\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2001(2001), No. 76, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \thanks{\copyright 2001 Southwest Texas State University.} \vspace{1cm}} \begin{document} \title[\hfilneg EJDE--2001/76\hfil Wave equation with a nonlocal condition] {Existence of solutions for one-dimensional wave equations with nonlocal conditions} \author[Sergei A. Beilin\hfil EJDE--2001/76\hfilneg] { Sergei A. Beilin } \address{Sergei A. Beilin \hfill\break Department of Mathematics\\ Samara State University\\ 1, Ac.Pavlov st.\\ 443011 Samara Russia} \email{sbeilin@narod.ru, awr@ssu.samara.ru} \date{} \thanks{Submitted August 21, 2001. Published December 10, 2001.} \subjclass[2000]{35L99, 35L05, 35L20} \keywords{Mixed problem, non-local conditions, wave equation} \begin{abstract} In this article we study an initial and boundary-value problem with a nonlocal integral condition for a one-dimensional wave equation. We prove existence and uniqueness of classical solution and find its Fourier representation. The basis used consists of a system of eigenfunctions and adjoint functions. \end{abstract} \maketitle \newtheorem{theorem}{Theorem}[section] % theorems numbered with section # \newtheorem{lemma}[theorem]{Lemma} \numberwithin{equation}{section} \renewcommand{\phi}{\varphi} \section{Introduction} Certain problems of modern physics and technology can be effectively described in terms of nonlocal problems for partial differential equations. These nonlocal conditions arise mainly when the data on the boundary cannot be measured directly. The first paper, devoted to second-order partial differential equations with non-local integral conditions goes back to Cannon \cite{Cannon}. Later, the problems with non-local integral conditions for parabolic equations were investigated by Kamynin \cite{Kamynin}, Ionkin \cite{Ionkin}, Yurchuk \cite{Yurchuk}, Bouziani \cite{Bouziani1}; problems for elliptic equations with operator nonlocal conditions were considered by Mikhailov and Guschin \cite{MikhailovGuschin}, Scubachevski \cite{Scubachevski}, Paneiah \cite{Paneiah}. Then, Gordeziani and Avalishvili \cite{GordezianiAvalishvili}, Bouziani \cite{Bouziani2} devoted a few papers to nonlocal problems for hyperbolic equations. Pulkina \cite{PulkinaEJDE,PulkinaDiffUr} studied the nonlocal analogue to classical Goursat problem. In this paper we investigate the nonlocal analogue to classical mixed problem, which involves initial, boundary and nonlocal integral conditions. In the rectangular domain $D=\{(x,t): 00$ the functions (\ref{eq13}) are not orthonormal with $X_0$. To construct a basis in $L_2$, we complete (\ref{eq13}) by using adjoint functions. Following M. Keldysh \cite{Keldysh}, we define an adjoint function $\tilde X_k$, corresponding eigenvalue $\lambda_k$ from (\ref{eq12}), as a solution to the boundary-valued problem $$\label{eq14} \tilde X''_k(x)+\lambda_k\tilde X_k(x)=-2\sqrt{\lambda_k}X_k(x), \quad \tilde X_k(0)=0, \quad \tilde X'_k(0)=\tilde X'_k(l).