Finding the Time-dependent Term in 2D Heat Equation from Nonlocal Integral Conditions

The aim of this paper is to find the time-dependent term numerically in a two-dimensional heat equation using initial and Neumann boundary conditions and nonlocal integrals as over-determination conditions. This is a very interesting and challenging nonlinear inverse coefficient problem with important applications in various fields ranging from radioactive decay, melting or cooling processes, electronic chips, acoustics and geophysics to medicine. Unique solvability theorems of these inverse problems are supplied. However, since the problems are still ill-posed (a small modification in the input data can lead to bigger impact on the ultimate result in the output solution) the solution needs to be regularized. Therefore, in order to obtain a stable solution, a regularized objective function is minimized in order to retrieve the unknown coefficient. The two-dimensional inverse problem is discretized using the forward time central space (FTCS) finite-difference method (FDM), which is conditionally stable and recast as a nonlinear least-squares minimization of the Tikhonov regularization function. Numerically, this is effectively solved using the MATLAB subroutine lsqnonlin. Both exact and noisy data are inverted. Numerical results for a few benchmark test examples are presented, discussed and assessed with respect to the FTCS-FDM mesh size discretisation, the level of noise with which the input data is contaminated, and the choice of the regularization parameter is discussed based on the trial and error technique.