数学物理方程-陈才生主编-课后习题答案1-3章

11.11.1.1(1),;(2);(3).1.1.2(D'Alembert)Green..1.1.31.()(1),(2),Newton,(3).2.()(1)Newton(F=ma).(2)Fourier().,.q(),q=¡kru;k,.qx=¡kux;qy=¡kuy;qz=¡kuz:(3)Newton.¡kruu¯¯¡u0,u0.¢2¢1(4)().()()().(5)(Fick)().,,...q().q=¡Kru;K,.qx=¡Kux;qy=¡Kuy;qz=¡Kuz:(6)Gauss.,¡1,Z@­E¢dS=1Z­½d­;,½.(7)(Hooke).,,f=¡kx,k(),(),.,=£:3.(),()..,.,(1),(2)(3),.1.1.4,1.2¢3¢,,.1.21.1L,,0,q(),12x(L¡x),.12x(L¡x),u(x;0)=12x(L¡x):,u(0;t)=0;q,kux(L;t)=q:8:ut=a2uxx;0xL;t0;u(x;0)=12x(L¡x);06x6L;u(0;t)=0;ux(L;t)=qk;t0:1.2L,x[0;L].u,½,T.,,,k.Á(x),0.x=0,x=L,k.u(x;t).x=L,k,Tux¯¯x=L=¡ku¯¯x=L;(Tux+ku)¯¯x=L=0:,,,k,F(x;t)=¡kut,[1]1.2.1¢4¢1utt=T½uxx¡k½ut:,8:utt=T½uxx¡k½ut;0xL;t0;u(x;0)=Á(x);ut(x;0)=0;06x6L;u(0;t)=0;(Tux+ku)¯¯¯x=L=0;t0:1.3­=f(x;y)j0x1;0y1g8:uxx¡uyy=0;(x;y)2­;u(x;0)=f1(x);u(x;1)=f2(x);06x61;u(0;y)=g1(y);u(1;y)=g2(y);06y61;f1;f2;g1;g2.??.,,.,u=u(x;y),v=u+sinnxsinny(n1)..1.4(ut=¡uxx;(x;t)2R1£(0;1);u(x;0)=1;x2R1.un(x;t)=1+1nen2tsinnx(n1),8:ut=¡uxx;(x;t)2R1£(0;1);u(x;0)=1+1nsinnx;x2R1:,n!+1supx2R0¯¯un(x;0)¡1¯¯=1n!0.,n!1supx2R1;t0¯¯un(x;t)¡1¯¯=supx2R1;t0¯¯¯¯1nen2tsinnx¯¯¯¯=supt0¯¯¯¯1nen2t¯¯¯¯1nen2!1;.1.31.5,Euler1.3¢5¢½vt+½(v¢r)v+rp=F;(1.3.1)@½@t+r¢(½v)=0;(1.3.2)p=f(½);(1.3.3),(1.3.1)vx;vy;vzv;p;½;F.F=0,.p0½0,S=½¡½0½0,½=½0(1+S).,.,Sv,½¼½0.(1.3.1)(1.3.2),(1.3.1)vt=¡1½rp;(1.3.4)(1.3.2)½t+½0r¢v=0;St+r¢v=0:(1.3.5),(1.3.3)p=p0(1+S)°,p=p0(1+°S);(1.3.6)°.(1.3.4)(1.3.6),vt=¡°p0½0rS:(1.3.7)(1.3.5)(1.3.7)Stt¡a2¢S=0;(1.3.8),a2=°p0½0.1.6,,½,E,u(x;t).@2u@t2=E½x2@@xµx2@u@x¶:(1.3.9),Hooke,½.¹u1.1[x;x+¢x],½S¢x,S¢6¢1.Newton,½S¢x@2¹u@t2=P(x+¢x;t)S(x+¢x)¡P(x;t)S(x)=@@x(PS)¢x;P(x;t)xS(x)tx.¢x!0,½S@2u@t2=@@x(PS);(1.3.10)Hooke,P=E@u@x.S=r2=(xtan®)2,PS(1.3.10),½x2tan2®@2u@t2=@@xµx2tan2®E@u@x¶;,(1.3.9)..1.2,½³1¡xh´2@2u@t2=E@@x·³1¡xh´2@u@x¸;(1.3.11)h.,S=r2;r=(h¡x)tan®.(1.3.10),(1.3.11).1.11.21.7Maxwell8:r¢E=0;r£E=¡1cHt;r¢H=0;r£H=1cEt:(1.3.12)1.3¢7¢(Ett=c2¢E;Htt=c2¢H;EH,c.(1.3.12)t,r£Ht=1cEtt:r£(1.3.12),r£r£E=¡1cr£Ht;r(r¢E)¡¢E=¡1cr£Ht;(1.3.12),Ett=c2¢E:,Htt=c2¢H:1.8.E(x),½(x),F(x;t),x().u(x;t)xt.[x;x+¢x].¹u(x;t)1.3.½S¢x,S.Newton,½S¢x@2¹u@t2=(p(x+¢x;t)¡p(x;t))S;p(x;t)xtx.,¢x!0,¹u!up(x+¢x;t)¡p(x;t)¢x!@p@x:½@2u@t2=@p@x:Hooke,.p=E@u@x;E.,E.½@2u@t2=E@2u@x2:¢8¢11.31.41.9L,!.,(1.4).:@2u@t2=12!2@@x·(L2¡x2)@u@x¸:u=u(x;t)xtx.,T.,x.,Newton,[x;x+¢x]½¢x@2¹u@t2=Tux(x+¢x;t)¡Tux(x;t);¹u,½,..xT=T(x)=ZLx!2s½ds=12½!2(L2¡x2);Tux¯¯x+¢x¡Tux¯¯x=12½!2@@xµ(L2¡x2)@u@x¶¢x:¢x!0,@2u@t2=12!2@@xµ(L2¡x2)@u@x¶:1.10L,,!..1.3¢9¢,x(1.5),u(x;t)(x),T(x;t),.