The dual reciprocity boundary integral equation technique to solve a class of the linear and nonlinear fractional partial differential equations

In this paper, we apply the boundary integral equation technique and the dual reciprocity boundary elements method (DRBEM) for the numerical solution of linear and nonlinear time‐fractional partial differential equations (TFPDEs). The main aim of the present paper is to examine the applicability and efficiency of DRBEM for solving TFPDEs. We employ the time‐stepping scheme to approximate the time derivative, and the method of linear radial basis functions is also used in the DRBEM technique. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity that appears in the nonlinear problems under consideration. To confirm the accuracy of the new approach, several examples are presented. The convergence of the DRBEM is studied numerically by comparing the exact solutions of the problems under investigation. Copyright © 2015 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 solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach

In this paper, a numerical method is presented to obtain and analyze the behavior of numerical solutions of distributed order fractional differential equations of the general form in the time domain with the Caputo fractional derivative. The suggested method is based on the Müntz–Legendre wavelet approximation. We derive a new operational vector for the Riemann–Liouville fractional integral of the Müntz–Legendre wavelets by using the Laplace transform method. Applying this operational vector and collocation method in our approach, the problem can be reduced to a system of linear and nonlinear algebraic equations. The arising system can be solved by the Newton method. Discussion on the error bound and convergence analysis for the proposed method is presented. Finally, seven test problems are considered to compare our results with other well‐known methods used for solving these problems. The results in the tabulated tables highlighted that the proposed method is an efficient mathematical tool for analyzing distributed order fractional differential equations of the general form.     

A fast second‐order difference scheme for the space–time fractional equation

In this paper, a fast second‐order accurate difference scheme is proposed for solving the space–time fractional equation. The temporal Caputo derivative is approximated by ?L2 ‐1_σ formula which employs the sum‐of‐exponential approximation to the kernel function appeared in Caputo derivative. The second‐order linear spline approximation is applied to the spatial Riemann–Liouville derivative. At each time step, a fast algorithm, the preconditioned conjugate gradient normal residual method with a circulant preconditioner (PCGNR), is used to solve the resulting system that reduces the storage and computational cost significantly. The unique solvability and unconditional convergence of the difference scheme are shown by the discrete energy method. Numerical examples are given to verify numerical accuracy and efficiency of the difference schemes.     

Finite element method for two‐dimensional time‐fractional tricomi‐type equations

In this article, we consider the finite element method (FEM) for two‐dimensional linear time‐fractional Tricomi‐type equations, which is obtained from the standard two‐dimensional linear Tricomi‐type equation by replacing the first‐order time derivative with a fractional derivative (of order α, with 1 <α< 2 ). The method is based on finite element method for space and finite difference method for time. We prove that the method is unconditionally stable, and the error estimate is presented. The comparison of the FEM results with the exact solutions is made, and numerical experiments reveal that the FEM is very effective. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Exact solutions of some systems of fractional differential‐difference equations

In this paper, the ‐expansion method is proposed to establish hyperbolic and trigonometric function solutions for fractional differential‐difference equations with the modified Riemann–Liouville derivative. The fractional complex transform is proposed to convert a fractional partial differential‐difference equation into its differential‐difference equation of integer order. We obtain the hyperbolic and periodic function solutions of the nonlinear time‐fractional Toda lattice equations and relativistic Toda lattice system. The proposed method is more effective and powerful for obtaining exact solutions for nonlinear fractional differential–difference equations and systems. Copyright © 2014 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.     

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.     

Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients

In this paper, we study the numerical solution to time‐fractional partial differential equations with variable coefficients that involve temporal Caputo derivative. A spectral method based on Gegenbauer polynomials is taken for approximating the solution of the given time‐fractional partial differential equation in time and a collocation method in space. The suggested method reduces this type of equation to the solution of a linear algebraic system. Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical analysis nonlinear multi‐term time fractional differential equation with collocation method via fractional B‐spline

In this work, we present numerical analysis for nonlinear multi‐term time fractional differential equation which involve Caputo‐type fractional derivatives for . The proposed method is based on utilization of fractional B‐spline basics in collocation method. The scheme can be readily obtained efficient and quite accurate with less computational work numerical result. The proposal approach transform nonlinear multi‐term time fractional differential equation into a suitable linear system of algebraic equations which can be solved by a suitable numerical method. The numerical experiments will be verify to demonstrate the effectiveness of our method for solving one‐ and two‐dimensional multi‐term time fractional differential equation.     

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.     

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.     

On the concept of exact solution for nonlinear differential equations of fractional‐order

In this article, the new exact travelling wave solutions of the nonlinear space‐time fractional Burger's, the nonlinear space‐time fractional Telegraph and the nonlinear space‐time fractional Fisher equations have been found. Based on a nonlinear fractional complex transformation, certain fractional partial differential equations can be turned into ordinary differential equations of integer order in the sense of the Jumarie's modified Riemann–Liouville derivative. The ‐expansion method is effective for constructing solutions to the nonlinear fractional equations, and it appears to be easier and more convenient by means of a symbolic computation system. Copyright © 2016 John Wiley & Sons, Ltd.     

