On the regularization of solution of an inverse ultraparabolic equation associated with perturbed final data

TL;DR

This paper addresses the ill-posed inverse problem for ultraparabolic equations by introducing a regularization method to stabilize solutions, supported by theoretical error estimates and numerical validation.

Contribution

It proposes a new regularization approach for ultraparabolic inverse problems, extending previous work and providing stability and error analysis.

Findings

The regularization method stabilizes solutions effectively.

Error estimates demonstrate the method's accuracy.

Numerical examples confirm the method's efficiency.

Abstract

In this paper, we study the inverse problem for a class of abstract ultraparabolic equations which is well-known to be ill-posed. We employ some elementary results of semi-group theory to present the formula of solution, then show the instability cause. Since the solution exhibits unstable dependence on the given data functions, we propose a new regularization method to stabilize the solution. then obtain the error estimate. A numerical example shows that the method is efficient and feasible. This work slightly extends to the earlier results in Zouyed et al. \cite{key-9} (2014).