Research Papers

A Comparative Study of Integer and Noninteger Order Wavelets for Fractional Nonlinear Fredholm Integro-Differential Equations

[+] Author and Article Information
F. Mohammadi

Department of Mathematics,
University of Hormozgan,
P. O. Box 3995,
Bandar Abbas 3995, Iran
e-mail: f.mohammadi62@hotmail.com

J. A. Tenreiro Machado

Department of Electrical Engineering,
Institute of Engineering,
Polytechnic of Porto,
Rua Dr. António Bernardino de Almeida,
Porto 4249-015, Portugal
e-mail: jtm@isep.ipp.pt

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 8, 2017; final manuscript received May 15, 2018; published online June 18, 2018. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 13(8), 081002 (Jun 18, 2018) (11 pages) Paper No: CND-17-1555; doi: 10.1115/1.4040343 History: Received December 08, 2017; Revised May 15, 2018

This paper compares the performance of Legendre wavelets (LWs) with integer and noninteger orders for solving fractional nonlinear Fredholm integro-differential equations (FNFIDEs). The generalized fractional-order Legendre wavelets (FLWs) are formulated and the operational matrix of fractional derivative in the Caputo sense is obtained. Based on the FLWs, the operational matrix and the Tau method an efficient algorithm is developed for FNFIDEs. The FLWs basis leads to more efficient and accurate solutions of the FNFIDE than the integer-order Legendre wavelets. Numerical examples confirm the superior accuracy of the proposed method.

Copyright © 2018 by ASME
Topics: Wavelets
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Grahic Jump Location
Fig. 1

(a) The exact and numerical solutions with M = 6, k = 1, and α=0.5 and (b) the absolute error of approximate solutions for several values of α and M=6,k=1 (Example 1)

Grahic Jump Location
Fig. 2

(a) Numerical solution for different values of ν=α with M = 6, k = 1 and (b) the absolute error of approximate solutions for ν = 1 and different values of the fractional order α, with M=6,k=1 (Example 2)

Grahic Jump Location
Fig. 3

The absolute value of the residual functions for different values of the fractional orders α and ν, with M=6,k=1: (a) ν=0.5, (b) ν=0.75, and (c) ν=0.95 (Example 2)

Grahic Jump Location
Fig. 4

(a) Numerical solution for different values of ν=α with M = 10, k = 1 and (b) the absolute error of approximate solutions for ν = 1 and different values of fractional oreder α with M=10,k=1 (Example 3)

Grahic Jump Location
Fig. 5

The absolute value of the residual function for various values of α, with M = 10 and k = 1: (a) ν=0.5, (b) ν=0.75, and (c) ν=0.95 (Example 3)

Grahic Jump Location
Fig. 6

(a) The approximate solutions for different values of ν=α and M = 14, k = 1 and (b) the absolute error for various ν=α and M = 14 and k = 1 (Example 4)

Grahic Jump Location
Fig. 7

The absolute value of the residual function for different values of α and M = 14 and k = 1: (a) ν=0.90 and (b) ν=0.95 (Example 4)




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