The dual reciprocity boundary elements method for the linear and nonlinear two‐dimensional time‐fractional partial differential equations

In this paper, we apply the dual reciprocity boundary elements method for the numerical solution of two‐dimensional linear and nonlinear time‐fractional modified anomalous subdiffusion equations and time‐fractional convection–diffusion equation. The fractional derivative of problems is described in the Riemann–Liouville and Caputo senses. We employ the linear radial basis function for interpolation of the nonlinear, inhomogeneous and time derivative terms. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity which appears in the nonlinear problems under consideration. The accuracy and efficiency of the proposed schemes are checked by five test problems. The proposed method is employed for solving some examples in two dimensions on unit square and also in complex regions to demonstrate the efficiency of the new technique. Copyright © 2016 John Wiley & Sons, Ltd.     

A linear,decoupled fractional time‐stepping method for the nonlinear fluid–fluid interaction

In this paper, a linear decoupled fractional time stepping method is proposed and developed for the nonlinear fluid–fluid interaction governed by the two Navier–Stokes equations. Partitioned time stepping method is applied to two‐physics problems with stiffness of the coupling terms being treated explicitly and is also unconditionally stable. As for each fluid, the velocity and pressure are respectively determined by just solving one vector‐valued quasi‐elliptic equation and the Possion equation with homogeneous Neumann boundary condition per time step. Therefore, the cost of the fluid–fluid interaction is dominant to solve four simple linear equations, which greatly reduces the computational cost of the whole system. The method exploits properties of the fluid–fluid system to establish its stability and convergence with the same results as the standard scheme. Finally, numerical experiments are presented to show the performance of the proposed method.     

Numerical simulation of the nonlinear generalized time‐fractional Klein–Gordon equation using cubic trigonometric B‐spline functions

In this paper, an efficient numerical procedure for the generalized nonlinear time‐fractional Klein–Gordon equation is presented. We make use of the typical finite difference schemes to approximate the Caputo time‐fractional derivative, while the spatial derivatives are discretized by means of the cubic trigonometric B‐splines. Stability and convergence analysis for the numerical scheme are discussed. We apply our scheme to some typical examples and compare the obtained results with the ones found by other numerical methods. The comparison shows that our scheme is quite accurate and can be applied successfully to a variety of problems of applied nature.     

Application of the modified operational matrices in multiterm variable‐order time‐fractional partial differential equations

In this paper, we present a novel discrete scheme based on Genocchi polynomials and fractional Laguerre functions to solve multiterm variable‐order time‐fractional partial differential equations (M‐V‐TFPDEs) in the large interval. In this purpose, the accurate modified operational matrices are constructed to reduce the problems into a system of algebraic equations. Also, the computational algorithm based on the method and modified operational matrices in the large interval is easily implemented. Furthermore, we discuss the error estimation of the proposed method. Ultimately, to confirm our theoretical analysis and accuracy of numerical approach, several examples are presented.     

Fractional Crank–Nicolson–Galerkin finite element scheme for the time‐fractional nonlinear diffusion equation

This article presents a finite element scheme with Newton's method for solving the time‐fractional nonlinear diffusion equation. For time discretization, we use the fractional Crank–Nicolson scheme based on backward Euler convolution quadrature. We discuss the existence‐uniqueness results for the fully discrete problem. A new discrete fractional Gronwall type inequality for the backward Euler convolution quadrature is established. A priori error estimate for the fully discrete problem in L²(Ω) norm is derived. Numerical results based on finite element scheme are provided to validate theoretical estimates on time‐fractional nonlinear Fisher equation and Huxley equation.     

The boundary integral equation approach for numerical solution of the one‐dimensional Sine‐Gordon equation

This article describes a numerical method based on the boundary integral equation and dual reciprocity method for solving the one‐dimensional Sine‐Gordon (SG) equation. The time derivative is approximated by the time‐stepping method and a predictor–corrector scheme is employed to deal with the nonlinearity which appears in the problem. Numerical results are presented for some problems to demonstrate the usefulness and accuracy of this approach. In addition, the conservation of energy in SG equation is investigated. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

An efficient pseudo‐spectral Legendre–Galerkin method for solving a nonlinear partial integro‐differential equation arising in population dynamics

The pseudo‐spectral Legendre–Galerkin method (PS‐LGM) is applied to solve a nonlinear partial integro‐differential equation arising in population dynamics. This equation is a competition model in which similar individuals are competing for the same resources. It is a kind of reaction–diffusion equation with integral term corresponding to nonlocal consumption of resources. The proposed method is based on the Legendre–Galerkin formulation for the linear terms and interpolation operator at the Chebyshev–Gauss–Lobatto (CGL) points for the nonlinear terms. Also, the integral term, which is a kind of convolution, is directly computed by a fast and accurate method based on CGL interpolation operator, and thus, the use of any quadrature formula in its computation is avoided. The main difference of the PS‐LGM presented in the current paper with the classic LGM is in treating the nonlinear terms and imposing boundary conditions. Indeed, in the PS‐LGM, the nonlinear terms are efficiently handled using the CGL points, and also the boundary conditions are imposed strongly as collocation methods. Combination of the PS‐LGM with a semi‐implicit time integration method such as second‐order backward differentiation formula and Adams‐Bashforth method leads to reducing the complexity of computations and obtaining a linear algebraic system of equations with banded coefficient matrix. The desired equation is considered on one and two‐dimensional spatial domains. Efficiency, accuracy, and convergence of the proposed method are demonstrated numerically in both cases. Copyright © 2012 John Wiley & Sons, Ltd.     

