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3 $B$h$j0lHL$NJ}Dx<0$HFC@-6J@~(B

2 $B@a$Ge$2$?FC@-6J@~(B $x=at+x_0$ $B$,M-8z$KF/$$$?$N$O!"$3(B $B$ND>@~$,(B

$$\frac{dx}{dt}=a$$

$B$H$$$&@-7) $B$N(B $u_x$ $B$N78?t$K(B $BEy$7$+$C$?$?$a$G$"$k$3$H$,!"(B(10) $B$N7W;;$K$h$j$o$+$k!#(B

$B$h$C$F!"$h$j0lHL$N(B 1 $B3,@~7AJPHyJ,J}Dx<0(B

$$u_t+\alpha(t,x)u_x=\beta(t,x)$$ (11)

$B$KBP$7$F!">oHyJ,J}Dx<0(B

$$\frac{d x}{d t}=\alpha(t,x)$$ (12)

$B$rK~$?$94X?t(B $x=x(t)$ $B$N$3$H$r!"DL>oJ}Dx<0(B (11) $B$NFC@-(B $B6J@~$H8F$s$G$$$k!#(B(7) $B$N>l9g$K$O$3$NJ}Dx<0$O(B

$$\frac{dx}{dt}=a$$

$B$H$J$k$N$G3N$+$K(B $x=at+$($BDj?t(B) $B$H$J$k!#(B

$t=0$ $B$G(B $x(0)=x_0$ $B$H$J$k(B (12) $B$N2r$O!"(B $\alpha(t,x)$ $B$,DL>o$NO"B3$J4X?t$G$"$l$P3N$+$KB8:_$9$k!#(B $B$3$3$G!"(B``$BB8:_$9$k(B'' $B$H$$$&8@MU$N0UL#$O!"Nc$($P!"(B$0\leq t\leq 1$ $B$N(B $BHO0O$G(B $\alpha(t,x)$ $B$,O"B3$G!"(B$\vert\alpha(t,x)\vert$ $B$N:GBgCM$,M-8B$JCM$J(B $B$i$P!"$I$s$J(B $x_0$ $B$KBP$7$F$b!"(B$x(0)=x_0$ $B$H$J$k(B (12) $B$N2r$O!"(B$0\leq t\leq 1$ $B$NHO0OFb$G$OL58BBg$K(B $BH/;6$9$k$h$&$J$3$H$O5/$3$i$J$$!"$H$$$&0UL#$G$"$k!#(B

$B?^(B 6: $BH/;6$9$k2r$N%0%i%U(B

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$B$3$N$3$H$O!"FC@-6J@~$N=8$^$j$,(B $(t,x)$ $BJ?LL$NNN0h$rKd$a?T$/$9$H$$$&$3$H(B $B$r0UL#$9$k!#(B

$B$5$i$K!"(B$\alpha(t,x)$ $B$,DL>o$N3j$i$+$J4X?t!"$b$&>/$7>\$7$/$$$&$H(B $\alpha_x(t,x)$ $B$,O"B3$J4X?t$J$i$P!"(B$x(0)=x_0$ $B$H$J$k(B (12) $B$N2r$O!"$?$@0l$D$7$+$J$$$3$H$,CN$i$l$F$$$k!#(B $B$3$l$O!"FC@-6J@~F1;N$,8r$o$k$3$H$O$J$$$3$H$r0UL#$9$k!#(B

$B$3$l$i$N@-11) $B$K$D$$$F$b!"(B2 $B@a$HF1MM$NJ}K!$G(B $BFC@-6J@~$rMQ$$$F2r$r5a$a$k$3$H$,$G$-$k!#$5$i$K!"J}Dx<0$N1&JU$N(B $\beta(t,x)$ $B$NItJ,$K(B $u$ $B$,4^$^$l$F(B $\beta(t,x,u)$ $B$N$h$&$J7A$G$"$C$F(B $B$bF1$8J}K!$G2r$r5a$a$k$3$H$,$G$-$k!#(B

$B0lHLE*$JJ}K!$N@bL@$O$o$:$i$o$7$$$N$G!"(B2 $B$DDxNc$r>e$2$F$*$/$N$_$H$9$k!#(B

$BNc(B 1

$$\left\{\begin{array}{ll} u_t+2tu_x=t+x & (t>0,-\infty<x<\i... ...rk ),\\ u(0,x)=f(x) & (-\infty<x<\infty) \end{array}\right.$$

