A fractional‐order Jacobi Tau method for a class of time‐fractional PDEs with variable coefficients

This paper presents a shifted fractional‐order Jacobi orthogonal function (SFJF) based on the definition of the classical Jacobi polynomial. A new fractional integral operational matrix of the SFJF is presented and derived. We propose the spectral Tau method, in conjunction with the operational matrices of the Riemann–Liouville fractional integral for SFJF and derivative for Jacobi polynomial, to solve a class of time‐fractional partial differential equations with variable coefficients. In this algorithm, the approximate solution is expanded by means of both SFJFs for temporal discretization and Jacobi polynomials for spatial discretization. The proposed tau scheme, both in temporal and spatial discretizations, successfully reduced such problem into a system of algebraic equations, which is far easier to be solved. Numerical results are provided to demonstrate the high accuracy and superiority of the proposed algorithm over existing ones. Copyright © 2015 John Wiley & Sons, Ltd.     

A collocation method of lines for two‐sided space‐fractional advection‐diffusion equations with variable coefficients

We present the method of lines (MOL), which is based on the spectral collocation method, to solve space‐fractional advection‐diffusion equations (SFADEs) on a finite domain with variable coefficients. We focus on the cases in which the SFADEs consist of both left‐ and right‐sided fractional derivatives. To do so, we begin by introducing a new set of basis functions with some interesting features. The MOL, together with the spectral collocation method based on the new basis functions, are successfully applied to the SFADEs. Finally, four numerical examples, including benchmark problems and a problem with discontinuous advection and diffusion coefficients, are provided to illustrate the efficiency and exponentially accuracy of the proposed method.     

A space‐time spectral method for one‐dimensional time fractional convection diffusion equations

In this paper, we propose a space‐time spectral method for solving a class of time fractional convection diffusion equations. Because both fractional derivative and spectral method have global characteristics in bounded domains, we propose a space‐time spectral‐Galerkin method. The convergence result of the method is proved by providing a priori error estimate. Numerical results further confirm the expected convergence rate and illustrate the versatility of our method. Copyright © 2016 John Wiley & Sons, Ltd.     

An extension of the Gegenbauer pseudospectral method for the time fractional Fokker‐Planck equation

The time fractional Fokker‐Planck equation has been used in many physical transport problems which take place under the influence of an external force field. In this paper we examine pseudospectral method based on Gegenbauer polynomials and Chebyshev spectral differentiation matrix to solve numerically a class of initial‐boundary value problems of the time fractional Fokker‐Planck equation on a finite domain. The presented method reduces the main problem to a generalized Sylvester matrix equation, which can be solved by the global generalized minimal residual method. Some numerical experiments are considered to demonstrate the accuracy and the efficiency of the proposed computational procedure.     

Polynomial spectral collocation method for space fractional advection–diffusion equation

This article discusses the spectral collocation method for numerically solving nonlocal problems: one‐dimensional space fractional advection–diffusion equation; and two‐dimensional linear/nonlinear space fractional advection–diffusion equation. The differentiation matrixes of the left and right Riemann–Liouville and Caputo fractional derivatives are derived for any collocation points within any given bounded interval. Several numerical examples with different boundary conditions are computed to verify the efficiency of the numerical schemes and confirm the exponential convergence; the physical simulations for Lévy–Feller advection–diffusion equation and space fractional Fokker–Planck equation with initial δ‐peak and reflecting boundary conditions are performed; and the eigenvalue distributions of the iterative matrix for a variety of systems are displayed to illustrate the stabilities of the numerical schemes in more general cases. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 514–535, 2014     

Numerical scheme with convergence for a generalized time‐fractional Telegraph‐type equation

In this work, we design and analyze a numerical scheme for solving the generalized time‐fractional Telegraph‐type equation (GTFTTE) which is defined using the generalized time fractional derivative (GTFD) proposed recently by Agrawal. The GTFD involves the scale and the weight functions, and reduces to the traditional Caputo derivative for a particular choice of the weight and the scale functions. The scale and the weight functions play an important role in describing the behavior of real‐life physical systems and thus we study the solution behavior of the GTFTTE by varying the weight and the scale functions in the GTFD. We investigate the solution profile of the GTFTTE under some of these choices. We also provide the stability and the convergence analysis of the proposed numerical scheme for the GTFTTE. We consider two test examples to perform numerical simulations.     

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     

Numerical solution for (2 + 1)‐dimensional time‐fractional coupled Burger equations using fractional natural decomposition method

The aim of the present work is to find the numerical solutions for time‐fractional coupled Burgers equations using a new novel technique, called fractional natural decomposition method (FNDM). Two examples are considered in order to illustrate and validate the efficiency of the proposed algorithm. The numerical simulation has been conducted to ensure the exactness of the present method, and the obtained solutions are offered graphically to reveal the applicability and reliability of the FNDM. The outcomes of the study reveal that the FNDM is computationally very effective and accurate to study the (2 + 1)‐dimensional coupled Burger equations of arbitrary order.     

