A Regularization for Time-Fractional Backward Heat Conduction Problem with Inhomogeneous Source Function

TL;DR

This paper extends a regularization method for the time-fractional backward heat conduction problem to inhomogeneous, higher-dimensional cases with general elliptic operators, providing theoretical error estimates and numerical validation.

Contribution

It generalizes a previous regularization approach to more complex inhomogeneous and higher-dimensional problems, with rigorous error analysis and numerical demonstrations.

Findings

Optimal order error estimates achieved

Regularization parameter chosen for best accuracy

Numerical experiments confirm theoretical results

Abstract

Recently, Nair and Danumjaya (2023) introduced a new regularization method for the homogeneous time-fractional backward heat conduction problem (TFBHCP) in a one-dimensional space variable, for determining the initial value function. In this paper, the authors extend the analysis done in the above referred paper to a more general setting of an inhomogeneous time-fractional heat equation involving the higher dimensional state variables and a general elliptic operator. We carry out the analysis for the newly introduced regularization method for the TFBHCP providing optimal order error estimates under a source condition by choosing the regularization parameter appropriately, and also carry out numerical experiments illustrating the theoretical results.