Pell polynomial solution of the nonlinear variable order space fractional PDEs

Document Type : Regular research papers

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Department of Mathematics, Faculty of Science, Pamukkale University, Denizli

Abstract

In this paper, multi-term, space fractional variable order nonlinear partial differential equations (VONPDEs) are considered. This type of the equations covers the form of all space fractional VONPDEs containing the first-order time derivative. Here fractional derivatives are defined in the Caputo sense. The presented method is a combination of the Pell collocation method and the finite difference method. Firstly, after discretization with respect to the time variable, the finite difference method is applied to the multi-term, space fractional VONPDE in time derivative. This leads to a space-fractional variable order nonlinear ordinary differential equation. Then, the approximate solution of the space fractional variable order nonlinear ordinary differential equation is expressed in the form of truncated Pell series with unknown coefficients. Finally, the Pell collocation method transform the fractional variable order nonlinear ordinary differential equation into a system of nonlinear equations. Thus, the approximate solution of the nonlinear system is obtained by using the Newton method and unknown coefficients of the truncated Pell series are computed. The error and convergence analysis of the method is studied. In addition, the accuracy of the method is also supported by numerical examples. The numerical results also confirm the convergence and computational efficiency of the presented method. All numerical results are obtained by building fast algorithms using Matlab programming.

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