数理方法第二版前三章习题-谷超豪

1.1(1).(2),,,.(3),.(4).(5),.(6).1.2x1.1.(),u(x;t)xt.,u(x;t)@@tµ½(x)@u@t¶=@@xµE@u@x¶;½,E.:,(x;x+¢x),@u@x(x;t)@u@x(x+¢x;t).S,E(x)@u@x(x;t)S(x)E(x+¢x)@u@x(x+¢x;t)S(x+¢x).E(x+¢x)@u@x(x+¢x;t)S(x+¢x)¡E(x)@u@x(x;t)S(x),.1-1¹x,@2u@t2(¹x;t),,½(¹x)S(¹x)¢x@2u@t2(¹x;t)=E(x+¢x)S(x+¢x)@u@x(x+¢x;t)¡E(x)S(x)@u@x(x;t)=@@xµE(x¤)S(x¤)@u@x(x¤;t)¶¢x-1-1.2x¤2(x;x+¢x).¢x¢x!0,@@tµ½(x)S(x)@u@t¶=@@xµE(x)S(x)@u@x¶S(x)@@tµ½(x)@u@t¶=@@xµE@u@x¶;2.,(1),(2),(3),.:0l.(1),u(0;t)=u(l;t)=0;(2),,E(x)@u@x(x;t)S=0,@u@x(0;t)=@u@x(l;t)=0;(3).k,x=0E(0)@u@x(0;t)S,x,E(0)@u@x(0;t)S=ku(0;t);µ¡@u@x+¾u¶¯¯¯¯x=0=0;¾=k/ES.,x=l,µ¡@u@x+¾u¶¯¯¯¯x=l=0:3.E@@xµ³1¡xh´2@u@x¶=½³1¡xh´2@2u@t2;h.:S(x)=S0³1¡xh´2,S0.1,.1-24.,,,.-2-:,½@2u@t2=@@xµT(x)@u@x¶T(x).,T(x)=½g(l¡x),@2u@t2=g@@xµ(l¡x)@u@x¶:5.,,.,.:F(x)=k@u@t,T@2u@t2¡½@2u@x2=¡k@u@tx=0,u(0;t)=0;x=l,µ@u@x+¾u¶¯¯¯¯x=l=0.6.F(»),G(»),F(x¡at),G(x+at)(1.11).:.7.u(x;y;t)=1pt2¡x2¡y2t2¡x2¡y20@2u@t2=@2u@x2+@2u@y2.:,@u@t=¡t(t2¡x2¡y2)3=2;@2u@t2=3t2(t2¡x2¡y2)5=2¡¡t2¡x2¡y2¢¡3=2,@2u@x2=3x2(t2¡x2¡y2)5=2+¡t2¡x2¡y2¢¡3=2;@2u@y2=3y2(t2¡x2¡y2)5=2+¡t2¡x2¡y2¢¡3=2;.x2.1.@@xµ³1¡xh´2@u@x¶=1a2³1¡xh´2@2u@t2;u(x;t)=F(x¡at)+G(x+at)h¡xh0,F,G:t=0:v=(h¡x)'(x);@v@t=(h¡x)Ã(x):(1)v(x;t)=(h¡x)u(x;t),v(x;t)@2v@t2=a2@2v@x2-3-1.2,,v(x;t)v(x;t)=F(x¡at)+G(x+at),u(x;t)=F(x¡at)+G(x+at)h¡x(2),v(x;t)t=0:v=(h¡x)'(x);@v@t=(h¡x)Ã(x)v(x;t)=12((h¡x+at)'(x¡at)+(h¡x¡at)'(x+at))+12aZx+atx¡at(h¡»)Ã(»)d»;u(x;t)=12(h¡x)µ((h¡x+at)'(x¡at)+(h¡x¡at)'(x+at))+1aZx+atx¡at(h¡»)Ã(»)d»¶:2.'(x)Ã(x),?:,G(x)='(x)2+12aZxx0Ã(»)d»+CG(x+at),,G(x)´,G0(x)=0,a'0(x)+Ã(x)´03.,(Goursat)8:@2u@t2=a2@2u@x2;ujx¡at=0='(x);ujx+at=0=Ã(x);('(0)=Ã(0))::u(x;t)=F(x¡at)+G(x+at),,ujx¡at=0=F(0)+G(2x)='(x)ujx+at=0=F(2x)+G(0)=Ã(x)F(x)=óx2´¡G(0);G(x)='³x2´¡F(0),u(0;0)='(0)=Ã(0),F(0)+G(0)='(0)=Ã(0).u(x;t)=F(x¡at)+G(x+at)='µx+at2¶+õx¡at2¶¡'(0):4.