An efficient method for fractional nonlinear differential equations by quasi‐Newton's method and simplified reproducing kernel method

An efficient method for nonlinear fractional differential equations is proposed in this paper. This method consists of 2 steps. First, we linearize the nonlinear operator equation by quasi‐Newton's method, which is based on Fréchet derivative. Then we solve the linear fractional differential equations by the simplified reproducing kernel method. The convergence of the quasi‐Newton's method is discussed for the general nonlinear case as well. Finally, some numerical examples are presented to illustrate accuracy, efficiency, and simplicity of the method.     

Extremal solutions for p‐Laplacian boundary value problems with the right‐handed Riemann‐Liouville fractional derivative

The paper is concerned with the solvability for several nonlinear boundary value problems of fractional p‐Laplacian differential equation involving the right‐handed Riemann‐Liouville derivative. By applying monotone iterative technique, lower and upper solutions method and the Banach fixed point theorem, sufficient conditions for existence and uniqueness of extremal solutions are obtained and they extend existing results. At last, two examples are provided to illustrate the results.     

Sine‐cosine wavelet method for fractional oscillator equations

Purpose In this article, a novel computational method is introduced for solving the fractional nonlinear oscillator differential equations on the semi‐infinite domain. The purpose of the proposed method is to get better and more accurate results. Design/methodology/approach The proposed method is the combination of the sine‐cosine wavelets and Picard technique. The operational matrices of fractional‐order integration for sine‐cosine wavelets are derived and constructed. Picard technique is used to convert the fractional nonlinear oscillator equations into a sequence of discrete fractional linear differential equations. Operational matrices of sine‐cosine wavelets are utilized to transformed the obtained sequence of discrete equations into the systems of algebraic equations and the solutions of algebraic systems lead to the solution of fractional nonlinear oscillator equations. Findings The convergence and supporting analysis of the method are investigated. The operational matrices contains many zero entries, which lead to the high efficiency of the method, and reasonable accuracy is achieved even with less number of collocation points. Our results are in good agreement with exact solutions and more accurate as compared with homotopy perturbation method, variational iteration method, and Adomian decomposition method. Originality/value Many engineers can utilize the presented method for solving their nonlinear fractional models.     

Numerical solutions of time fractional degenerate parabolic equations by variational iteration method with Jumarie‐modified Riemann–Liouville derivative

In this article, the fractional variational iteration method is employed for computing the approximate analytical solutions of degenerate parabolic equations with fractional time derivative. The time‐fractional derivatives are described by the use of a new approach, the so‐called Jumarie modified Riemann–Liouville derivative, instead in the sense of Caputo. The approximate solutions of our model problem are calculated in the form of convergent series with easily computable components. Moreover, the numerical solution is compared with the exact solution and the quantitative estimate of accuracy is obtained. The results of the study reveal that the proposed method with modified fractional Riemann–Liouville derivatives is efficient, accurate, and convenient for solving the fractional partial differential equations in multi‐dimensional spaces without using any linearization, perturbation or restrictive assumptions. Copyright © 2011 John Wiley & Sons, Ltd.     

A new approach by two‐dimensional wavelets operational matrix method for solving variable‐order fractional partial integro‐differential equations

In this paper, a new computational scheme based on operational matrices (OMs) of two‐dimensional wavelets is proposed for the solution of variable‐order (VO) fractional partial integro‐differential equations (PIDEs). To accomplish this method, first OMs of integration and VO fractional derivative (FD) have been derived using two‐dimensional Legendre wavelets. By implementing two‐dimensional wavelets approximations and the OMs of integration and variable‐order fractional derivative (VO‐FD) along with collocation points, the VO fractional partial PIDEs are reduced into the system of algebraic equations. In addition to this, some useful theorems are discussed to establish the convergence analysis and error estimate of the proposed numerical technique. Furthermore, computational efficiency and applicability are examined through some illustrative examples.     

A high‐order compact scheme for the nonlinear fractional Klein–Gordon equation

In this article, a high‐order finite difference scheme for a kind of nonlinear fractional Klein–Gordon equation is derived. The time fractional derivative is described in the Caputo sense. The solvability of the difference system is discussed by the Leray–Schauder fixed point theorem, while the stability and L_∞ convergence of the finite difference scheme are proved by the energy method. Numerical examples are provided to demonstrate the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 706–722, 2015     

Fractional‐order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro‐differential equations

In this paper, a new two‐dimensional fractional polynomials based on the orthonormal Bernstein polynomials has been introduced to provide an approximate solution of nonlinear fractional partial Volterra integro‐differential equations. For this aim, the fractional‐order orthogonal Bernstein polynomials (FOBPs) are constructed, and its operational matrices of integration, fractional‐order integration, and derivative in the Caputo sense and product operational matrix are derived. These operational matrices are utilized to reduce the under study problem to a nonlinear system of algebraic equations. Using the approximation of FOBPs, the convergence analysis and error estimate associated to the proposed problem have been investigated. Finally, several examples are included to clarify the validity, efficiency, and applicability of the proposed technique via FOBPs approximation.