On a Generalization of the Initial-Boundary Problem for the Vibrating String Equation

Abstract

In the present paper, we study a generalization of the initial-boundary problem for the inhomogeneous vibrating string equation. The initial conditions include the higher order derivatives of the unknown function. The problem is studied under homogeneous boundary conditions of the first kind. The uniqueness and existence of a regular solution of the problem are proved. To prove the main result we use the spectral decomposition method.

Об Одной Краевой Задаче Для Уравнения Четвертого Порядка В Частных Производных

“…Substituting the Fourier series (10) and (11) into the given inhomogeneous differential equation (1), we obtain a first-order ordinary differential equation…”

“…Sokolovsky [8]. A mixed problem with a high derivative in the initial condition for the heat equation was studied by D. Amanov [9]; for the equation of string vibration, the mixed problem with high derivatives in the initial conditions was studied in [10]- [16]. Authors of [17], [18]- [19] studied boundary value problems for many types of high-order partial differential equations.…”

On a nonlocal boundary value problem for the equation fourth-order in partial derivatives

“…Для уравнения теплопроводности смешанная задача с высокой производной в начальном условии изучена в [10], а для уравнения колебания струны смешанная задача с высокими производными в начальных условиях изучена в [11]. Смешанные задачи для уравнений четвертого порядка изучены в [12]- [14].…”