# Inverse Problem of Finding the Coefficient of the Lowest Term in Two-dimensional Heat Equation with Ionkin-type Boundary Condition

## TL;DR

This paper addresses an inverse problem for a 2D heat equation with Ionkin boundary conditions, proposing a new discretization method and demonstrating its effectiveness through numerical examples.

## Contribution

It introduces a novel combination of uniform and non-uniform finite difference methods with numerical integration for solving the inverse problem.

## Key findings

- The method successfully reconstructs the lowest order coefficient.
- Numerical examples validate the accuracy and stability of the proposed approach.
- The approach handles Ionkin boundary conditions effectively.

## Abstract

We consider an inverse problem of determining the time-dependent lowest order coefficient of two-dimensional (2D) heat equation with Ionkin boundary and total energy integral overdetermination condition. The well-posedness of the problem is obtained by generalized Fourier method combined by the Banach fixed poind theorem. For obtaining a numerical solution of the inverse problem, we propose the discretization method from a new combination. On the one hand, it is known the traditional method of uniform finite difference combined with numerical integration on a uniform grid (trapezoidal and Simpson's), on the other hand, we give the method of non-uniform finite difference is combined by a numerical integration on a non-uniform grid (with Gauss-Lobatto nodes). Numerical examples illustrate how to implement the method.