detection of a time-dependent forcing term in a one-dimensional wave equation with a dynamic-type boundary condition

In the current paper, we study an inverse problem of identifying a time-dependent forcing term in the one-dimensional wave equation. we have the information of the wave displacement at two different instants of time and two sensor locations of space along with a dynamic type boundary condition. we prove the unique solvibility of the problem under some regularity and consistency conditions. then, an approximate solution of the given inverse problem based upon deploying the ritz technique along with the the collocation method is presented which converts the problem to a linear system of algebraic equations. the method takes advantage of the tikhonov regularization technique to solve the linear system of equations that is not well-conditioned in order to achieve stable solutions. numerical findings are also included to support the claim that the presented method is reliable in finding accurate and stable solutions.