$B$3$NJ}Dx<0$N>l9g!"FC@-6J@~$rM?$($kJ}Dx<0$O(B

$$\left\{\begin{array}{l} \displaystyle \frac{d x}{d t}=2t \hspace{1zw}(t>0), \\ [1zh] x(0)=x_0 \end{array}\right.$$

$B$G$"$k$N$G!"$3$l$r2r$$$F(B $x=t^2+x_0$. $B$3$N6J@~$K1h$C$?(B $u$ $B$NJQ2=$rD4(B $B$Y$k$H(B

$$\begin{eqnarray*} \frac{d }{d t}u(t,t^2+x_0) & = & \frac{\partial u}{\partial t... ..._0} \\ & = & \left. (x+t)\right\vert _{x=t^2+x_0} = t^2+t+x_0 \end{eqnarray*}$$

$B$h$j!"$3$NN>JU$r(B $0$ $B$+$i(B $t$ $B$^$G@QJ,$7$F(B

$$u(t,t^2+x_0) - u(0,x_0) = \int_0^t(s^2+s+x_0)\ ds =\frac{t^3}{3}+\frac{t^2}{2}+x_0t$$

$B$h$C$F(B

$$u(t,t^2+x_0)=\frac{t^3}{3}+\frac{t^2}{2}+x_0t+u(0,x_0) =\frac{t^3}{3}+\frac{t^2}{2}+x_0t+f(x_0)$$

$B$H$J$k!#(B$x=t^2+x_0$ $B$H$9$k$H(B $x_0=x-t^2$ $B$J$N$G$3$l$rBeF~$7$F(B

$$u(t,x)=\frac{t^3}{3}+\frac{t^2}{2}+(x-t^2)t+f(x-t^2) =-\frac{2}{3}t^3+\frac{t^2}{2}+xt+f(x-t^2)$$

$B$,F@$i$l$k!#(B

$B?^(B 7: $BFC@-6J@~(B $x=t^2+x_0$

$BNc(B 2

$$\left\{\begin{array}{ll} u_t+xu_x=tu+e^t & (t>0,-\infty<x<\infty),\\ u(0,x)=f(x) & (-\infty<x<\infty) \end{array}\right.$$

$BFC@-6J@~$O(B

$$\left\{\begin{array}{l} \displaystyle \frac{d x}{d t}=x \hspace{1zw}(t>0), \\ [1zh] x(0)=x_0 \end{array}\right.$$

$B$h$j(B ($BJQ?tJ,N%7A(B)$B!"(B$x=x_0e^t$ $B$G$"$k$3$H$,$o$+$k!#$3$l$K1h$C$F(B

$$\begin{eqnarray*} \frac{d }{d t}u(t,x_0e^t) & = & u_t(t,x_0e^t)+u_x(t,x_0e^t)x_... ... = & \left. (tu+e^t)\right\vert _{x=x_0e^t} = tu(t,x_0e^t)+e^t \end{eqnarray*}$$

$B$H$J$k$N$G!"(B $v(t)=u(t,x_0e^t)$ $B$H$9$k$H(B $v=v(t)$ $B$O(B

$$\frac{d v}{d t}=tv+e^t$$

$B$N!"Hs@FoHyJ,J}Dx<0$rK~$?$9!#$h$C$F$3$l$r2r$$$F(B

$$v(t)=e^{t^2/2}\int_0^t e^{s-s^2/2}ds + v(0)e^{t^2/2},$$

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

$$u(t,x_0e^t)=e^{t^2/2}\int_0^t e^{s-s^2/2}ds + u(0,x_0)e^{t^2/2} =e^{t^2/2}\int_0^t e^{s-s^2/2}ds + f(x_0)e^{t^2/2}$$

$B$,F@$i$l$k!#(B$x=x_0e^t$ $B$h$j(B $x_0=xe^{-t}$ $B$@$+$i7k6I(B

$$u(t,x)=e^{t^2/2}\int_0^t e^{s-s^2/2}ds + f(xe^{-t})e^{t^2/2}$$

$B$H$J$k!#(B