Abstract:
The problem of conjugation of two equations of hyperbolic type of the third order – the Haller equation in the positive part of the domain with an operator from the wave operator in the negative part – is investigated in the mixed domain. The problem consists in finding a regular solution of the equation under consideration when boundary conditions of the second kind are given on the positive part of the boundary of the region, and Cauchy conditions on one of the characteristics are given in the negative part. In addition, continuous gluing of the sought solution and its derivative on the conjugate line is required. A theorem on the existence of a single regular solution of the problem under study in the mixed domain is proved. To prove the single-valued solvability of the problem, we use the Tricomi method, according to which we obtain the corresponding fundamental relations between the traces of the desired solution and its derivative, transferred from the positive and negative parts of the mixed region to the conjugate line. From the obtained fundamental relations, we come to the second boundary value problem for an inhomogeneous ordinary differential equation of the second order with respect to the trace of the derivative of the desired solution, the solution of which is found and written out in explicit form. Then the solution of the investigated problem is written out in explicit form as the solution of the second boundary value problem for the Haller equation in the positive part of the mixed region and as the solution of the Darbu’s problem for the model equation of hyperbolic type of the third order in the negative part. Sufficient conditions on the given functions, providing regularity of the obtained solutions of the investigated problem in the mixed domain, are found.
Keywords:Hallaire equation, hyperbolic equations, local problem, regular solution
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