Abstract
The aim of this paper is to prove the existence and uniqueness of smooth solutions to the Dirichlet type problem for one class of third-order equations that do not belong to any of the classic types. One of the main classes of non-classical equations is third-order composite type equations, the operator of which is a composition of first-order hyperbolic operator and an elliptic operator in the main part. A number of boundary value problems for the model composite type equations with the Laplace operator were investigated by T.D. Dzhuraev. Many studies have proved the existence of solutions to boundary value problems upon fulfillment of conditions of the convexity of area boundary. The method of proof used in this paper is similar to the method used in the research paper of the author mentioned above. For the research of composite type linear equations a combination of the method of potentials (Green's function) and integral identities is applied. The research method is based on reducing the studied problem with the help of the Green's function to an integral equation, the proof of its solubility and thus - the proof of the solvability of original problem. Upon fulfillment of certain conditions on given functions, a third-order equation reduces to a second-order equation of elliptic type with an unknown right-hand side and the boundary function. With the help of Green's functions for elliptic equations, the studied problem is reduced to a second-order equivalent integral equation, the solvability follows from Fredholm alternative and the theorem of uniqueness of the solution of the original problem
Highlights
The aim of this paper is to prove the existence and uniqueness of smooth solutions to the Dirichlet type problem for one class of third-order equations that do not belong to any of the classic types
Задачи Коши и Гурса для уравнения 3-го порядка / В.В
Summary
В данной работе рассматривается задача типа Дирихле для линейного уравнения третьего порядка составного типа g(x, y) где α , β – заданные постоянные, причем α 2 + β 2 ≠ 0 , а L – линейное дифференциальное выражение вида Для уравнения (1) изучается следующая краевая задача типа Дирихле: Задача Dαβ . Предположим, что существуют две функции u1(x, y) и u2 (x, y), удовлетворяющие условиям задачи (1)–(2). Тогда для функции v(x, y) получим следующую задачу Дирихле:
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