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

In this paper, we investigate inverse initial problem for a fourth-order differential equation that becomes degenerate along the boundary of a rectangular domain. By applying the method of separation of variables, we reduce the problem to a spectral problem for an ordinary differential equation. We then construct the Green’s function for this spectral problem, which allows us to transform it into a Fredholm integral equation of the second kind with a symmetric kernel. Using the theory of integral equations with symmetric kernels, we establish the existence and some properties of the eigenvalues and eigenfunctions of the spectral problem. The solution to the original problem is expressed as a Fourier series expansion in terms of these eigenfunctions, and we prove the uniform convergence of this series.