Stable numerical evaluation of Grünwald–Letnikov fractional derivatives applied to a fractional IHCP

Abstract

The computation of Grünwald–Letnikov fractional derivatives from noisy data is considered as an ill-posed problem and treated by mollification techniques. It is shown that, with the appropriate choice of the radius of mollification, the method is a regularizing algorithm. Next, the recovery of the boundary temperature and heat flux functions from one measured transient temperature data at some interior point of a one-dimensional semi-infinite conductor when the governing diffusion equation is of fractional type is discussed. A simple algorithm based on space marching mollification techniques and Grünwald–Letnikov fractional derivatives is introduced for the numerical solution of this inverse ill-posed problem. In all cases, stability and error estimates are included together with numerical examples of interest.