Solutions for non-homogeneous wave equations subject to unusual and Neumann boundary conditions

This paper examines the accuracy, stability, and convergence of the forward and inverse solutions for non-homogeneous wave equations in different cases. The first case is when the wave governing equation is subject to the so-called unusual (non-local/non-classical) boundary condition. Where there is a non-local displacement and flux tension boundary data are placed at the very left end of the examined domain (string or rob), combined with Dirichlet and Neumann data at blue the left end of the domain. We explore the ideal value for the rational stiffness coefficient to be used for the second case which concerns moving Dirichlet data to the right and adding over-determination Neumann boundary conditions to the right or left. The forward solutions for the given problems, represented in the flux tension values, are investigated using the finite difference method (FDM) combined with the separation variable method. We also test the effect of the rotational stiffness coefficient on the effectiveness and convergence of those direct solutions. On the other hand, to obtain stable numerical results for the corresponding inverse problems (concerning retrieval of the force source function), the zeroth, the first, and the second-order of the Tikhonov regularization parameters are tested and the minimum error norm is calculated accordingly to find the best estimate for that space-dependent force source. Because such inverse problems are ill-posed, the existence and uniqueness of their solutions are tolerated and verified based on existing theorems in literature. This work includes numerical examples which illustrate the usefulness of the approach for solving non-homogeneous wave equations under novel kinds of boundary conditions.