Rothe's method in direct and time-dependent inverse source problems for a semilinear pseudo-parabolic equation

TL;DR

This work addresses the inverse problem of identifying a time-dependent source in a semilinear pseudo-parabolic equation with variable coefficients by leveraging a weighted integral measurement of the solution. The authors establish existence and uniqueness of a weak solution through Rothe’s time-discretisation, accommodating the additional complexity from the pseudo-parabolic term and a Dirichlet boundary condition under a smallness condition on the data. They reformulate the inverse problem as a coupled direct problem, derive a priori estimates, and prove existence and uniqueness of the solution $(u,h)$. Numerically, they propose a Crank–Nicolson-type scheme that decouples the computation of $u$ and the source factor $h$ at each time step and validate it on manufactured-solution tests, demonstrating high accuracy for $u$ and good accuracy for $h$ within practical resolutions. The results advance the analysis and computation of inverse-source problems for nonlinear pseudo-parabolic PDEs with nonlocal measurements and variable coefficients, with potential extensions to relax the uniqueness condition and to provide a rigorous numerical analysis of the scheme.

Abstract

In this paper, we investigate the inverse problem of determining an unknown time-dependent source term in a semilinear pseudo-parabolic equation with variable coefficients and a Dirichlet boundary condition. The unknown source term is recovered from additional measurement data expressed as a weighted spatial average of the solution. By employing Rothe's time-discretisation method, we prove the existence and uniqueness of a weak solution under a smallness condition on the problem data. We also present a numerical scheme for computations.

Figures (5)

• Figure 1: Graphical visualisation of $\omega(x)$ for Test Case 1 and Case 2.
• Figure 2: Comparison of exact and numerical solutions of $u(t,x)$ and $h(t)$ at selected time steps for Test Case 1.
• Figure 3: Maximum and $L^2$-norm errors of the numerical solution for $u(t,x)$ and $h(t)$ over the entire time interval for Test Case 1.
• Figure 4: Comparison of exact and numerical solutions of $u(t,x)$ and $h(t)$ at selected time steps for Test Case 2.
• Figure 5: Maximum and $L^2$-norm errors of the numerical solution for $u(t,x)$ and $h(t)$ over the entire time interval for Test Case 2.

Theorems & Definitions (20)

• Remark 2.1
• Remark 2.2
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Lemma 2.1
• proof
• Theorem 2.5
• proof