Computational technique for multi-dimensional non-linear weakly singular fractional integro-differential equation

Abstract

In this article, our aim is to construct two new schemes for numerically solving the nonlinear weakly singular integro-fractional differential equation (WSIFDE) in 1D and 2D. Scheme-I uses 1D and 2D shifted Legendre polynomials (SLP) as basis functions whereas scheme-II uses 1D and 2D interpolating basis function (IBF) as basis functions. The main characteristic of these schemes is that it reduces the original equation into the system of nonlinear algebraic equations and thus greatly simplifying the problem. This system is then solved to find the unknown coefficients. The working of proposed schemes is illustrated on several test examples and the obtained numerical results confirm the desired accuracy and efficiency of the schemes. Further, we investigated the convergence analysis and also established the error bounds for the proposed schemes. Finally, it is found that both the schemes are easy to implement, but scheme-I produces better numerical results and also takes less CPU time in comparison to scheme-II. • Approximate solutions of nonlinear weakly singular fractional integro-differential equations in 1D and 2D are discussed. • Two new schemes based on the shifted Legendre polynomials (SLP) and interpolating basis functions (IBF) are constructed. • Error bounds for function approximations are established for presented schemes. • Convergence analysis is rigorously discussed for both the schemes. • The accuracy and efficiency of the presented schemes are illustrated by performing numerical experiments on several test functions.