Kernel Determination Problem in the Third Order 1D Moore-Gibson-Thompson Equation with Memory

In this study, we address the inverse problem of determining the convolution kernel function in the third-order Moore--Gibson--Thompson (MGT) equation, which is commonly used to model fluid motion with memory effects. Specifically, we focus on the determination of the unknown kernel, which governs the memory term in the equation. Initially, we employ the Fourier spectral method to solve the direct initial-boundary-value problem for a non-homogeneous MGT equation that with the memory term. The Fourier spectral method allows us to leverage the problem's inherent linearity and spatial homogeneity, leading to an efficient and explicit construction of the solution. The direct problem is analyzed under appropriate initial and boundary conditions, which are carefully specified to ensure mathematical consistency. To solve the inverse problem, we introduce an additional condition-typically a form of observational data such as at certain points-which provides the necessary constraints for determining the kernel. We prove local existence and uniqueness theorems for solution of the problem.