Theoretical and Numerical Approach for Solving the Inverse Problem: A Fourth Order Pseudohyperbolic Equation

OpenAlex İngilizce

ABSTRACT This paper addresses an inverse problem (IP) involving the determination of time dependent potential and force coefficients in a fourth‐order pseudohyperbolic equation, based on nonlocal integral observations. To establish the unique solvability of the IP, the problem is first transformed into a system of Volterra integral equations. The existence and uniqueness of the solution are then demonstrated using the contraction mapping principle over a short time interval. Since the underlying equation is ill‐posed, solution regularization is necessary. To achieve a stable solution, a regularized cost functional is introduced to estimate the unknown parameters. The quintic B‐spline (QnB‐spline) and finite difference (FD) with the Crank–Nicolson scheme are used as direct solvers. The resulting inverse problem is formulated as a nonlinear least‐squares problem with Tikhonov regularization. This optimization is numerically solved using MATLAB's function. The paper presents and analyzes numerical results for two benchmark test cases, along with a discussion on the von Neumann stability analysis of the proposed QnB‐spline numerical scheme.