# A numerical method for an inverse source problem for parabolic equations and its application to a coefficient inverse problem

## TL;DR

This paper introduces a two-stage numerical method combining derivative equations and quasi-reversibility to solve inverse source and nonlinear coefficient inverse problems for parabolic equations, with demonstrated numerical results.

## Contribution

It develops a novel two-stage numerical approach for inverse source problems and applies it iteratively to solve nonlinear coefficient inverse problems, advancing computational methods in this area.

## Key findings

- Successful reconstruction of sources from observations.
- Effective iterative solution for nonlinear coefficient inverse problems.
- Numerical validation demonstrating method accuracy.

## Abstract

Two main aims of this paper are to develop a numerical method to solve an inverse source problem for parabolic equations and apply it to solve a nonlinear coefficient inverse problem. The inverse source problem in this paper is the problem to reconstruct a source term from external observations. Our method to solve this inverse source problem consists of two stages. We first establish an equation of the derivative of the solution to the parabolic equation with respect to the time variable. Then, in the second stage, we solve this equation by the quasi-reversibility method. The inverse source problem considered in this paper is the linearization of a nonlinear coefficient inverse problem. Hence, iteratively solving the inverse source problem provides the numerical solution to that coefficient inverse problem. Numerical results for the inverse source problem under consideration and the corresponding nonlinear coefficient inverse problem are presented.