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8.3 $BFC@-J}8~$N:9J,(B

$BA0@a$G=R$Y$?J}K!$O!"JPHyJ,$r$=$l$>$l=D2#$NJ}8~$K:9J,2=$7$F9M$($k(B $BJ}K!$G$"$k$,!"J}Dx<0(B (35) $B$N:8JU$r!"FC@-(B $B6J@~$NJ}8~$K2 $B@a$G8+$?$h$&$K!"(B

$$\frac{d}{dt}u(t,at+x_0) = \left. (u_t+ a u_x)\right\vert _{x=at+x_0}$$

$B$G$"$C$?$,!"$3$N<0$N:9J,2=$r9T$($P(B

$$\frac{u(t+h,a(t+h)+x_0)-u(t,at+x_0)}{h}\approx \left. (u_t+ a u_x)\right\vert _{x=at+x_0},$$

$B$9$J$o$A!"(B$at+x_0$ $B$r(B $x$ $B$H$9$l$P(B

$$\frac{u(t+h,x+ah)-u(t,x)}{h}\approx u_t(t,x)+ a u_x(t,x)$$

$B$,@.$jN)$D$3$H$,$o$+$k!#$3$l$rMQ$$$k$HJ}Dx<0(B (35) $B$O(B

$$\frac{u(t+h,x+ah)-u(t,x)}{h}\approx f(t,x)$$

$B$9$J$o$A(B

$$u(t+h,x+ah)\approx u(t,x)+hf(t,x)$$

$B$H$J$k$N$G!"(B$\Delta t=h$, $\Delta x=ah$ $B$H$7!"(B $u^n_j=u(n\Delta t,j\Delta x)$ $B$H$9$l$P(B

$$\begin{eqnarray*} u^{n+1}_{j+1} & = & u((n+1)\Delta t,(j+1)\Delta x) \\ & = &... ... = & u^n_j + \Delta t f^n_j \ \ (f^n_j = f(n\Delta t,j\Delta x)) \end{eqnarray*}$$

$B$H7W;;$G$-$k$3$H$K$J$k!#(B

$B$3$l$r$5$i$KH/E8$5$;$F!"GHF0J}Dx<0$N=i4|CMLdBj(B

$$\left\{\begin{array}{ll} u_{tt}-a^2 u_{xx} = f(t,x) & (t>0... ...y) \\ u_t(0,x)= h(x) & (-\infty<x<\infty) \end{array}\right.$$ (39)

$B$N:9J,6a;w$r9M$($F$_$k(B ($a>0$)$B!#(B 1 $B@a$G8+$?$h$&$K(B

$$u_{tt}-a^2 u_{xx} = \left(\frac{\partial }{\partial t}+a\fra... ...rtial }{\partial t}-a\frac{\partial }{\partial x}\right)u(t,x)$$

$B$G$"$j!"$h$C$F(B $v(t,x)=u_t(t,x)-au_x(t,x) =(\partial/\partial t-a\partial/\partial x)u(t,x)$ $B$H$*$1$P(B

$$u_{tt}-a^2 u_{xx}=\left(\frac{\partial }{\partial t}+a\frac{\partial }{\partial x}\right)v$$

$B$H$J$k$N$G!"J}Dx<0$O(B

$$v_t(t,x)+av_x(t,x)=f(t,x)$$

$B$H$J$k!#$3$l$O(B $v$ $B$K4X$9$k0\N.J}Dx<0$J$N$G!"$3$l$rFC@-J}8~$K(B $B:9J,2=$9$k$H(B

$$\frac{v(t+\Delta t,x+a\Delta t)-v(t,x)}{\Delta t}\approx f(t,x)$$ (40)

$B$H$J$k!#0lJ}!"(B$u_t-au_x=v$ $B$b!"(B$u$ $B$K4X$9$k0\N.J}Dx<0$H$_$l$P!"(B $BFC@-B.EY$,(B $a$ $B$+$i(B $-a$ $B$KJQ$o$C$F$$$k$@$1$J$N$G!"(B

$$\frac{u(t+\Delta t,x-a\Delta t)-u(t,x)}{\Delta t}\approx v(t,x)$$ (41)

