Space-dependent heat source determination problem with nonlocal periodic boundary conditions

Abstract

The purpose of this paper is to identify the space-dependent heat source coefficient numerically, for the first time, in the third-order pseudo-parabolic equation with initial and nonlocal periodic boundary conditions from nonlocal integral observation. This problem emerges significantly in the modelling of numerous phenomena in physics, engineering, mechanics and science. Although, the inverse source problem considered in this article is ill-posed by being sensitive to noise but has a unique solution. For the numerical realization, we apply the finite difference method (FDM) for discretizing the forward problem and the Tikhonov regularization for finding a stable and accurate solution. The resulting nonlinear minimization problem is solved computationally using the MATLAB subroutine lsqnonlin. Numerical results presented for two benchmark test examples with linear and nonlinear source functions show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data. Furthermore, the von Neumann stability analysis is also discussed.