The Tikhonov regularization method for the inverse source problem of time fractional heat equation in the view of ABC-fractional technique

This research considers an inverse source problem for fractional diffusion equation that containing fractional derivative with non-singular and non-local kernel, namely, Atangana-Baleanu-Caputo fractional derivative. In our study, an explicit solution set is acquired via the expansion method and the overdetermination condition at a final time. The problem is ill-posed in the meaning of Hadamard and thus the solution does not continuously depend on the input data. We have applied the Tikhonov regularization method to regularize the unstable solution. For the estimation of convergence between the exact and the regularized solutions, we focus on two parameter choice rules, a-priori and a-posteriori parameter. In the end, a simulation example is utilized and discussed to affirm the presented theoretical results.

Automated summary

This research examines an inverse source problem for a fractional diffusion equation containing the Atangana-Baleanu-Caputo fractional derivative, obtaining an explicit solution set through expansion method and overdetermination condition. Due to the ill-posed nature of the problem, Tikhonov regularization method is applied to stabilize the solution, with a focus on two parameter choice rules. A simulation example is used to validate the presented theoretical results.