Approximate solutions of inverse problems for nonlinear space fractional diffusion equations with randomly perturbed data

TL;DR

This paper develops new regularization methods for solving inverse problems in nonlinear space fractional diffusion equations with noisy data, providing convergence rates for the regularized solutions.

Contribution

It introduces truncation and quasi-reversibility methods tailored for nonlinear space fractional diffusion inverse problems with noisy data, including convergence analysis.

Findings

Regularized solutions converge to the true solution under certain conditions.

New regularization techniques improve stability for noisy inverse problems.

Convergence rates are established for the proposed methods.

Abstract

This paper is concerned with backward problem for nonlinear space fractional diffusion with additive noise on the right-hand side and the final value. To regularize the instable solution, we develop some new regularized method for solving the problem. In the case of constant coefficients, we use the truncation methods. In the case of perturbed time dependent coefficients, we apply a new quasi-reversibility method. We also show the convergence rate between the regularized solution and the sought solution under some a priori assumption on the sought solution.