A multigrid compact finite difference method for solving the one‐dimensional nonlinear sine‐Gordon equation

The aim of this paper is to propose a multigrid method to obtain the numerical solution of the one‐dimensional nonlinear sine‐Gordon equation. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a compact finite difference scheme of fourth‐order for discretizing the spatial derivative and the standard second‐order central finite difference method for the time derivative. The proposed method uses the Richardson extrapolation method in time variable. The obtained system has been solved by V‐cycle multigrid (VMG) method, where the VMG method is used for solving the large sparse linear systems. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional sine‐Gordon equation. Copyright © 2014 John Wiley & Sons, Ltd.     

A spectral element method using the modal basis and its application in solving second‐order nonlinear partial differential equations

We present a high‐order spectral element method (SEM) using modal (or hierarchical) basis for modeling of some nonlinear second‐order partial differential equations in two‐dimensional spatial space. The discretization is based on the conforming spectral element technique in space and the semi‐implicit or the explicit finite difference formula in time. Unlike the nodal SEM, which is based on the Lagrange polynomials associated with the Gauss–Lobatto–Legendre or Chebyshev quadrature nodes, the Lobatto polynomials are used in this paper as modal basis. Using modal bases due to their orthogonal properties enables us to exactly obtain the elemental matrices provided that the element‐wise mapping has the constant Jacobian. The difficulty of implementation of modal approximations for nonlinear problems is treated in this paper by expanding the nonlinear terms in the weak form of differential equations in terms of the Lobatto polynomials on each element using the fast Fourier transform (FFT). Utilization of the Fourier interpolation on equidistant points in the FFT algorithm and the enough polynomial order of approximation of the nonlinear terms can lead to minimize the aliasing error. Also, this approach leads to finding numerical solution of a nonlinear differential equation through solving a system of linear algebraic equations. Numerical results for some famous nonlinear equations illustrate efficiency, stability and convergence properties of the approximation scheme, which is exponential in space and up to third‐order in time. Copyright © 2014 John Wiley & Sons, Ltd.     

Reconstructing unknown nonlinear boundary conditions in a time‐fractional inverse reaction–diffusion–convection problem

This paper has focused on unknown functions identification in nonlinear boundary conditions of an inverse problem of a time‐fractional reaction–diffusion–convection equation. This inverse problem is generally ill‐posed in the sense of stability, that is, the solution of problem does not depend continuously on the input data. Thus, a combination of the mollification regularization method with Gauss kernel and a finite difference marching scheme will be introduced to solve this problem. The generalized cross‐validation choice rule is applied to find a suitable regularization parameter. The stability and convergence of the numerical method are investigated. Finally, two numerical examples are provided to test the effectiveness and validity of the proposed approach.     

A method based on meshless approach for the numerical solution of the two‐space dimensional hyperbolic telegraph equation

In the current article, we investigate the RBF solution of second‐order two‐space dimensional linear hyperbolic telegraph equation. For this purpose, we use a combination of boundary knot method (BKM) and analog equation method (AEM). The BKM is a meshfree, boundary‐only and integration‐free technique. The BKM is an alternative to the method of fundamental solution to avoid the fictitious boundary and to deal with low accuracy, singular integration and mesh generation. Also, on the basis of the AEM, the governing operator is substituted by an equivalent nonhomogeneous linear one with known fundamental solution under the same boundary conditions. Finally, several numerical results and discussions are demonstrated to show the accuracy and efficiency of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

On the numerical solution of the Klein‐Gordon equation

A predictor–corrector (P–C) scheme based on the use of rational approximants of second‐order to the matrix‐exponential term in a three‐time level reccurence relation is applied to the nonlinear Klein‐Gordon equation. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. Both the predictor and the corrector scheme are analyzed for local truncation error and stability. The proposed method is applied to problems possessing periodic, kinks and single, double‐soliton waves. The accuracy as well as the long time behavior of the proposed scheme is discussed. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Amplitude equation for the stochastic reaction‐diffusion equations with random Neumann boundary conditions

In this paper, we consider a quite general class of reaction‐diffusion equations with cubic nonlinearities and with random Neumann boundary conditions. We derive rigorously amplitude equations, using the natural separation of time‐scales near a change of stability and investigate whether additive degenerate noise and random boundary conditions can lead to stabilization of the solution of the stochastic partial differential equation or not. The nonlinear heat equation (Ginzburg–Landau equation) is used to illustrate our result. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability and convergence of optimum spectral non‐linear Galerkin methods

Our objective in this article is to present some numerical schemes for the approximation of the 2‐D Navier–Stokes equations with periodic boundary conditions, and to study the stability and convergence of the schemes. Spatial discretization can be performed by either the spectral Galerkin method or the optimum spectral non‐linear Galerkin method; time discretization is done by the Euler scheme and a two‐step scheme. Our results show that under the same convergence rate the optimum spectral non‐linear Galerkin method is superior to the usual Galerkin methods. Finally, numerical example is provided and supports our results. Copyright © 2001 John Wiley & Sons, Ltd.     

