3.3.1 Neumann$B6-3&>r7o$G$N2rK!(B

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3.3.1 Neumann$B6-3&>r7o$G$N2rK!(B

$Br7o$G9M;!$9$k!#(B $0\le x\le 1$ , $t\ge 0$ $B$GDj5A$5$l$?(B

$B$KBP$9$kGHF0J}Dx<0(B

$\displaystyle \frac{\rd^2 u}{\rd t^2}=\frac{\rd^2 u}{\rd x^2}$ $\displaystyle \mbox{($0<x<1$, $t>0$)}$

$B$r=i4|>r7o(B

$\begin{displaymath} \left. \begin{array}{l} u(x,0)=\varphi(x)\\ \dsp\frac{\rd u}{\rd t}=\psi(x) \end{array}\right\}\qquad (0\le x\le 1) \end{displaymath}$

$B$G(B Neumann $B6-3&>r7o(B

$\displaystyle \frac{\rd }{\rd x}u(0,t)=0, \quad \frac{\rd }{\rd x}u(1,t)=0\qquad (t>0)$

$B$N2<$G2r$/!#$3$NLdBj$r(B(

)$B$H$9$k!#(B

Dirichlet $B6-3&>r7o$N;~$HF1MM$K!"(B, $B$rMQ$$$F)$B$rF3$/!#(B

(3.4)

$\displaystyle \left\{\begin{array}{lll} \dsp\frac{\rd }{\rd t}\left(\begin{arra... ...{\rd x}v(0,t)&=&\dsp\frac{\rd }{\rd x}v(1,t)=0\qquad (t>0). \end{array}\right .$

(

)$B$N<0$r!"%U%j!<%I%j%/%9!J(BFriedrichs$B!K$N:9J,K!$r:NMQ$7$FN%;62=$7!"(B Dirichlet $B6-3&>r7o$N;~$HF1MM$K!"(B $\tau$ , $\lambda$ $B$rMQ$$!"@.J,$G=q$/$H!"(B

(3.5)	$\displaystyle v_{i,j+1}$	$\displaystyle =$	$\displaystyle \frac{1}{2}\{(v_{i+1,j}+v_{i-1,j})+\lambda(w_{i+1,j}-w_{i-1,j})\}$
(3.6)	$\displaystyle w_{i,j+1}$	$\displaystyle =$	$\displaystyle \frac{1}{2}\{\lambda(v_{i+1,j}-v_{i-1,j})+(w_{i+1,j}+w_{i-1,j})\} \qquad(0<i<N,\ \ j=0,1,2,\ldots).$

$B$H$J$k!#=i4|>r7o!"6-3&>r7o$K$D$$$F$O

$\displaystyle v_{i,0}$	$\displaystyle =$	$\displaystyle \psi(ih)\qquad( 0\le i\le N)$
$\displaystyle w_{i,0}$	$\displaystyle =$	$\displaystyle \varphi'(ih)\qquad( 0\le i\le N)$
$\displaystyle w_{0,j}$	$\displaystyle =$	$\displaystyle w_{N,j}=0\qquad(j=1,2,\ldots).$

$B$3$3$G!"(B $v_{0,j+1}$ ( $j=0,1,2,\ldots$ ) $B$K$D$$$F9M;!$9$k!#(B

$\displaystyle \dsp\frac{\rd }{\rd x}v(0,t)=\dsp\frac{\rd^2 }{\rd x \rd t}u(0,t)= \dsp\frac{\rd^2 }{\rd t \rd x}u(0,t)=\dsp\frac{\rd }{\rd t}w(0,t)=0$

$B$J$N$G!"2>A[3J;RE@(B $x_{-1}$ $B$rF3F~$7(B $\dsp\frac{\rd }{\rd x}v(0,t_{j})$ $B$rCf?4:9J,>&$G6a;w$9$k!#(B

$$\displaystyle \dsp\frac{\rd }{\rd x}v(0,t_{j})$B$

$B$h$j(B,

$\displaystyle v_{-1,j}=v_{1,j}\qquad(j=0,1,2,\ldots)$

$B$=$3$G!"(B(

), (

)$B$N<0$K(B

$B$GMQ$$$k$H(B

	$\displaystyle v_{0,j+1}=\frac{1}{2}\{(v_{1,j}+v_{-1,j})+\lambda(w_{1,j}-w_{-1,j})\}$
	$\displaystyle w_{0,j+1}=\frac{1}{2}\{\lambda(v_{1,j}-v_{-1,j})+(w_{1,j}+w_{-1,j})\}$
	$\displaystyle \qquad(j=0,1,2,\ldots).$

