Mathematics and Mechanics

Problem with data on parallel characteristics for the loaded wave equation

A. Kh. Attaev

Institute of Applied Mathematics and Automation – branch of Kabardino-Balkarian Scientific Center of the Russian Academy of Sciences, 360000, Russia, Nalchik, 89 A Shortanov street

Abstract: The theory of loaded equations is very relevant both in theoretical terms and in its numerous practical applications in various fields of modern natural science. This explains a huge number of works devoted to research and application of loaded equations in the last fifty years. The main objective of the study is to present the loaded equations as a method for setting new correct boundary value problems. The proof for the correctness of the problem is based on the d'Alembert's formula obtained for the studied wave equation. The paper considers a loaded hyperbolic equation with two loaded terms. The load traces are related to different characteristic manifolds of a one-dimensional wave operator. Our goal is to study the problem with data on non-intersecting characteristics. We prove the existence and uniqueness of the solution, which is presented in an explicit form. The distinguishing feature of the problem is that it is ill-posed in the absence of loaded terms.

Keywords: loaded equation, one-dimensional wave operator, characteristic, ill-posed problem, d'Alembert formula

UDC: 517

MSC: 35L02

Received: 18.10.2024
Revised: 04.12.2024
Accepted: 10.12.2024

DOI: 10.35330/1991-6639-2024-26-6-13-18