$B$H:9J,2=$G$-!"$h$C$F!"$3$N<0$G(B $t$ $B$r(B $t+\Delta t$$B!"(B$x$ $B$r(B $x+a\Delta t$ $B$H$9$l$P(B

$$\frac{u(t+2\Delta t,x)-u(t+\Delta t,x+a\Delta t)}{\Delta t} \approx v(t+\Delta t,x+a\Delta t)$$ (42)

$B$H$$$&<0$bF@$i$l$k!#4JC1$N$?$a(B $P^s_y=(t+s,x+y)$ $B$H$9$k$H(B (41),(42) $B$N:8JU$O(B $B$=$l$>$l(B

$$\frac{u(P^{\Delta t}_{-a\Delta t})-u(P^0_0)}{\Delta t},\ \ \frac{u(P^{2\Delta t}_0)-u(P^{\Delta t}_{a\Delta t})}{\Delta t}$$

$B$J$N$G!"(B (41), (42) $B$r(B $B<0(B (40) $B$KBeF~$9$k$H(B

$$\begin{eqnarray*} && \frac{ \displaystyle \frac{u(P^{2\Delta t}_{0}) -u(P^{\De... ...t}_{-a\Delta t})+u(P^0_0)}{\Delta t^2} \\ & \approx & f(P^0_0) \end{eqnarray*}$$

$B$H$J$k!#$3$l$K$h$j!"GHF0J}Dx<0(B (39) $B$O(B

$$u(P^{2\Delta t}_{0})\approx u(P^{\Delta t}_{a\Delta t}) +u(P^{\Delta t}_{-a\Delta t})-u(P^0_0) + \Delta t^2 f(P^0_0),$$

$B$9$J$o$A!"(B $\Delta x=a\Delta t$ $B$H$7$F!"(B $u^n_j=u(n\Delta t,j\Delta x)$ $B$H$9$l$P(B

$$u^{n+2}_j=u^{n+1}_{j+1}+u^{n+1}_{j-1}-u^n_j+\Delta t^2 f^n_j \ \ (f^n_j=f(n\Delta t,j\Delta x))$$

$B$H6a;w7W;;$G$-$k$3$H$K$J$k!#(B

$B$3$N<0$K$h$k7W;;$O(B $n+2=2$$B!"$9$J$o$A(B $t=2\Delta t$ $B$KBP$9$k(B $u$ $B$N(B $B6a;wCM$r5a$a$k$3$H$+$i;O$^$k$N$G(B $B!"(B $u^1_j\ (t=\Delta t)$, $u^0_j\ (t=0)$ $B$NCM$OJL$K5a$a$kI,MW$,$"$k$,!"(B $B$3$l$i$O=i4|CM$h$j7W;;$G$-!"(B$u^0_j$ $B$O(B

$$u^0_j = u(0,j\Delta x) = g(j\Delta x)$$

$B$+$i!"$^$?!"(B $u^1_j=u(\Delta t,j\Delta x)$ $B$O(B

$$\frac{u(\Delta t,j\Delta x)-u(0,j\Delta x)}{\Delta t}\approx u_t(0,j\Delta x)=h(j\Delta x)$$

$B$H9M$($l$P(B

$$\frac{u^1_j-u^0_j}{\Delta t}=h(j\Delta x)$$

$B$K$h$j5a$a$i$l$k!#(B

$BGHF0J}Dx<0(B (39) $B$N?tCM7W;;$K$O!"$[$+$K$b!"=D(B $B2#$N:9J,$rMQ$$$kJ}K!(B

$$\frac{u^{n+1}_j-2u^n_j+u^{n-1}_j}{\Delta t^2} -a^2\frac{u^n_{j+1}-2u^n_j+u^n_{j-1}}{\Delta x^2}=f^n_j$$

$B$b$"$j$3$l$b$h$/;H$o$l$k!#$3$N:9J,$O(B 2 $B3,HyJ,(B $F''(x)$ $B$,(B

$$\begin{eqnarray*} F''(x) & = & \lim_{h\rightarrow 0} \frac{\displaystyle \fra... ...\\ & = & \lim_{h\rightarrow 0} \frac{F(x+h)-2F(x)+F(x-h)}{h^2} \end{eqnarray*}$$

$B$H=q$1$k$3$H$K4p$E$$$F$$$k!#(B