Unique Solvability of a Mixed Problem for a Fourth Order Time-Fractional Space Degenerate Partial Differential Equation

In this work, in a rectangular domain, we study a mixed problem for a fourth order differential equation degenerating on the bound of the domain. By applying the method of separation of variables to the considered problem, a spectral problem for an ordinary differential equation has been obtained. Then, the Green’s function of the spectral problem has been constructed, with the help of which it is equivalently reduced to the second kind Fregholm integral equation with a symmetric kernel. Using the theory of integral equations with symmetric kernels the existence and some properties of the eigenfunctions and eigenvalues of this spectral problem has been studied. The solution of the original problem has been written as the sum of a Fourier series with respect to the system of eigenfunctions of the spectral problem. Uniformly convergence of this series has been proved.