Construction of functional separable solutions in implicit form for non-linear Klein–Gordon type equations with variable coefficients

Polyanin, A.D.

Автор: Polyanin A.D.
Журнал: International Journal of Non-Linear Mechanics
Том: 114
Год издания: 2019
Издательство: Pergamon Press Ltd.
Местоположение издательства: United Kingdom
Первая страница: 29
Последняя страница: 40
DOI: 10.1016/j.ijnonlinmec.2019.04.005
Аннотация: The paper deals with non-linear Klein-–Gordon type equations $$ c(x)u_{tt}=[a(x)f(u)u_x]_x+b(x)g(u). $$ The direct method for constructing functional separable solutions in implicit form to non-linear PDEs is used. This effective method is based on the representation of solutions in the form $$ \int h(u)\,du=\xi(x)\omega(t)+\eta(x), $$ where the functions $h(u)$, $\xi(x)$, $\eta(x)$, and $\omega(t)$ are determined further by analyzing the resulting functional-differential equations. Examples of specific Klein--Gordon type equations and their exact solutions are given. The main attention is paid to non-linear equations of a fairly general form, which contain several arbitrary functions dependent on the unknown $u$ and/or the spatial variable $x$ (it is important to note that exact solutions of non-linear PDEs, that contain arbitrary functions and therefore have significant generality, are of great practical interest for testing various numerical and approximate analytical methods for solving corresponding initial-boundary value problems). Many new generalized traveling-wave solutions and functional separable solutions (in closed form) are described. Solutions of several Klein--Gordon equations with delay are also given.
Добавил в систему: Полянин Андрей Дмитриевич

	ИСТИНА	Войти в систему Регистрация
	ИПМех РАН
	Главная Поиск Статистика О проекте Помощь

ИСТИНА

ИПМех РАН

Construction of functional separable solutions in implicit form for non-linear Klein–Gordon type equations with variable coefficientsстатья