\documentstyle[amssymb,12pt]{article} \def\baselinestretch{1.2} %\textwidth=18cm %\oddsidemargin=-1cm \begin{document} \begin{center} {\bf ASSIGNED EXERCISES OF MATH544: PDE set 4} {\it March, 2001} \end{center} \vspace{0.4cm} \noindent {\bf INITIAL AND BOUNDARY VALUE PROBLEMS OF \\ SOME HYPERBOLIC TYPE OF EQUATIONS}\\ Ronald B Guenther and John W Lee, Partial Differential Equations of Mathematical Physics and Integral Equations \vspace{0.3cm} \noindent {\bf 1}.\, Homogeneous wave equations with initial conditions: Let $f(x)$ and $g(x)$ be some arbitrary functions and $$u_{tt}-c^2\,u_{xx}=0,$$ $$u(x,0)=f(x),~~u_{t}(x,0)=g(x)$$ \vspace{0.3cm} \noindent {\bf Theorem 1}. {\it Let $f(x)$ have a second derivative and $g(x)$ have a first derivative for $-\infty < x < \infty $. Then the initila value problem given above is well-posed. Its solution is given by} $$u(x,t)={1 \over 2}\,[f(x+ct)+f(x-ct)]+{1 \over 2c}\, \int^{x+ct}_{x-ct}\, g(\xi)\,d \xi$$ \vspace{0.3cm} \noindent {\bf 2}.\, Homogeneous wave equation with initial and boundary value problem. Let $$u_{tt}-c^2\,u_{xx}=0,$$ $$u_{x,0}=f(x),~~u_{t}(x,0)=g(x), ~~~ 0 \le x \le L$$ $$u(0,t)=u(L,t)=0,~~~t \ge 0$$ \vspace{0.3cm} \noindent {\bf Theorem 2}. {\it Let $f(x)=$ have a continuous fourth derivative and $g(x)$ have a continuous third derivative for $ 0 \le x \le L$, and $f(0)=f(L)=0$ , $f^{\prime \prime}(0)=f^{\prime \prime}(L)=0$, and $g(0)=g(L)=0$. Then the initial boundary value problem above has a solution given by $$u(x,t)=\sum_{n=1}^{\infty}\, [A_{n}\, \cos(\lambda_{n} ct)+B_{n}\, \sin (\lambda_{n} ct)]\, \sin (\lambda_{n} x),$$ $$A_{n}={2 \over L}\, \int_{0}^{L}\, f(x) \sin ({n\pi x \over L}) dx,$$ $$B_{n}={2 \over \lambda_{n} c L}\, \int_{0}^{L}\, g(x) \sin({n\pi x \over L}) dx.$$ where $\lambda_{n}={n\pi \over L}$.} \vspace{0.3cm} \noindent {\bf Comment }. Uniform convergence requires $$|\lambda_{n}^2 A_{n}| \le {C \over n^2},~~~ |\lambda_{n}^2 B_{n}| \le {C \over n^2}$$ where $C=max \{ C_{1}, C_{2} \}$, here $$C_{1}={2l \over \pi^2}\, \int_{0}^{L}\,|f^{4}(x)| dx,$$ $$C_{2}={2L \over c\pi^2}\,[|g^{\prime \prime}(0)|+|g^{\prime \prime}(L)|+ \int_{0}^{L}\, |g^{\prime \prime \prime}(0)| dx]$$ \vspace{0.3cm} \noindent {\bf 3}.\, {\bf Theorem 3}.\, {\it Let the initila data $f$ and $g$ in the previous problem satisfy the hypothesis of Theorem 2 (above). Then this problem is well posed.} \vspace{0.3cm} \noindent {\bf Hint}. Let $(f_{1},g_{1})$ and $(f_{2},g_{2})$ are the data for the solutions $u_{1}$ and $u_{2}$ respectively. Then prove that $$|u_{1}(x,t)-u_{2}(x,t)| \le D \, max \{ ||f_{1}-f_{2}||_{2}, ||g_{1}-g_{2}||_{1} \}$$ where $D$ is a constant and $$||f_{1}-f_{2}||_{2}=max\{||f_{1}-f_{2}||, ||f_{1}^{\prime}-f_{2}^{\prime}||, ||f_{1}^{\prime \prime}-f_{2}^{\prime \prime}|| \}$$ $$||g_{1}-g_{2}||_{1}=max \{||g_{1}-g_{2}||, ||g_{1}^{\prime}-g_{2}^{\prime}|| \}$$ Here the norm is defined as $||h(x)||=max |h(x)|$, for all $ x \in [0,L]$. The way one proves this theorem: Let $u=u_{1}-u_{2}$, $f=f_{1}-f_{2}, g=g_{1}-g_{2}$. Use the compatibility conditions and integration by parts to find $$|\lambda_{n} B_{n}| \le {C_{1} \over n^2},~~~ |A_{n}| \le {C_{2} \over n^2}$$ where $A_{n}$ and $B_{n}$ are Fourier coefficients based on $f=f_{1}-f_{2}$ and $g=g_{1}-g_{2}$ and $$C_{1}={2L \over c\pi^2}\, \int_{0}^{L}\,|g^{\prime}(x)| dx,~~~C_{2}= {2L \over \pi^2}\, \int_{0}^{L}\, |f^{\prime \prime}|\,dx$$ \noindent Using the solution given in the problem 2 above one can find the estimate given above at the beginning of this problem. \vspace{0.3cm} \noindent {\bf 4}.\, Inhomogeneous Initial and Boundary value problem: (k>0) $$u_{tt}+ku_{t}-c^2 u_{xx}=F(x,t), ~~~ 0 \le x \le L, ~~t>0$$ $$u(x,0)=f(x),~~~u_{t}(x,0)=g(x), ~~~ 0 \le x \le L$$ $$u(0,t)=r(t),~~u(L,t)=s(t), ~~t \ge 0$$ \vspace{0.3cm} \noindent {\bf Theorem 4}. There exists at most one solution to the above inhomogeneous initial boundary value problem. \vspace{0.3cm} \noindent Let the data $f=0,g=0,r=0,s=0$. Then we have \vspace{0.3cm} \noindent {\bf Theorem 5}. Let $F(x,t)$ have continous third order partial deriveatives for $0 \le x \le L$ and $t \ge 0$. Assume that $F(0,t)=F(L,t)=F_{xx}(0,t)= F_{xx}(L,t)=0$ for all $t \ge 0$. Then the above inhomegenous initial and boundary value problem has a unique solution given by $$u(x,t)=\sum_{0}^{\infty}\, u_{n}(t)\, sin(\lambda_{n} x)$$ $$u_{n}(t)={1 \over c \lambda_{n}}\, \int_{0}^{t}\, F_{n}(\tau)\, G(t,\tau) d\tau$$ where$G(t,\tau)$ is the Green's function corresponding to the equation $u_{n}(t)$ $$u_{n}^{\prime \prime}+k u_{n}^{\prime}-c^2 \lambda_{n}^2\,u_{n}=F_{n}(t)$$ We assumed $F(x,t)=\sum_{0}^{\infty}\,F_{n}(t) \sin(\lambda_{n} x)$ with $$F_{n}(t)={2 \over L}\, \int_{0}^{L}\, F(x,t) \sin (\lambda_{n} x) dx.$$ \end{document}