A Pseudo-spectral Method for Time Distributed Order Two-sided Space Fractional Differential Equations

Abstract

Time distributed order two-sided space differential equations of arbitrary order offer a robust approach to modelling complex dynamical systems. In this study, we describe a scheme for obtaining the numerical solutions of time distributed order multidimensional two-sided space fractional differential equations. The numerical discretization scheme is a hybrid scheme, comprising a Newton-Cotes quadrature formula and a spectral collocation method. The time distributed order fractional differential operator is approximated using the composite Simpson's rule, and the solution of the resulting differential equation is expressed as a linear combination of shifted Chebyshev polynomials in all variables. Convergence analysis of the numerical scheme is presented. Some one-and two-dimensional time distributed order two-sided space fractional differential equations, such as the fractional advection-dispersion and diffusion equations, are presented to demonstrate the accuracy and computational efficiency of the numerical scheme, and numerical solutions are compared with the exact solutions, where these are available.

Efficient Modified Adomian Decomposition Method for Solving Nonlinear Fractional Differential Equations

“…They are increasingly used in such fields as fluid flow, control theory of dynamical systems, diffusive transport akin to diffusion, probability, and statistics [3,4]. Moreover, since most fractional differential equations do not have exact analytical solutions, approximate and numerical techniques are used extensively to treat the fractional differential models, including, for example, the homotopy analysis method [5], homotopy perturbation approach [6], variational iteration technique [7], Chebyshev spectral method [8], orthogonal polynomial method [9], Grunwald-Letnikov method [10], fractional Adams method [11], and several other methods; read [12,13] and the given references therewith.…”