Simultaneous identification of the solely time‐dependent potential and source control terms from known moisture moments

Abstract

Pseudo‐parabolic equations are commonly used as mathematical models in mechanics, biology, and physics to address various applied problems. One particular application involves describing moisture transfer dynamics in subsoil layers using pseudo‐parabolic equations. This manuscript examines the inverse problem (IP) of identifying the moisture transfer function, along with the time‐varying potential and source control terms, in a linear pseudo‐parabolic equation from known moisture moments. We prove the existence of a unique solution to the inverse problem for sufficiently small times by employing the contraction principle. The inverse problem is reformulated as a nonlinear least‐squares minimization problem, with the unknowns subjected to simple bounds. To guarantee stability, the Tikhonov regularization technique is utilized. For the numerical discretization, we develop the Crank–Nicolson finite‐difference method to solve the direct problem. To solve the nonlinear least‐squares minimization problem, we utilize the built‐in subroutine lsqnonlin from the MATLAB optimization toolbox. We present and thoroughly discuss numerical outcomes for a benchmark test example, providing insights into the performance and effectiveness of the proposed methodology.