A single‐node characteristic collocation method for unsteady‐state convection‐diffusion equations in three‐dimensional spaces

We develop a nonconventional single‐node characteristic collocation method with piecewise‐cubic Hermite polynomials for the numerical simulation to unsteady‐state advection‐diffusion transport partial differential equations. This method greatly reduces the number of unknowns in the conventional collocation method, and generates accurate numerical solutions even if very large time steps are taken. The reduction of number of nodes has great potential for problems defined on high space dimensions, which appears in such problems as quantification of uncertainties in subsurface porous media. The method developed here is easy to formulate. Numerical experiments are presented to show the strong potential of the method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 786–802, 2011     

A high‐order B‐spline collocation scheme for solving a nonhomogeneous time‐fractional diffusion equation

A high‐accuracy numerical approach for a nonhomogeneous time‐fractional diffusion equation with Neumann and Dirichlet boundary conditions is described in this paper. The time‐fractional derivative is described in the sense of Riemann‐Liouville and discretized by the backward Euler scheme. A fourth‐order optimal cubic B‐spline collocation (OCBSC) method is used to discretize the space variable. The stability analysis with respect to time discretization is carried out, and it is shown that the method is unconditionally stable. Convergence analysis of the method is performed. Two numerical examples are considered to demonstrate the performance of the method and validate the theoretical results. It is shown that the proposed method is of order O(Δx⁴ + Δt^{2 ? α}) convergence, where α ∈ (0,1) . Moreover, the impact of fractional‐order derivative on the solution profile is investigated. Numerical results obtained by the present method are compared with those obtained by the method based on standard cubic B‐spline collocation method. The CPU time for present numerical method and the method based on cubic B‐spline collocation method are provided.     

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.     

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

In this paper, shifted Legendre polynomials will be used for constructing the numerical solution for a class of multiterm variable‐order fractional differential equations. In the proposed method, the shifted Legendre operational matrix of the fractional variable‐order derivatives will be investigated. The fundamental problem is reduced to an algebraic system of equations using the constructed matrix and the collocation technique, which can be solved numerically. The error estimate of the proposed method is investigated. Some numerical examples are presented to prove the applicability, generality, and accuracy of the suggested method.     

A modified Chebyshev ϑ‐weighted Crank–Nicolson method for analyzing fractional sub‐diffusion equations

This study presents a robust modification of Chebyshev ? ‐weighted Crank–Nicolson method for analyzing the sub‐diffusion equations in the Caputo fractional sense. In order to solve the problem, by discretization of the sub‐fractional diffusion equations using Taylor's expansion a linear system of algebraic equations that can be analyzed by numerical methods is presented. Furthermore, consistency, convergence, and stability analysis of the suggested method are discussed. In this framework, compact structures of sub‐diffusion equations are considered as prototype examples. The main advantage of the proposed method is that, it is more efficient in terms of CPU time, computational cost and accuracy in comparing with the existing ones in open literature.     

Orthogonal spline collocation scheme for multiterm fractional convection‐diffusion equation with variable coefficients

The orthogonal spline collocation (OSC) technique is an efficient way to solve a wide variety of problems that are modeled by ordinary and partial differential equations. In this article, by using OSC method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established for a class of two‐dimensional multiterm fractional convection‐diffusion reaction equation with variable coefficients. The optimal estimates in H^j (j = 0, 1, 2) norms at each time step are derived. Also, estimate in space is provided. At last, we provide some numerical results to verify the accuracy and efficiency of the proposed algorithm.     

Solving 2D time‐fractional diffusion equations by a pseudospectral method and Mittag‐Leffler function evaluation

Two‐dimensional time‐fractional diffusion equations with given initial condition and homogeneous Dirichlet boundary conditions in a bounded domain are considered. A semidiscrete approximation scheme based on the pseudospectral method to the time‐fractional diffusion equation leads to a system of ordinary fractional differential equations. To preserve the high accuracy of the spectral approximation, an approach based on the evaluation of the Mittag‐Leffler function on matrix arguments is used for the integration along the time variable. Some examples along with numerical experiments illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

Superconvergence of the direct discontinuous Galerkin method for convection‐diffusion equations

This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for one‐dimensional linear convection‐diffusion equations. We prove, under some suitable choice of numerical fluxes and initial discretization, a 2k‐th and ‐th order superconvergence rate of the DDG approximation at nodes and Lobatto points, respectively, and a ‐th order of the derivative approximation at Gauss points, where k is the polynomial degree. Moreover, we also prove that the DDG solution is superconvergent with an order k + 2 to a particular projection of the exact solution. Numerical experiments are presented to validate the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 290–317, 2017     

Computational methods for delay parabolic and time‐fractional partial differential equations

This article is concerned with ?‐methods for delay parabolic partial differential equations. The methodology is extended to time‐fractional‐order parabolic partial differential equations in the sense of Caputo. The fully implicit scheme preserves delay‐independent asymptotic stability and the solution continuously depends on the time‐fractional order. Several numerical examples of interest are included to demonstrate the effectiveness of the method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Variational iteration method for solving the space‐ and time‐fractional KdV equation

This paper presents numerical solutions for the space‐ and time‐fractional Korteweg–de Vries equation (KdV for short) using the variational iteration method. The space‐ and time‐fractional derivatives are described in the Caputo sense. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers in the functionals can be identified optimally via variational theory. The iteration method, which produces the solutions in terms of convergent series with easily computable components, requiring no linearization or small perturbation. The numerical results show that the approach is easy to implement and accurate when applied to space‐ and time‐fractional KdV equations. The method introduces a promising tool for solving many space–time fractional partial differential equations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

Solving time‐periodic fractional diffusion equations via diagonalization technique and multigrid

This paper addresses numerical computation of time‐periodic diffusion equations with fractional Laplacian. Time‐periodic differential equations present fundamental challenges for numerical computation because we have to consider all the discrete solutions once in all instead of one by one. An idea based on the diagonalization technique is proposed, which yields a direct parallel‐in‐time computation for all the discrete solutions. The major computation cost is therefore reduced to solve a series of independent linear algebraic systems with complex coefficients, for which we apply a multigrid method using the damped Richardson iteration as the smoother. Such a linear solver possesses mesh‐independent convergence factor, and we make an optimization for the damping parameter to minimize such a constant convergence factor. Numerical results are provided to support our theoretical analysis.