Higher order approximations for fractional order integro-parabolic partial differential equations on an adaptive mesh with error analysis

Abstract

This work deals with a higher order numerical approximation for analyzing a class of multi-term time fractional partial integro-differential equations involving Volterra integral operators. The solutions to these problems have a mild singularity at the initial time, due to which an initial layer appears, which becomes more sharper as the highest order time fractional derivative decreases. This behaviour reduces the rate of convergence by standard approaches. We start the present work by considering the existence and uniqueness of a class of generalized partial integro-differential equations and then, present the L1 discretization on a graded mesh in time which is adapted towards the initial time level. This discretization leads to a higher order accuracy than the solutions obtained on a time uniform mesh. The convergence analysis corresponding to the Volterra integral operator is nontrivial as it uses a repeated quadrature rule. This analysis can also be extended for weakly singular kernels. The stability analysis of the present scheme with a sharp error estimation is also provided. The analysis with extensive experiments shows that a higher rate of accuracy can be attained for several suitable choices of the grading parameters for solving several classes of time fractional integro-differential equations.