The Cauchay problem for a loaded partial differential equation of the first order

A. Kh. Attaev

Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 44 (2023) no. 3, pp. 9-18

Résumé

As is well known, the presence of characteristics is very significant in the study of the Cauchy problem for partial differential equations regardless of its order. In the case where the partial differential equation is loaded, additional conditions dependent on the type of load arise for the unique solvability of the Cauchy problem. These conditions arise even for the simplest first and higher order partial differential equations. The main purpose of this paper is to illustrate the effects arising from the study of the Cauchy problem for the linear loaded first-order partial differential equation. Since the correctness of the Cauchy problem is equivalently reduced to the integral equation of the second kind, the basic method is used to prove its solvability – method of successive substitutions. The main conclusion is that the solvability of the Cauchy problem for a loaded partial derivative equation essentially depends on the choice of the load. In the case when the solvability of the Cauchy problem is proven, it turns out that the area of influence of the Cauchy data is not limited to the characteristics only, but new non-characteristic lines appear, beyond which the Cauchy data cannot clearly be extended.

Détail

Keywords: differential equations, loaded differential equation, Cauchy problem, integral equation, method of successive substitutions, characteristics of a differential equation, well-posed problem.