Solution of the Mixed Problem for a Third-Order Partial Differential Equation

We consider the initial-boundary value problem for a third-order partial differential equation with highest mixed derivative. An abstract Cauchy problem for a first-order algebraic-differential equation in a Banach space with a distinguished time variable is solved. It is proved that, by equivalent replacements, the original problem is reduced to the Cauchy problem for an algebraic-differential equations. To solve the stated problem, the Fredholm property of the operator before the highest derivative is used. Conditions under which the solution of the problem exists and is unique are determined, and this solution is found in analytical form.