\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 48, pp. 1--6.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/48\hfil Initial-boundary value problems] {Initial-boundary value problems for the \\ wave equation} \author[T. Sh. Kalmenov, D. Suragan \hfil EJDE-2014/48\hfilneg] {Tynysbek Sh. Kalmenov, Durvudkhan Suragan} % in alphabetical order \address{Tynysbek Sh. Kalmenov \newline Institute of Mathematics and Mathematical Modeling, 28 Shevchenko str., 050010 Almaty, Kazakhstan} \email{kalmenov.t@mail.ru} \address{Durvudkhan Suragan \newline Institute of Mathematics and Mathematical Modeling, 28 Shevchenko str., 050010 Almaty, Kazakhstan} \email{suragan@list.ru} \thanks{Submitted January 9, 2014. Published February 19, 2014.} \subjclass[2000]{35M10} \keywords{Hyperbolic equation; wave potential; \hfill\break\indent initial boundary value problems} \begin{abstract} In this work we consider an initial-boundary value problem for the one-dimensional wave equation. We prove the uniqueness of the solution and show that the solution coincides with the wave potential. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \allowdisplaybreaks \section{Introduction} In $\Omega=(0,1)$ consider the one-dimensional potential $$u(x)=\int_0^1-\frac{1}{2}|x-y|f(y)dy, \label{eq1}$$ where $f$ is an integrable function in $(0,1)$. The kernel of the potential is a fundamental solution of the second order differential equation $$-\varepsilon''(x-y)=\delta(x-y),\label{eq2}$$ where $\varepsilon(x-y)=-\frac{1}{2}|x-y|$ and $\delta$ is the Dirac delta function. Hence the potential \eqref{eq1} satisfies the equation $$-u''(x)=f(x), x\in\Omega.\label{eq3}$$ On the other hand, integrating by part, we obtain \begin{align*} u(x)&=\int_0^1-\frac{1}{2}|x-y|f(y)dy=\int_0^1\frac{1}{2}|x-y|u''(y)dy\\ &=\int_0^{x}\frac{1}{2}(x-y)u''(y)dy+\int_{x}^1\frac{1}{2}(y-x)u''(y)dy\\ &=u(x)-x\frac{u'(0)+u'(1)}{2}-\frac{-u'(1)+u(0)+u(1)}{2}, \quad \forall x\in (0,1); \end{align*} i.e., $$x(u'(0)+u'(1))+(-u'(1)+u(0)+u(1))=0, \quad \forall x\in (0,1).$$ Therefore, the self-adjoint boundary conditions for the potential \eqref{eq1} are $$u'(0)+u'(1)=0,\quad -u'(1)+u(0)+u(1)=0.\label{eq4}$$ Hence if we solve the equation \eqref{eq3} with the boundary conditions \eqref{eq4}, then we find an unique solution of this boundary value problem in the form \eqref{eq1}. The simple method finds equivalent boundary value problems of ODE for one dimensional potential integrals. However this task becomes tedious for PDE, and we obtained boundary conditions of the volume potentials for elliptic equations and showed some their applications in works \cite{k1,k2,k3}. In particular in \cite{k1}, by using a new non-local boundary value problem, which is equivalent to the Newton potential, we found explicitly all eigenvalues and eigenfunctions of the Newton potential in the 2-disk and the 3-ball. The aim of this paper is to give an analogy of the boundary value problem \eqref{eq3}-\eqref{eq4} for the wave potential. Unlike elliptic and parabolic cases, where we obtained non-local boundary conditions for the corresponding volume potentials, and some other nonclassic non-local boundary initial boundary value problems of hyperbolic equations (see, for example, \cite{p1,p2}) we get a local initial boundary value problem for the wave potential. \section{Main result and their proof} In the bounded domain $\Omega\equiv\{(x,t):(0,l)\times(0,T)\}$ we consider the wave potential $$u(x,t)=\int_{\Omega}\varepsilon(x-\xi,t-\tau)f(\xi,\tau)d\xi d\tau,\label{e1}$$ where $\varepsilon(x-\xi,t-\tau)=\frac{1}{2}\theta(t-\tau-|x-\xi|)$ is a fundamental solution of Cauchy problem for the wave equation; i.e., \begin{gather*} \frac{\partial^{2}\varepsilon(x-\xi,t-\tau)}{\partial t^{2}}-\frac{\partial^{2}\varepsilon(x-\xi,t-\tau)}{\partial x^{2}}=\delta(x-\xi,t-\tau), \\ \frac{\partial^{2}\varepsilon(x-\xi,t-\tau)}{\partial \tau^{2}}-\frac{\partial^{2}\varepsilon(x-\xi,t-\tau)}{\partial \xi^{2}}=\delta(x-\xi,t-\tau), \\ \varepsilon(x-\xi,t-\tau)|_{\tau=t}=\frac{\partial\varepsilon(x-\xi,t-\tau)}{\partial t}|_{\tau=t}= \frac{\partial\varepsilon(x-\xi,t-\tau)}{\partial\tau}|_{\tau=t}=0 \end{gather*} if $f(x,t)\in L_{2}(\Omega)$ then $u(x,t)\in W_{2}^1(\Omega)\cap W_{2}^1(\partial\Omega)$ and the wave potential \eqref{e1} satisfies to the equation \cite{v1} $$\frac{\partial^{2}u(x,t)}{\partial t^{2}}-\frac{\partial^{2}u(x,t)}{\partial x^{2}}=f(x,t), \quad (x,t)\in\Omega, \label{e2}$$ with the initial conditions u(x,0)=u_{t}(x,0)=0,\quad 0