Eur. Phys. J. Plus (2019) 134: 368
https://doi.org/10.1140/epjp/i2019-12743-6

Regular Article

Legendre wavelets for fractional partial integro-differential viscoelastic equations with weakly singular kernels^⋆

¹ School of Mathematical Science, Nanjing Normal University, 210023, Nanjing, China
² Department of Mathematics, Shiraz University of Technology, Shiraz, Iran
³ Engineering School (DEIM), University of Tuscia, Viterbo, Italy

^* e-mail: cattani@unitus.it

Received: 15 January 2019
Accepted: 23 April 2019
Published online: 31 July 2019

Abstract

This study deals with a new class of fractional partial integro-differential equations (FPI-DEs) characterized by the presence of weakly singular kernel and a Newtonian viscoelasticity factor. To numerically solve such equations, a hybrid method is established by combining the Legendre wavelets (LWs), the collocation method, and a new operational matrix of fractional integration (OMFI). More precisely, the unknown solution is expanded by the LWs with unknown coefficients. Then, the OMFI and the collocation method are utilized to extract a system of algebraic equations whose solution is an approximation for the problem’s solution. Convergence and error estimation of the LWs expansion in two dimensions are investigated. Moreover, the efficiency and accuracy of the proposed method are demonstrated by solving some concrete examples. The obtained results confirm the presented approach is very accurate to provide satisfactory solutions.