Forward and inverse problems for a time-fractional pseudo-parabolic equation with variable coefficients

Technical Summary: Forward and Inverse Problems for a Time-Fractional Pseudo-Parabolic Equation with Variable Coefficients

Problem Statement
This work investigates forward and inverse problems associated with a time-fractional pseudo-parabolic equation defined in a separable Hilbert space $H$. The governing equation is given by:
$$D^\rho_t [u(t) + \mu Au(t)] + \sigma(t)Au(t) = r(t)g, \quad t \in (0, T]$$
subject to the initial condition $u(0) = \phi$. Here, $A$ is an unbounded, positive, self-adjoint operator with a compact inverse; $\mu > 0$ is a constant; $0 < \rho < 1$ is the order of the Caputo fractional derivative; $\sigma(t)$ is a time-dependent coefficient; $r(t)$ is a scalar source function; and $g, \phi \in H$.

The study addresses two distinct problems:

1. The Forward Problem: Determining the solution $u(t)$ given known coefficients $\sigma(t)$, $r(t)$, and source term $g$.
2. The Inverse Problem: Determining the pair of functions $\{u(t), r(t)\}$ when $r(t)$ is unknown, subject to an additional overdetermination condition of the form $F[u(t)] = \Phi(t)$, where $F: H \to \mathbb{R}$ is a bounded linear functional and $\Phi(t)$ is a given scalar function.

Methodology
The authors employ a spectral approach based on the Fourier method, utilizing the eigenfunctions $\{v_k\}$ and eigenvalues $\{\lambda_k\}$ of the operator $A$.

• Forward Problem Analysis: The solution $u(t)$ is represented as an infinite series $u(t) = \sum u_k(t)v_k$. This reduces the partial differential equation to a sequence of scalar Cauchy problems for the coefficients $u_k(t)$. The existence and uniqueness of these scalar solutions are established using properties of the Mittag-Leffler function and integral estimates. Global existence and uniqueness for the full problem are proved by deriving norm estimates for $u(t)$, $Au(t)$, and the fractional derivatives $D^\rho_t u(t)$ and $D^\rho_t (Au(t))$ in the space $C([0, T]; H)$.
• Inverse Problem Analysis: To solve for the unknown source $r(t)$, the functional $F$ is applied to the governing equation. This yields an explicit integral representation for $r(t)$ involving the solution $u(t)$. The existence of a solution to the inverse problem is established by formulating the problem as a fixed-point equation $r(t) = B[r](t)$. The authors apply Schauder's fixed point theorem, demonstrating that the operator $B$ maps a closed, convex, and bounded subset of $C[0, T]$ into itself and that the image set is relatively compact (verified via the Arzelà-Ascoli theorem). Uniqueness is proved by showing that the difference between two potential solutions must vanish.
• Numerical Scheme: For the specific case where $A$ is a second-order differential operator ($Au = -u_{xx}$), a finite difference scheme is developed. The Caputo derivative is approximated using a non-uniform L1 scheme on a graded time mesh to handle the singularity at $t=0$, while spatial derivatives are handled via central differences. The resulting linear algebraic system is tridiagonal and solved using the Thomas algorithm.

Key Contributions and Results

1. Generalization of the Forward Model: Unlike previous studies which often assumed a constant coefficient $\sigma(t)$, this work investigates the case where $\sigma(t)$ is a time-dependent function. The authors prove the global existence and uniqueness of the solution under the condition that $\sigma(t)$ is continuous and strictly positive.
2. Generalized Inverse Problem Formulation: The paper addresses an inverse problem for determining a time-dependent source $r(t)$ under a general linear functional condition $F[u(t)] = \Phi(t)$. This generalizes previous works that were restricted to specific functionals (e.g., point evaluation at a final time or integral conditions). The authors provide specific examples of admissible functionals and verify the necessary conditions for existence and uniqueness.
3. Theoretical Proofs: Rigorous proofs are provided for the existence and uniqueness of solutions for both the forward and inverse problems. The inverse problem's existence relies on Schauder's fixed point theorem, while uniqueness is derived from energy estimates.
4. Numerical Validation: An efficient computational algorithm is presented for the forward problem. Numerical experiments are conducted with a specific test case where the exact solution is known. The results demonstrate the accuracy of the proposed finite difference scheme, with numerical solutions closely matching the exact solution.

Significance
The authors claim that the primary novelty of this work lies in the generalization of the mathematical model to include a time-dependent coefficient $\sigma(t)$ in the forward problem, a scenario not previously studied in the context of time-fractional pseudo-parabolic equations. Furthermore, the inverse problem is treated with a highly general functional $F$, broadening the class of applicable inverse problems beyond those restricted to specific boundary or integral conditions. The paper establishes a rigorous theoretical framework for these generalized problems and provides a practical numerical method for their solution, thereby extending the existing literature on fractional pseudo-parabolic equations.