An Inverse Initial Problem for a Fourth Order Time-Fractional Space Degenerate Partial Differential Equation

Abstract

This study focuses on an inverse initial value problem involving a fourth-order differential equation that degenerates along the boundary of a rectangular domain. By employing the method of separation of variables, the problem is transformed into a spectral problem for an ordinary differential equation. To analyze this spectral problem, we construct its Green’s function, which enables us to reformulate it as a Fredholm integral equation of the second kind with a symmetric kernel. Utilizing the theory of integral equations with symmetric kernels, we demonstrate the existence and certain characteristics of the eigenvalues and eigenfunctions. The solution to the original problem is then represented as a Fourier series in terms of these eigenfunctions, and we establish the uniform convergence of the resulting series.