Efficient numerical techniques for fractional integro-differential equations with weakly singular kernel

Abstract This paper addresses two finite difference approaches for computing the numerical solution to a Volterra-type fractional integro differential equation with a weakly singular kernel. The problem is discretized on a uniform mesh using the L1 technique for the Caputo fractional order derivative, and the integral component is solved using the product left rectangular rule. The approach proves to be first-order convergent. To improve the order of accuracy, a post-processing approach namely the Richardson extrapolation scheme is used, and the convergence rate is greatly increased to second order. Another difference scheme is developed to address the model problem by using the product trapezoidal rule for the weakly singular integral and the L1 technique for the fractional derivative. The product trapezoidal rule has a greater convergence rate than the product rectangular rule, hence the latter method provides more accurate results even without extrapolation. The comoutational cost is also reduced in the second scheme which can be seen from the tabular results. The error analysis is performed for both the suggested numerical algorithms, and some numerical experiments are performed to confirm the theoretical results.