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Research Papers

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 66177-15177, Iran
e-mail: sh.esmaeili@uok.ac.ir

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 12, 2015; final manuscript received May 4, 2017; published online June 16, 2017. Assoc. Editor: Brian Feeny.

J. Comput. Nonlinear Dynam 12(5), 051020 (Jun 16, 2017) (7 pages) Paper No: CND-15-1335; doi: 10.1115/1.4036710 History: Received October 12, 2015; Revised May 04, 2017

Since the solutions of the fractional differential equations (FDEs) have 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 reflecting 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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Figures

Grahic Jump Location
Fig. 1

Nonpolynomial basis ϕjk with Sα={0.3,0.6,1,1.3,1.6} and n = 4

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Fig. 2

Indicator function (top) and fractional integral of indicator function (bottom)

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Fig. 3

Analytical and numerical solution of problem (20); n = 100, for α=0.3 (left) and α=1.8 (right) (Example 2)

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Fig. 4

Analytical and numerical solution of problem (21); n = 10, for α=0.3 (left) and α=1.8 (right) (Example 3)

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Fig. 5

Numerical solution of Duffing equation for α=1.95 and a = 1 with n = 200 (Example 5)

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