Direct operational matrix approach for weakly singular Volterra integro-differential equations: application in theory of anomalous diffusion

Abstract

In the current paper, we present an efficient direct scheme for weakly singularVolterra integro-differential equations arising in theory of anomalous diffusion. The behav-ior of the system demonstrating the anomalous diffusion is significant for small times. Themethod is based on operational matrices of Chebyshev and Legendre polynomials with sometechniques to reduce the total errors of the already existing scheme. The proposed schemeconverts these equations into linear systems of algebraic equations. The main advantagesof the method are high accuracy, simplicity of performing, low storage requirement. Themain focus of this study is to obtain an analytical explicit expression to estimate the error. Numerical results confirm the superiority and applicability of our scheme in comparisonwith the other methods in literature.