Stability of the superposition of rarefaction wave and contact discontinuity for the non‐isentropic Navier–Stokes–Poisson system

This paper is devoted to the study of the nonlinear stability of the composite wave consisting of a rarefaction wave and a viscous contact discontinuity wave of the non‐isentropic Navier–Stokes–Poisson system with free boundary. We first construct the composite wave through the quasineutral Euler equations and then prove that the composite wave is time asymptotically stable under small perturbations for the corresponding initial‐boundary value problem of the non‐isentropic Navier–Stokes–Poisson system. Only the strength of the viscous contact wave is required to be small. However, the strength of the rarefaction wave can be arbitrarily large. In our analysis, the domain decomposition plays an important role in obtaining the zero‐order energy estimates. By introducing this technique, we successfully overcome the difficulty caused by the critical terms involved with the linear term, which does not satisfy the quasineural assumption for the composite wave. Copyright © 2016 John Wiley & Sons, Ltd.     

Gauss‐Lobatto‐Legendre‐Birkhoff pseudospectral approximations for the multi‐term time fractional diffusion‐wave equation with Neumann boundaryconditions

A second‐order finite difference/pseudospectral scheme is proposed for numerical approximation of multi‐term time fractional diffusion‐wave equation with Neumann boundary conditions. The scheme is based upon the weighted and shifted Grünwald difference operators approximation of the time fractional calculus and Gauss‐Lobatto‐Legendre‐Birkhoff (GLLB) pseudospectral method for spatial discretization. The unconditionally stability and convergence of the scheme are rigorously proved. Numerical examples are carried out to verify theoretical results.     

The comparison of two reliable methods for accurate solution of time‐fractional Kaup–Kupershmidt equation arising in capillary gravity waves

In this paper, we compared two different methods, one numerical technique, viz Legendre multiwavelet method, and the other analytical technique, viz optimal homotopy asymptotic method (OHAM), for solving fractional‐order Kaup–Kupershmidt (KK) equation. Two‐dimensional Legendre multiwavelet expansion together with operational matrices of fractional integration and derivative of wavelet functions is used to compute the numerical solution of nonlinear time‐fractional KK equation. The approximate solutions of time fractional Kaup–Kupershmidt equation thus obtained by Legendre multiwavelet method are compared with the exact solutions as well as with OHAM. The present numerical scheme is quite simple, effective, and expedient for obtaining numerical solution of fractional KK equation in comparison to analytical approach of OHAM. Copyright © 2016 John Wiley & Sons, Ltd.     

High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation

In this study, new high‐order backward semi‐Lagrangian methods are developed to solve nonlinear advection–diffusion type problems, which are realized using high‐order characteristic‐tracking strategies. The proposed characteristic‐tracking strategies are second‐order L‐stable and third‐order L(α)‐stable methods, which are based on a classical implicit multistep method combined with a error‐correction method. We also use backward differentiation formulas and the fourth‐order finite‐difference scheme for diffusion problem discretization in the temporal and spatial domains, respectively. To demonstrate the adaptability and efficiency of these time‐discretization strategies, we apply these methods to nonlinear advection–diffusion type problems such as the viscous Burgers' equation. Through simulations, not only the temporal and spatial accuracies are numerically evaluated but also the proposed methods are shown to be superior to the compared existing characteristic‐tracking methods under the same rates of convergence in terms of accuracy and efficiency. Finally, we have shown that the proposed method well preserves the energy and mass when the viscosity coefficient becomes zero.     

Fourth‐order compact scheme for the one‐dimensional sine‐Gordon equation

Finite difference scheme to the generalized one‐dimensional sine‐Gordon equation is considered in this paper. After approximating the second order derivative in the space variable by the compact finite difference, we transform the sine‐Gordon equation into an initial‐value problem of a second‐order ordinary differential equation. Then Padé approximant is used to approximate the time derivatives. The resulting fully discrete nonlinear finite‐difference equation is solved by a predictor‐corrector scheme. Both Dirichlet and Neumann boundary conditions are considered in our proposed algorithm. Stability analysis and error estimate are given for homogeneous Dirichlet boundary value problems using energy method. Numerical results are given to verify the condition for stability and convergence and to examine the accuracy and efficiency of the proposed algorithm. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009