In this article, for one degenerate equation of the fourth order, containing the fractional derivative of Caputo, in rectangular domain, a initial-boundary problem is formulated and investigated. The existenceand uniqueness of the solution of the problem have been proved. At the same time, by applying the method of separation of variables to the considered problem, a spectral problem for an ordinary differential equation has been obtained. Next, the Green's function of the spectral problem was constructed, with the help of which it is equivalently reduced to an the second kind Fredholm integral equation with a symmetric kernel. The solution of the considered problem has been written as the sum of a Fourier series with respect to the system of eigenfunctions of the spectral problem.
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