, , CAUCHY PROBLEM FOR A LOADED DEGENERATE HYPERBOLIC EQUATION // BSU bulletin. Mathematics, Informatics. - 2018. №1. . - С. 95-99.

CAUCHY PROBLEM FOR A LOADED DEGENERATE HYPERBOLIC EQUATION

The article studies the unique solvability of the Cauchy problem for a loaded equation with Lavrentyev-Bitsadze’s operator in the principal part. The load is defined at the fixed points of the area of desired solutions. The solution region is restricted by the characteristics of lines and abscissa intercept AB, where A (0; 0), B (1; 0). We consider a regular solution of the problem. This is a solution from the class of continuous in closure region, doubly continuously differentiable inside this domain. The theorem of existence and uniqueness of such a solution is proved. The problem has been equivalently reduced (using d'Alembert’s formula) to a system of algebraic equations. The lemma of the unique solvability for this problem has been proved by mathematical induction. The unambiguous criterion for solvability of the problem is given. We have separately considered the case of constant coefficients and gave the example of violating the solvability conditions for the problem. The procedure for it deci- sion is also proposed in the article.

loaded differential equation; Cauchy problem; regular solution; existence and uniqueness.

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Polubarinova-Kochina P. Ya. Teoriya dvizheniya gruntovykh vod

[Theory of Ground Water Movement]. Moscow: Nauka Publ., 1977. 664 p.

Moiseev E. I. Uravneniya smeshannogo tipa so spektral'nym parametrom [Equations of Mixed Type with a Spectral Parameter]. Moscow: Moscow State University Publ., 1988. 150 p.

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