\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \AtBeginDocument{{\noindent\small \emph{ Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 29, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/29\hfil Reproducing kernel methods] {Reproducing kernel methods for solving linear initial-boundary-value problems} \author[L.-H. Yang, Y. Lin\hfil EJDE-2008/29\hfilneg] {Li-Hong Yang, Yingzhen Lin} \address{Li-Hong Yang \newline College of Science, Harbin Engineering University, 150001, China} \email{lihongyang@hrbeu.edu.cn} \address{Yingzhen Lin \newline Department of Mathematics, Harbin Institute of Technology (WEIHAI), 264209, China} \email{liliy55@163.com } \thanks{Submitted January 26, 2008. Published February 28, 2008.} \thanks{Supported by grants 60572125 from the National Natural Science Foundation of China, and \hfill\break\indent HEUF707061 from the Basic Scientific Research Foundation of Harbin Engineering University.} \subjclass[2000]{35A35, 35A45, 35G05, 65N99} \keywords{Hyperbolic equation; linear initial-boundary conditions; \hfill\break\indent reproducing kernel space} \begin{abstract} In this paper, a reproducing kernel with polynomial form is used for finding analytical and approximate solutions of a second-order hyperbolic equation with linear initial-boundary conditions. The analytical solution is represented as a series in the reproducing kernel space, and the approximate solution is obtained as an n-term summation. Error estimates are proved to converge to zero in the sense of the space norm, and a numerical example is given to illustrate the method. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{property}[theorem]{Property} \section{Introduction} A reproducing kernel Hilbert space is a useful framework for constructing approximate solutions of partial differential equations (PDE). Many numerical methods have been proposed for solving linear and nonlinear PDEs, but we did not find a method that uses reproducing kernels. In this paper, we focus on the exact and approximate solutions to PDEs with linear initial-boundary conditions. A reproducing kernel with polynomial form in the corresponding Hilbert space is given and the space completion is proved. We consider the following second-order one-dimensional hyperbolic equation in a reproducing kernel space: \label{e1.1} \frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}+f(x,t),\quad 0y\,. \end{cases} By the definition of $W_2^m[a,b]$ and \eqref{e2.8}, the coefficients $c_i,d_i$, $i=1,\dots,2m$ satisfy $$\label{e2.10} \begin{gathered} \frac{\partial^i lR_m(y,y)}{\partial x^i} =\frac{\partial^i rR_m(y,y)}{\partial x^i},\quad i=0,1,\dots,2m-2\\ (-1)^m(\frac{\partial^{2m-1} rR_m(y+,y)}{\partial x^{2m-1}}-\frac{\partial^{2m-1} lR_m(y-,y)}{\partial x^{2m-1}})=1\\ \frac{\partial^{i} R_m(a,y)}{\partial x^{i}}-(-1)^{m-i-1}\frac{\partial^{2m-i-1} R_m(a,y)}{\partial x^{2m-i-1}}=0,\quad i=0,1,\dots,m-1\\ \frac{\partial^{2m-i-1} R_m(b,y)}{\partial x^{2m-i-1}}=0,\quad i=0,1,\dots,m-1\,. \end{gathered}$$ Then the solution of \eqref{e2.10} yields the expression of the reproducing kernel $R_m(x,y)$. In this paper we consider the case $m=3$ and $[a,b]=[0,1]$, the corresponding kernel space is defined as $W_2^3[0,1]$ are function $f$ such that $f^{(2)}(x)$ is absolutely continuous on $[0,1]$ and $f^{(3)}(x)\in L^2[0,1],x\in[0,1]$, and the reproducing kernel