An extension of the spectral Tau method for numerical solution of multi-order fractional differential equations with convergence analysis

AbstractThe main purpose of this paper is to provide an efficient numerical approach for the fractional differential equations (FDEs) based on a spectral Tau method. An extension of the operational approach of the Tau method with the orthogonal polynomial bases is proposed to convert FDEs to its matrix–vector multiplication representation. The fractional derivatives are described in the Caputo sense. The spectral rate of convergence for the proposed method is established in the L2 norm. We tested our procedure on several examples and observed that the obtained numerical results confirm the theoretical prediction of the exponential rate of convergence

A finite difference method for fractional diffusion equations with Neumann boundary conditions

A finite difference numerical method is investigated for fractional order diffusion problems in one space dimension. The basis of the mathematical model and the numerical approximation is an appropriate extension of the initial values, which incorporates homogeneous Dirichlet or Neumann type boundary conditions. The well-posedness of the obtained initial value problem is proved and it is pointed out that each extensions is compatible with the original boundary conditions. Accordingly, a finite difference scheme is constructed for the Neumann problem using the shifted Grünwald--Letnikov approximation of the fractional order derivatives, which is based on infinite many basis points. The corresponding matrix is expressed in a closed form and the convergence of an appropriate implicit Euler scheme is proved

Shifted Jacobi spectral collocation method with convergence analysis for solving integro-differential equations and system of integro-differential equations

This article addresses the solution of multi-dimensional integro-differential equations (IDEs) by means of the spectral collocation method and taking the advantage of the properties of shifted Jacobi polynomials. The applicability and accuracy of the present technique have been examined by the given numerical examples in this paper. By means of these numerical examples, we ensure that the present technique is simple and very accurate. Furthermore, an error analysis is performed to verify the correctness and feasibility of the proposed method when solving IDE

A survey on fuzzy fractional differential and optimal control nonlocal evolution equations

We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Applied Mathematics', ISSN: 0377-0427. Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication 20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515

On a Legendre tau method for boundary value problems with a Caputo derivative

In this paper, we revisit a Legendre-tau method for two-point boundary value problems with a Caputo fractional derivative in the leading term, and establish an L2 error estimate for smooth solutions. Further, we apply the method to the Sturm-Liouville problem. Numerical experiments indi- cate that for the source problem, it converges steadily at an algebraic rate even for nonsmooth data, and the convergence rate enhances with problem data regularity, whereas for the Sturm-Liouville problem, it always yields excellent convergence for eigenvalue approximations