Analysis of a second-order numerical scheme for time-fractional partial integro-differential equations with a weakly singular kernel

Abstract

In this paper, a second-order numerical scheme is developed and analyzed to approximate the solution of a class of time-fractional Volterra integro-differential equations with a weakly singular kernel on a rectangular domain. In the present scheme, we first discretize the domain on a uniform grid. The L1-2 technique is used to discretize the Caputo derivative of order α with α∈(0,1) in the temporal direction. A weakly singular kernel makes it more challenging to approximate the integral and reduces the accuracy of the classical numerical integration method. In order to improve the convergence rate, the composite product trapezoidal formula is employed to approximate the fractional integral. Secondly, a classical central difference technique is used for spatial discretization. Using a maximum norm, the stability and convergence analysis is carried out under suitable regularity assumptions. It is proved that the scheme achieves second-order accuracy. In addition, we extend the proposed difference scheme to solve the corresponding semilinear problem. The well-known Newton’s linearization technique is used to deal with semilinearity. Numerical examples demonstrate the efficiency and applicability of the proposed scheme.