as $$R_3(x,y)=\begin{cases} \frac{1}{120}(120+x^5+120xy-5x^4y+30x^2y^2+10x^3y^2), &xy) \end{cases}$$ Other reproducing kernel spaces needed in this paper are described similarly. The space $W_{2,1}^3[0,1]$ is a subspace of $W_2^3[0,1]$ with $f(0)=f'(0)=0$, and the reproducing kernel is $$R_{31}(x,y)=\begin{cases} \frac{1}{120}x^2(x^3-5x^2y+30y^2+10xy^2), & xy. \end{cases}$$ The space $W_{2,2}^3[0,1]$ is a subspace of $W_2^3[0,1]$ with $f(0)+w_1 f'(0)=0$, $f(1)+w_2 f'(1)=0$, and the reproducing kernel is \begin{align*} &R_{31}(x,y)\\ &=\begin{cases} \frac{1}{8400}\Big(-350x^4y+x^5(46-48y+30y^2+10y^3-y^5) +24(46+92y\\ +30y^2+10y^3-y^5) +48x(46+92y+30y^2+10y^3-y^5) +30x^2(24+48y\\ +40y^2-10y^3+y^5) +10x^3(24+48y+40y^2-10y^3+y^5)\Big),\\ \text{ if } xy. \end{cases} \end{align*} \subsection*{Two-dimensional reproducing kernel space} We construct the two-dimen\-sional reproducing kernel spaces as in \cite{Yang, Aronszajn}. Let \begin{align*} P(\Omega)& = \overline{W_{2,1}^3[0,1]\otimes W_{2,2}^3[0,1]} & =\{\sum_{i,j=1}^{\infty}c_{i,j}g_i^{(1)}(x)g_j^{(2)}(t): \sum_{i,j=1}^{\infty}|c_{i,j}|^2<\infty\} \end{align*} where $\{g_i^{(k)}\}$ is a complete orthonormal sequence in the space $W_{2,k}^3$, $k=1,2$, and endowed with the inner product % \label{e2.12} \begin{align*} (u(x,t),v(x,t))_{P} &=(\sum_{k,l=1}^{\infty}c_{k,l}^{(1)}g^{(1)}_k(x)g^{(2)}_l(t), \sum_{p,q=1}^{\infty}c^{(2)}_{p,q}g^{(1)}_p(x)g^{(2)}_q(t))\\ &=\sum_{k,l=1}^{\infty}c_{k,l}^{(1)} \sum\lrcorner_{p,q=1}^{\infty}c^{(2)}_{p,q}(g^{(1)}_k(x) g^{(2)}_l(t),g^{(1)}_p(x)g^{(2)}_q(t))\\ &=\sum_{k,l=1}^{\infty}c^{(1)}_{k,l}c^{(2)}_{k,l}\\ \end{align*} and the norm $% \label{e2.12} \|u\|_{P}=\sqrt{(u,u)_{P}}=(\sum_{k,l=1}^{\infty}c_{k,l}^2)^{1/2}$ According to \cite{Russell}, the space $P(\Omega)$ is a Hilbert space with the norm $\| \cdot \|_{P}$, and possesses the reproducing kernel $%\label{e2.13} \bar{R}((\xi,\eta),(x,t))=R_{31}(\xi,x)\cdot R_{32}(\eta,t).$ It is easy to prove that the following properties hold. \begin{property} For $u(x)\in W_{3,1}[0,1],\,v(y)\in W_{32}[0,1]$, it follows that $u(x)\cdot v(y)\in P(\Omega)$ . \end{property} \begin{property} $(u_1(x)\cdot v_1(y),\,u_2(x)\cdot v_2(y))_P=\langle u_1(x),\,u_2(x)\rangle _{W_{31}}\cdot \langle v_1(y),\,v_2(y)\rangle_{W_{32}}$ holds for any $u_1,\,u_2\in W_{3,1}[0,1],\,v_1,v_2\in W_{32}[0,1]$. \end{property} Similarly, the other two-dimensional reproducing kernel space can be defined as \begin{align*} P_1(\Omega)& = \overline{W_{2}^1[0,1]\otimes W_{2}^1[0,1]}\\ & =\{\sum_{i,j=1}^{\infty}c_{i,j}g_i(x)g_j(t): \sum_{i,j=1}^{\infty}|c_{i,j}|^2<\infty\}, \end{align*} where $\{g_i\}$ is a complete orthonormal sequence in the space $W_{2}^1$. Its reproducing kernel function $\tilde{R}((\xi,\eta),(x,t))$ can be obtain from the reproducing kernel function $R_1(x,y)$ of the space $W_2^1[0,1]$, that is, $\tilde{R}((\xi,\eta),(x,t))=R_{1}(\xi,x)\cdot R_{1}(\eta,t)$. \section{Solution of Equation \eqref{e1.1}} In this section, we consider the second-order one-dimensional hyperbolic equation \eqref{e1.1} with initial-value conditions \eqref{e1.4}--\eqref{e1.5} and mixed boundary-value conditions \eqref{e1.2}--\eqref{e1.3}. Without the loss of generality, we discuss equation \eqref{e1.1} with homogeneous conditions, that is, \label{e3.1} \begin{gathered} \frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}+f(x,t),\quad 0