Vestnik КRAUNC. Fiz.-Mat. nauki. 2023. vol. 44. no. 3. P. 9-18. ISSN 2079-6641

MATHEMATICS               

https://doi.org/10.26117/2079-6641-2023-44-3-9-18

Research Article                                             

Full text in Russian

MSC 35L02

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The Cauchay Problem for a Loaded Partial Differential Equation of the First Order

A. Kh. Attaev^\ast

Institute of Applied Mathematics and Automation KBSC RAS, 89А Shortanova St., Nalchik, 360000, Russia

Abstract. 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.

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

Received: 04.10.2023; Revised: 12.10.2023; Accepted: 30.10.2023; First online: 02.11.2023

For citation. Attaev A. Kh. The Cauchay problem for a loaded partial differential equation of the first order. Vestnik
KRAUNC. Fiz.-mat. nauki. 2023, 44: 3, 9-18. EDN: ZIXUUH. https://doi.org/10.26117/2079-6641-2023-44-3-9-18.

Funding. The study was carried out without support from foundations.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. The author participated in the writing of the article and is fully responsible for
submitting the final version of the article to the press.

^\astCorrespondence: E-mail: attaev.anatoly@yandex.ru

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Attaev A. Kh., 2023

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)

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