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

Sa？etak In the current paper, we present an efficient direct scheme for weakly singular Volterra integro-differential equations arising in the theory of anomalous diffusion. The behavior of the system demonstrating the anomalous diffusion is significant for small times. The method is based on operational matrices of Chebyshev and Legendre polynomials with some techniques to reduce the total errors of the already existing schemes. The proposed scheme converts these equations into a linear system of algebraic equations. The main advantages of the method are high accuracy, simplicity of performing, and low storage requirement. The main 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 comparison with other methods in the literature