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

TL;DR

This paper investigates forward and inverse problems for a generalized time-fractional pseudo-parabolic equation with variable coefficients, proving existence, uniqueness, and developing numerical methods.

Contribution

It extends previous work by analyzing a time-dependent coefficient in the model and introduces a novel inverse problem with a general functional.

Findings

Proved global existence and uniqueness of solutions for the forward problem.

Developed a numerical scheme and computational algorithm for second-order differential operators.

Established existence and uniqueness for the inverse problem using Schauder's fixed point theorem.

Abstract

In this work, forward and inverse problems for a time-fractional pseudo-parabolic equation $D_t^{\rho} [u(t) + \mu Au(t)] + \sigma(t) Au(t) = r(t)g$ are investigated in a Hilbert space, where $A$ is an unbounded, positive, self-adjoint operator. According to the known papers, the forward problem has been studied only in the case $\sigma(t) = const$. The main novelty of the forward problem in this work is that the model is further generalized and investigated for a time-dependent coefficient $\sigma(t)$. To determine the solution of the forward problem, the Fourier method is employed, and the global existence and uniqueness of the solution are proved. Moreover, when the operator $A$ is a second-order differential operator, a numerical scheme and an efficient computational algorithm are developed. The inverse problem of determining a time-dependent source function is considered under the overdetermination condition of the form $F[u(t)] = \Phi(t)$. The functional $F$ is taken in a general form, and such an inverse problem has not been considered before. The global existence of the solution to the inverse problem is proved by applying Schauder's fixed point theorem, and its uniqueness is established. Furthermore, several examples related to the operator $A$ and the functional $F$ are provided.