A time-dependent inverse source problem for a semilinear pseudo-parabolic equation with Neumann boundary condition

TL;DR

This work addresses the time-dependent inverse source problem for a semilinear pseudo-parabolic equation with Neumann boundary conditions, aiming to recover a time-varying source $h(t)$ from the domain-wide measurement $m(t)$. By reformulating $h(t)$ in terms of the state $u$ and data, the authors establish existence and uniqueness of a weak solution using Rothe's time discretisation and develop a convergent time-stepping algorithm. They implement a numerically efficient scheme with polynomial regularisation to handle noisy data and validate it through 1D and 2D simulations using finite elements, demonstrating first-order convergence in the noise-free case and robust reconstruction under moderate noise. The results confirm the method's applicability to higher dimensions and underline its physical motivation, with future work pointing to Dirichlet or local measurements as potential extensions.

Abstract

In this paper, we study the inverse problem for determining an unknown time-dependent source coefficient in a semilinear pseudo-parabolic equation with variable coefficients and Neumann boundary condition. This unknown source term is recovered from the integral measurement over the domain $Ω$. Based on Rothe's method, the existence and uniqueness of a weak solution, under suitable assumptions on the data, is established. A numerical time-discrete scheme for the unique weak solution and the unknown source coefficient is designed, and the convergence of the approximations is proven. Numerical experiments are presented to support the theoretical results. Noisy data is handled through polynomial regularisation.

Figures (11)

• Figure 1: Experiment 1 (1D): (a) rate of convergence of $u_1$; (b) rate of convergence for $h_1$, for noise-free data $m_1$.
• Figure 2: Experiment 1 (1D): (a) Polynomial approximations of noisy measurement data $m_1^\epsilon(t)$ for $\epsilon = 0.005$, and (b) corresponding absolute errors.
• Figure 3: Experiment 1: (a) The exact source and its numerical approximation, and (b) its corresponding absolute error, obtained for various levels of noise.
• Figure 4: Experiment 1 (1D): (a) The exact solution at final time $T=1$ and its numerical approximation, and (b) its corresponding absolute error, obtained for various levels of noise.
• Figure 5: Experiment 2 (1D): (a) Polynomial approximations of noisy measurement data $m_2^\epsilon(t)$ for $\epsilon = 0.03$, and (b) corresponding absolute errors. Due to the form of $m_2^\epsilon(t)$, only even degrees are considered.

Theorems & Definitions (12)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Theorem 3.1
• proof
• Theorem 4.1
• proof
• Lemma 4.1
• proof
• Theorem 4.2