$$ We obtain $$\tilde X_k(x)=x\cos{\frac{2\pi kx}{l}}, \quad k=1,2,\ldots$$ Rewrite now a system of eigenvalue and adjoint functions of (\ref{eq10}) as $$\label{eq15} X_0=x, \quad X_{2k-1}(x)= x\cos{\frac{2\pi kx}{l}}, \quad X_{2k}(x)=\sin{\frac{2\pi kx}{l}}.$$ In a similar way we find the system of eigenvalue and adjoint functions (\ref{eq11}): $$\label{eq16} Y_0(x)=\frac{2}{l^2}, \quad Y_{2k-1}(x)=\frac{4}{l^2}\cos{\frac{2\pi kx}{l}}, \quad Y_{2k}(x)=\frac{4(l-x)}{l^2}\sin{\frac{2\pi kx}{l}}\,,$$ where for every $\lambda_k$ with $k>0$, $X_{2k}(x)$, $Y_{2k}(x)$ are eigenvalue functions, $X_{2k-1}(x)$, $Y_{2k-1}(x)$ are adjoint functions of the problems (\ref{eq10}) and (\ref{eq11}) respectively. Direct calculations show that (\ref{eq15}) and (\ref{eq16}) form a biorthogonal system for $x\in (0,l)$: $$(X_i,Y_j)=\int_0^lX_i(x)Y_j(x)\,dx=\delta_{ij}.$$ As it was shown in \cite{Ilyin} the system (\ref{eq15}) is complete and forms a basis in $L_2(0,l)$. Hence, an arbitrary function $f(x)\in L_2(0,l)$ may be expanded as $$f(x)=A_0X_0(x)+\sum_{k=1}^{\infty}(A_{2k}X_{2k}(x)+A_{2k-1}X_{2k-1}(x)),$$ where $$\label{eq17} A_i=\int_0^lf(x)Y_i(x)\,dx.$$ Returning to the separation variables technique, for $T(t)$ we obtain $$T_k(t)=a_k\sin{\frac{2\pi kt}{l}}+b_k\cos{\frac{2\pi kt}{l}}.$$ We assume now that a solution to $\mathcal{H}$ is of the form $$\label{solutionH} u(x,t)=A_0X_0+\sum_{k=1}^{\infty}\left( (A_{2k}X_{2k}+A_{2k-1}X_{2k-1})T_k- \frac{lt}{2\pi k}A_{2k-1}X_{2k}T'_k\right).$$ Substitute $T_k(t)$ and rewrite the coefficients. Then \begin{aligned} u(x,t)=&C_0X_0+\sum_{k=1}^{\infty}( X_{2k}(C_{2k}\sin{\frac{2\pi kt}{l}}+ D_{2k}\cos{\frac{2\pi kt}{l}}) \\ &+X_{2k-1}(C_{2k-1}\sin{\frac{2\pi kt}{l}}+D_{2k-1}\cos{\frac{2\pi kt}{l}}) \\ &-tX_{2k}(C_{2k-1}\cos{\frac{2\pi kt}{l}}-D_{2k-1}\sin{\frac{2\pi kt}{l}})). \end{aligned} The initial data (\ref{eq7}) give us the following two equalities \begin{gather*} \varphi(x)=C_0X_0+\sum_{k=1}^{\infty}(D_{2k}X_{2k}+D_{2k-1}X_{2k-1}),\\ \psi(x)=\sum_{k=1}^{\infty}\left( (\frac{2\pi k}{l}C_{2k}-C_{2k-1})X_{2k} +\frac{2\pi k}{l}C_{2k-1}X_{2k-1}\right), \end{gather*} and the coefficients can be found via formula (\ref{eq17}). Assume a solution to the problem $\mathcal{N}\mathcal{H}$ is of the form $$\label{solutionNH} u(x,t)=V_0(t)X_0(x)+\sum_{k=1}^{\infty} \left( V_{2k}(t)X_{2k}(x)+V_{2k-1}(t)X_{2k-1}(x) \right),$$ where $V_i(t)$ are unknown coefficients satisfying the initial conditions $V_i(0)=V'_i(0)=0$. Substitute (\ref{solutionNH}) into the equation $u_{tt}-u_{xx}=g(x,t)$, where $g(x,t)$ has been expanded as a biorthogonal series: $$g(x,t)=g_0(t)X_0(x)+\sum_{k=1}^{\infty}(g_{2k}(t)X_{2k}(x) +g_{2k-1}(t)X_{2k-1}(x)),$$ with coefficients $$g_i(t)=\int_0^lg(x,t)Y_i(x)\,dx, \quad i=0,1,\dots$$ We obtain \begin{eqnarray*} \lefteqn{ V''_0(t)x+\sum_{k=1}^{\infty}\left(V''_{2k}(t) +\frac{4\pi^2k^2}{l^2}V_{2k}(t)\right)\sin{\frac{2\pi kx}{l}} }\\ \lefteqn{+\sum_{k=1}^{\infty}\left(V''_{2k-1}(t)+\frac{4\pi^2k^2}{l^2}V_{2k-1}(t)\right) x\cos{\frac{2\pi kx}{l}} }\\ \lefteqn{ +\sum_{k=1}^{\infty}V_{2k-1}(t)\frac{4\pi k}{l}\sin{ \frac{2\pi kx}{l}} }\\ &=& g_0(t)X_0(x)+\sum_{k=1}^{\infty}(g_{2k}(t)X_{2k}(x)+g_{2k-1}(t) X_{2k-1}(x)). \end{eqnarray*} Thus we have a Cauchy problem for the system of ordinary differential equations \begin{gather*} V''_0(t)=g_0(t) \\ V''_{2k}+\frac{4\pi k}{l}(\frac{\pi k}{l}V_{2k}(t)+V_{2k-1}(t))=g_{2k}(t) \\ V''_{2k-1}(t)+\frac{4\pi^2k^2}{l^2}V_{2k-1}(t)=g_{2k-1}(t) \end{gather*} with initial data $$V_0(0)=V'_0(0)=0, \quad V_{2k}(0)=V'_{2k}(0)=0, \quad V_{2k-1}(0)=V'_{2k-1}(0)=0,$$ which has a unique solution \begin{gather*} V_0(t) = \int_0^t (t-\tau) g_0(\tau) \,d\tau, \\ V_{2k-1}(t) = {1\over k\pi} \int_0^t g_{2k-1}(\tau) \sin {k\pi (t-\tau) \over l} \,d\tau, \\ V_{2k}(t) = {1\over k\pi} \int_0^t \left( g_{2k}(\tau) - 4\pi k V_{2k-1}(\tau) \right) \sin {k\pi (t-\tau) \over l}\,d\tau. \end{gather*} \begin{theorem} \label{thm2} Let: \begin{enumerate} \item $g(x,t) \in C^2(D)$, $g_x(x,t) \in C[0,l]$ for all $t\in (0,T)$, $|g(x,t)| \leq P, (x,t) \in D$ \item $\varphi \in C[0,l] \cap C^2(0,l)$, $\psi \in C[0,l]$, $\phi(0)=0$, $\phi'(0)=\phi'(l)$, $\psi(0) = 0$. \end{enumerate} Then there exists the solution to (\ref{eq6})--(\ref{eq9}), $$u(x,t) \in C(\bar D) \cap C^1(\bar D \setminus \{t=T\}) \cap C^2(D)$$ which has the form of a sum of (\ref{solutionH}) and (\ref{solutionNH}). \end{theorem} \noindent{\bf Series Proof.} It is sufficient to prove uniform convergence of the series (\ref{solutionH}) and (\ref{solutionNH}) and the series, obtained with formal differentiation. Let $| \phi'(x)|\leq M_1$, $|\phi''(x)|\leq M_2$, $|\psi(x)|\leq N$, $|\psi'(x)|\leq N_1$, $|g_x|\leq P_1$, $|g_{xx}| \leq P_2$. Integrating $C_i, \ D_i, \ V_i$ by parts and taking in account the abovementioned assumptions, we obtain: \begin{gather*} |D_{2k}|\leq {1\over k^2} {l(lM_2+2M_1)\over \pi^2}, \quad |D_{2k-1}|\leq {1\over k^2} {M_2 l \over \pi^2}, \\ |C_{2k}|\leq {1\over k^2} {l(N_1+2N) \over 2\pi^2}, \quad |C_{2k-1}|\leq {1\over k^2} {N_1 l \over \pi^2}, \\ |V_{2k}|\leq {1\over k^2} {4T^2(2p_1+P_2l) \over \pi^2}, \quad |V_{2k-1}|\leq {1\over k^2} {2TP_1 \over \pi^2}, \end{gather*} and hence the series (\ref{solutionH}) and (\ref{solutionNH}) and the series, obtained with formal differentiation, converge uniformly. \hfill$\Box$ \begin{thebibliography}{00} \frenchspacing \bibitem{Bizadze} A. V. Bitzadze, {\itshape Urawnenija matematicheskoj fiziki,} M.,Nauka'', 1976. \bibitem{Bouziani1} A. 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