Identifying an unknown heat source term in the third-order pseudo-parabolic equation from nonlocal integral observation

Abstract

The aim of this paper is to find the time-dependent heat source term numerically in the third-order pseudo-parabolic equation with initial and boundary conditions supplemented by nonlocal integral observation. This is a very interesting and challenging nonlinear inverse source problem with important applications in various fields of mechanics and physics. Unique solvability theorem of this inverse problem is supplied. However, since the problem is still ill-posed (small errors in the integral input data cause large errors in the output force) the solution needs to be regularized. Therefore, to obtain a stable solution, a regularized objective function is minimized to retrieve the unknown heat source coefficient. The third-order pseudo-parabolic problem is discretized using the finite difference method (FDM) and recast as nonlinear least-squares minimization of the Tikhonov regularization function with simple bounds on the unknown force coefficient. Numerically, this is effectively solved using the MATLAB subroutine lsqnonlin. Both analytical and perturbed (noisy) data are inverted. Numerical results for three benchmark test examples are presented and discussed. In addition, the von Neumann stability analysis is also discussed.