# A new algorithm to determine the creation or depletion term of parabolic equations from boundary measurements

## TL;DR

This paper introduces a novel numerical method for determining creation or depletion coefficients in parabolic equations from boundary data, avoiding the need for prior knowledge of the true solution and demonstrating effectiveness through numerical tests.

## Contribution

A new approach that solves nonlinear inverse problems for parabolic equations without requiring an initial guess of the true coefficient.

## Key findings

- Method successfully recovers coefficients from boundary measurements
- Numerical experiments confirm robustness and accuracy
- Avoids reliance on advanced prior knowledge of solutions

## Abstract

We propose a robust numerical method to find the coefficient of the creation or depletion term of parabolic equations from the measurement of the lateral Cauchy information of their solutions. Most papers in the field study this nonlinear and severely ill-posed problem using optimal control. The main drawback of this widely used approach is the need of some advanced knowledge of the true solution. In this paper, we propose a new method that opens a door to solve nonlinear inverse problems for parabolic equations without any initial guess of the true coefficient. This claim is confirmed numerically. The key point of the method is to derive a system of nonlinear elliptic equations for the Fourier coefficients of the solution to the governing equation with respect to a special basis of L^2. We then solve this system by a predictor-corrector process, in which our computation to obtain the first and second predictors is effective. The desired solution to the inverse problem under consideration follows.