polarized PLZT plate is developed for the first time. The equilibrium equations are derived from a variational formulation based on the potential energy principle and the new modified couple stress theory. The Galerkin method is adopted to derive the nonlinear algebraic equations from governing differential equations. And then the nonlinear algebraic equations are solved by using Newton-Raphson method. After simplification, the new model includes only a material length scale parameter. In addition, numerical examples are carried out to study the effect of material length scale parameter on the nonlinear bending of a simply supported pure 0-3 polarized PLZT plate subjected to light illumination and uniform distributed load. The results indicate the new model is able to capture the size effect and geometric nonlinearity.

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

OpenAIRE

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

2008-01-01

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

An overview of adaptive model theory: solving the problems of redundancy, resources, and nonlinear interactions in human movement control.

Science.gov (United States)

Neilson, Peter D; Neilson, Megan D

2005-09-01

Adaptive model theory (AMT) is a computational theory that addresses the difficult control problem posed by the musculoskeletal system in interaction with the environment. It proposes that the nervous system creates motor maps and task-dependent synergies to solve the problems of redundancy and limited central resources. These lead to the adaptive formation of task-dependent feedback/feedforward controllers able to generate stable, noninteractive control and render nonlinear interactions unobservable in sensory-motor relationships. AMT offers a unified account of how the nervous system might achieve these solutions by forming internal models. This is presented as the design of a simulator consisting of neural adaptive filters based on cerebellar circuitry. It incorporates a new network module that adaptively models (in real time) nonlinear relationships between inputs with changing and uncertain spectral and amplitude probability density functions as is the case for sensory and motor signals.

An approximation theory for nonlinear partial differential equations with applications to identification and control

Science.gov (United States)

Banks, H. T.; Kunisch, K.

1982-01-01

Approximation results from linear semigroup theory are used to develop a general framework for convergence of approximation schemes in parameter estimation and optimal control problems for nonlinear partial differential equations. These ideas are used to establish theoretical convergence results for parameter identification using modal (eigenfunction) approximation techniques. Results from numerical investigations of these schemes for both hyperbolic and parabolic systems are given.

Detection of Differential Item Functioning with Nonlinear Regression: A Non-IRT Approach Accounting for Guessing

Czech Academy of Sciences Publication Activity Database