Construction of a solution to a boundary-value problem a third-order nonhomogeneous equation using Green's function

In this paper, we consider the third boundary-value problem for a third-order nonhomogeneous equation with multiple characteristics in three-dimensional space. We prove the uniqueness of the solution using the energy integral method, and we prove the existence of the solution using the separation of variables method. We present the solution explicitly using the constructed Green's function. We find the conditions on the given functions that ensure the regularity of the solution. Proving the uniform convergence, we establish that the ``small denominator'' is nonzero.

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