An operational matrix based scheme for numerical solutions of nonlinear weakly singular partial integro-differential equations

Abstract

In this article, we introduce an operational matrix scheme based on two-dimensional wavelets for the Volterra weakly singular nonlinear partial integro-differential equations. By implementing two-dimensional wavelets approximations and its operational matrices of integration and differentiation along with collocation points, the weakly singular partial integro-differential equations are reduced into the system of nonlinear algebraic equations. Moreover, Bernoulli wavelet approximation and Legendre wavelet approximation have been used for inspecting the errors and convergence analysis of the given problems. Some numerical examples are included to establish the accuracy of the proposed scheme via Bernoulli wavelet approximation and Legendre wavelet approximation respectively. Additionally, comparisons of error values between the two wavelets have been presented.