$B>e$N<0$K(B $v_{-1,j}=v_{1,j}$ $B!"(B $w_{0,j+1}=0$ $B$rBeF~$9$k$H!"(B

$\displaystyle 0=\frac{1}{2}\{\lambda$ $B!&(B $\displaystyle 0+(w_{-1,j}+w_{1,j})\}$

$B$h$j!"(B

$\displaystyle w_{-1,j}=-w_{1,j}\qquad(j=0,1,2,\ldots).$

$B$f$($K!"(B

$\displaystyle v_{0,j+1}=\frac{1}{2}\{(v_{1,j}+v_{-1,j})+\lambda(w_{1,j}-w_{-1,j})\}= v_{1,j}+\lambda w_{1,j}\qquad(j=0,1,2,\ldots).$

$BF1MM$K(B $v_{N,j+1}$ ( $j=0,1,2,\ldots$ )$B$K$D$$$F$b9T$J$&!#(B

$\displaystyle \dsp\frac{\rd }{\rd x}v(1,t)=\dsp\frac{\rd^2 }{\rd x \rd t}u(1,t)= \dsp\frac{\rd^2 }{\rd t \rd x}u(1,t)=\dsp\frac{\rd }{\rd t}w(1,t)=0$

$B$J$N$G!"2>A[3J;RE@(B $x_{N+1}$ $B$rF3F~$7(B $\dsp\frac{\rd }{\rd x}v(1,t_{j})$ $B$rCf?4:9J,>&$G6a;w$9$k!#(B

$$\displaystyle \dsp\frac{\rd }{\rd x}v(1,t_{j})$B$

$B$h$j(B,

$\displaystyle v_{N-1,j}=v_{N+1,j}\qquad(j=0,1,2,\ldots).$

$B$=$3$G!"(B(

), (

)$B$N<0$K(B

$B$GMQ$$$k$H(B

	$\displaystyle v_{N,j+1}=\frac{1}{2}\{(v_{N+1,j}+v_{N-1,j})+\lambda(w_{N+1,j}-w_{N-1,j})\}$
	$\displaystyle w_{N,j+1}=\frac{1}{2}\{\lambda(v_{N+1,j}-v_{N-1,j})+(w_{N+1,j}+w_{N-1,j})\}$
	$\displaystyle \qquad(j=0,1,2,\ldots).$

$B>e$N<0$K(B $v_{N-1,j}=v_{N+1,j}$ $B!"(B $w_{N,j+1}=0$ $B$rBeF~$9$k$H!"(B

$\displaystyle 0=\frac{1}{2}\{\lambda$ $B!&(B $\displaystyle 0+(v_{N+1,j}+v_{N-1,j})\}$

$B$h$j!"(B

$\displaystyle w_{N+1,j}=-w_{N-1,j}\qquad(j=0,1,2,\ldots).$

$B$f$($K!"(B

$\displaystyle v_{N,j+1}=\frac{1}{2}\{(v_{N+1,j}+v_{N-1,j})+\lambda(w_{N+1,j}-w_{N-1,j})\}= v_{N-1,j}-\lambda w_{N-1,j}\qquad(j=0,1,2,\ldots)$

$B$HI=$;$k!#(B

()$B$NLdBj$G5a$a$?$N$O!"(B $v_{i,j}$ , $w_{i,j}$ ( $0\le i\le N$ , $j=0,1,2,\ldots$ ) $B$G$"$k!#(B $B$=$3$GLdBj(B()$B$G5a$a$?$$2r$r(B $u_{i,j}$ ( $0\le i\le N$ , $j=0,1,2,\ldots$ ) $B$H$9$k$H!"(B $u_{i,j}$ $B$O!"

(3.7)	$\displaystyle u_{i,j+1}$	$\displaystyle =$	$\displaystyle \frac{1}{2}(u_{i+1,j}+u_{i-1,j})+\tau v_{i,j} \qquad($ $\displaystyle \mbox{$0< i<N$, $j=0,1,2,\ldots$}$ $\displaystyle )$
(3.8)	$\displaystyle u_{i,0}$	$\displaystyle =$	$\displaystyle \varphi(ih)\qquad\qquad\qquad\qquad\qquad (0\le i\le N)$