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research-article

A piecewise nonpolynomial collocation method for fractional differential equations

[+] Author and Article Information
Shahrokh Esmaeili

Department of Applied Mathematics University of Kurdistan P.O. Box 416, Sanandaj, Iran
sh.esmaeili@uok.ac.ir

1Corresponding author.

ASME doi:10.1115/1.4036710 History: Received October 12, 2015; Revised May 04, 2017

Abstract

Since the solutions of the fractional differential equations have, in general, unbounded derivatives at zero, their numerical solutions by piecewise polynomial collocation method on uniform meshes will lead to poor convergence rates. This paper presents a piecewise nonpolynomial collocation method for solving such equations which reflects the singularity of the exact solution. The entire domain is divided into several small subdomains, and the nonpolynomial pieces are constructed using a block-by-block scheme on each subdomain. The method is applied to solve linear and nonlinear fractional differential equations. Numerical examples are given and discussed to illustrate the effectiveness of the proposed approach.

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