Towards the Oblique Derivative Problem for the Lavrentyev – Bitsadze Equation in the Half-Plane

The article solves the boundary value problem with an oblique derivative for the Lavrentyev – Bitsadze equation, which is an equation of the mixed elliptic-hyperbolic type. The mixed type equations are used in transonic gas dynamics. A restricted solution to the problem is sought in the half plane, consisting of the upper half plane (where the equation is elliptic) and adjacent band (where the equation is hyperbolic). At the boundary of the domain an oblique derivative towards one of the characteristics is specified. At the boundary of the upper half plane, which is a line of equation type alteration, the matching conditions of the fourth kind are set. In the hyperbolicity band the solution is represented by d’Alembert formula, and in the upper half plane, where the equation is elliptic, the restricted solution is represented by Poisson integral of unknown density. For unknown Poisson integral density a singular integral equation is obtained, which is reduced to the Riemann boundary value problem for holomorphic functions. The solution of this problem is obtained in an explicit form. Thus, the solution to the problem with an oblique derivative for the Lavrentyev – Bitsadze equation was obtained in an explicit form for the case of the half plane accurate to a constant summand. An example of solving the problem to prove the theoretical calculations is provided at the end of the article.

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