(2.5),(2.6),:f(x;t),(1)x[x1;x2],[x1;x2];(2)x[x1;x2][x1;x2].:(1),f(x;t)jt0;x1¡at6x6x2+atg,(x;t)xx1¡atxx2+at[x1;x2].,[x1;x2];-4-(2),[x1;x2]'(x)=0,Ã(x)=0,[x1;x2](x;t)u(x;t)=0.,(x;t)x1+atxx2¡at,x¡atx1x+atx2,u(x;t)=12('(x¡at)+'(x+at))+12aZx+atx¡atÃ(»)d»:x16x6x2'(x)=Ã(x)=0,..1-35.8:utt¡a2uxx=0;x0;t0;ujt=0='(x);utjt=0=0;(ux¡kut)jx=0=0;k.:u(x;t)=F(x¡at)+G(x+at)x¡at0,u(x;t)=12(Á(x¡at)+Á(x+at));xat;x¡at0,,ux¡kutjx=0=(1+ak)F0(¡at)+(1¡ak)G0(at)=0,u(x;t)=12Á(x+at)+1¡ak2(1+ak)Á(at¡x)+ak1+akÁ(0);0xat6.8:utt¡uxx=0;0tkx;k1;ujt=0='0(x);x0;utjt=0='1(x);x0;utjt=kx=Ã(x);'0(0)=Ã(0).:u(x;t)=F(x¡t)+G(x+t)-5-1.2x¡t0,,u(x;t)=12('0(x+t)+'0(x¡t))+12Zx+tx¡t'1(»)d»t=x,,F(0)+G(2x)=12('0(2x)+'0(0))+12Z2x0'1(»)d»t=kx,,F((1¡k)x)+G((1+k)x)=Ã(x)FG,u(x;t)=õx¡t1¡k¶+12µ'0(x+t)¡'0µ1+k1¡k(x¡t)¶¶+12Zx+t1+k1¡k(x¡t)'1(»)d»;xtkx7.8:utt¡uxx=0;0tf(x);ujt=x='(x);ujt=f(x)=Ã(x);'(0)=Ã(0),t=f(x),x=tx=¡t,x,f0(x)6=1:u(x;t)=F(x¡t)+G(x+t)ujt=x=F(0)+G(2x)='(x)G(x)='³x2´¡F(0)ujt=f(x)=F(x¡f(x))+G(x+f(x))=F(x¡f(x))+'µx+f(x)2¶¡F(0)=Ã(x)y=x¡f(x),f0(x)6=1,x=h(y).,F(y)=Ã(h(y))¡Á³h(y)¡y2´+F(0)u(x;t)=Áµx+t2¶¡Áµh(x¡t)¡x¡t2¶+Ã(h(x¡t))8.8:@2u@t2¡@2u@x2=tsinxujt=0=0;@u@t¯¯¯¯t=0=sinx:,a=1,'=0,Ã=sinx,f=tsinx,u(x;t)=12Zx+tx¡tsin»d»+12Zt0Zx+(t¡¿)x¡(t¡¿)¿sin»d»d¿=tsinx-6-9.8:utt=a2uxx+tx(1+x2)2;ujt=0=0;utjt=0=11+x2::,'=0,Ã=11+x2,f=11+x2,u(x;t)=12aZx+atx¡at11+»2d»+12aZt0Zx+a(t¡¿)x¡a(t¡¿)»¿(1+»2)2d»d¿=18a30B@4atarctan(x)+ln1+(x+t)21+(x¡t)2¡2¡x+at¡2a2¢arctan(x+t)+2¡x¡at¡2a2¢arctan(x¡t)1CAx3.1.:(1)8:@2u@t2=a2@2u@x2ujt=0=sin3¼xl;@u@t¯¯¯¯t=0=x(l¡x);(0xl);u(0;t)=u(l;t)=0(2)8:@2u@t2¡a2@2u@x2=0;u(0;t)=0;@u@x(0;t)=0;u(x;0)=hlx;@u@t(x;0)=0::(1),Ak=2lZl0sin3¼»lsink¼»ld»=½1;k=3;0;k6=3:Bk=2k¼aZl0»(l¡»)sink¼»ld»=8:8l3(2n¡1)4¼4a;k=2n¡10;k=2nn=1;2;3;¢¢¢,u(x;t)=cos3¼atlsin3¼xl+8l2¼4a1Xn=11(2n¡1)4sin(2n¡1)¼atlsin(2n¡1)¼xl(2)u(x;t)=1Xn=1µAncos(n+1=2)¼atl+Bnsin(n+1=2)¼atl¶sin(n+1=2)¼xl-7-1.2An=2lZl0hl»sin(n+1=2)¼»ld»=(¡1)n2h(n+1=2)2¼2;Bn=0;n=1;2;3;¢¢¢2.,,8:@2u@t2=a2@2u@x2;u(0;t)=0;u(l;t)=Asin!t;u(x;0)=@u@t(x;0)=0:.:v=u¡Axlsin!t,v8:@2v@t2¡a2@2v@x2=Axl!2sin!t;v(0;t)=v(l;t)=0;v(x;0)=0;vt(x;0)=¡A!lx:v1v2v=v1+v2.,v1=1Xn=1Bnsinn¼atlsinn¼xlBn=¡2A!n¼2Zl0»lsinn¼xld»=(¡1)n2A!l(n¼)2av2=1Xn=1Zt0Bn(¿)sinn¼a(t¡¿)ld¿sinn¼xlBn(¿)=2n¼aA!2lsin!¿Zl0»sinn¼xld»=(¡1)n+12A!2l(n¼)2aZt0Bn(¿)sinn¼a(t¡¿)ld¿=8:(¡1)n2A!2l(!2l2¡n2¼2a2)n2¼2aµn¼asin!t¡!lsinn¼atl¶;n6=!l¼a(¡1)n+1A!2ln2¼2aµ1!sin!t¡tcos!t¶;n=!l¼au(x;t)=v1+v2+Axlsin!t3.utt¡a2uxx=0;0xl;t0:(1)ujx=0=uxjx=l=0;ujt=0=sin32l¼x;utjt=0=sin52l¼x:-8-(2)uxjx=0=uxjx=l=0;ujt=0=x;utjt=0=0::(1)u(x;t)=1Xn=1µAncos(n+1=2)¼atl+Bnsin(n+1=2)¼atl¶sin(n+1=2)¼xlAn=2lZl0sin3¼»2lsin(n+1=2)¼»ld»=(1;n=10;n6=1Bn=l(n+1=2)¼a2lZl0sin5¼»2lsin(n+1=2)¼»ld»=8:2l5¼a;n=20;n6=2u(x;t)=cos3¼at2lsin3¼x2l+2l5¼asin5¼at2lsin5¼x2l(2)u(x;t)=1Xn=0µAncosn¼atl+Bnsinn¼atl¶cosn¼xlA0=1lZl0'(»)d»=1lZl0»d»=l2An=2lZl0'(»)cosn¼»ld»=2lZl0»cosn¼»ld»=2l((¡1)n¡1)n2¼2(n6=0)Bn=04.:8:utt¡a2uxx=g;0xl;t0;ujx=0=uxjx=l=0;ujt=0=0;utjt=0=sin¼x2l:g.:u=v¡g2a2x(x¡2l),v8:vtt¡a2vxx=0;0xl;t0;vjx=0=vxjx=l=0;vjt=0=g2a2x(x¡2l);vtjt=0=sin¼x2l:v(x;t)=1Xn=0µAncos2n+12l¼at+Bnsin2n+12l¼at¶sin2n+12l¼x;,An=2lRl0'(»)sin2n+12l¼»d»;Bn=4(2n+1)¼aRl0Ã(»)sin2n+12l¼»d»:-9-1.2An=¡16l2g(2n+1)3a2¼3