[x;x+¢x],T(x;t).||,x.,x½g(L¡x)(½,),T(x;t)=½g(L¡x).,F(x;t)=½!2u(x;t)¢x.Newton,1.5½¢x@2¹u@t2=½g(L¡x)ux¯¯x+¢x¡½g(L¡x)ux¯¯x+½!2¢x¹u;¹u.,¢x!0,@2u@t2=g@@xµ(L¡x)@u@x¶+!2u:@2u@t2¡a2@2u@x2=¡g@u@x+!2u;a(x)=pg(L¡x)..,,@2u@t2=g@@xµx@u@x¶+!2u:1.11,R,Newton.u(x;t)@u@t=kc½@2u@x2¡2Hc½R(u¡u0);k,c,½,HNewton,u0.[x;x+¢x](1.6).Fourier,()¡k@u@x,¢V(¡kux¯¯x¡(¡kux)¯¯x+¢x)R2=¡@@x(¡kux)R2¢x=kuxxR2¢x:¢10¢1Newton,¢V(2R¢x)¡H(u¡u0)(2R¢x).¢V.c½ut¢V=c½utR2¢x;c½utR2¢x=kuxxR2¢x¡2RH(u¡u0)¢x;c½¢xR2.1.61.12,Oxy.(Oxy{u).u(x;y;t)(x;y)t.u(x;y;t).½.,½.,,.,1,.(x;y);(x+¢x;y);(x+¢x;y+¢y);(x;y+¢y).Hooke,.T.t,1.7.ABT¢x,Ty®;BCT¢y,x°;CDT¢x,y¯;DAT¢y,x±.Oxy,Oxy.Newton,¡T¢xsin®+T¢xsin¯¡T¢ysin±+T¢ysin°=(½¢x¢y)utt:(1.3.13),8:sin®¼tan®=uy(x;y;t);sin¯¼tan¯=uy(x;y+¢y;t);sin±¼tan±=ux(x;y;t);sin°¼tan°=ux(x+¢x;y;t):1.3¢11¢(1.3.13),T¢x[uy(x;y+¢y;t)¡uy(x;y;t)]+T[ux(x+¢x;y;t)¡ux(x;y;t)]=½¢x¢yutt;T·¢uy¢y+¢ux¢x¸=½utt:¢x!0;¢y!0,T(uxx+uyy)=½utt:a2=T=½,utt¡a2(uxx+uyy)=0:(1.3.14).u,()F(x;y:t),f(x;y;t)=F(x;y;t)=½,(1.3.14)utt¡a2(uxx+uyy)=f(x;y;t):(1.3.15).1.71.13­=f(x;y)jx2R1;y0g.(¢u+u=0;(x;y)2­;u(x;0)=Á(x);uy(x;0)=Ã(x);x2R1;Á(x);Ã(x)R1.¢12¢1.,un(x;y)=1n2enysin(pn2+1x).,n!1,supx2R1jÁ(x)j+supx2R1jÃ(x)j=(n¡2+n¡1)supx2R0jsin(pn2+1x)j!0:,n!1,sup(x;y)2­jun(x;y)j!1:1.14..,(),(),.u(x;t)xt.[x0;x],1.8.[x0;x]1.8,M(t)=Zxx0u(y;t)dy;,M0(t)=Zxx0ut(y;t)dy.,,.FickM0(t)=¡=Kux(x;t)¡Kux(x0;t);Zxx0ut(y;t)dy=Kux(x;t)¡Kux(x0;t):x,ut=(Kux)x:,K.­,D½­.Fick,ZZDZutdxdydz=ZZ@DK@u@ndS=ZZDZdiv(Kru)dxdydz:D,ut=div(Kru)=@@xµK@u@x¶+@@yµK@u@y¶+@@zµK@u@z¶:.K=a2,ut=div(Kru)=a2µ@2u@x2+@2u@y2+@2u@z2¶=a2¢u:1.3¢13¢1.15L,,.,u0,..x=0,,ux(0;t)=0.,.u(L;t)=u0.8:ut=Duxx;0xL;t0;ux(0;t)=0;u(L;t)=u0;t0;u(x;0)=0;06x6L:1.16,u0,t,Newton..1.9,r1,r2,,urt.ut=D¢u=D(urr+r¡1ur);r1rr2;t0;u(r;0)=u0;1.9u(r1;t)=at+b;a;b.u(r;0)=u0,b=u0.u(r1;t)=at+u0:u1,Newton¡kur(r2;t)=H(u(r2;t)¡u1);(u+hur)¯¯r=r2=u1;h=k=H,kH..1.17L,,x=0.x=L.(1)x=L;¢14¢1(2)x=LF(t)=¡ku(L;t),k,u(L;t)x=L.(1)Hooke,ux¯¯x=L=F(t)ES;E,S.(2)F(t)=¡ku(L;t),ux¯¯x=L=¡kESu(L;t);(ux+hu)¯¯x=L=0;h=k=ES:22.12.1.11.x;ya11uxx+2a12uxy+a22uyy+a1ux+b1uy+cu=f(x;y);(x;y)2­(2.1.1)a11(x;y);a12(x;y);a22(x;y)­,¢=a212¡a11a22.(1)(x0;y0),¢0,(2.1.1)(x0;y0);(2)(x0;y0),¢=0,(2.1.1)(x0;y0);(3)(x0;y0),¢0,(2.1.1)(x0;y0).,(2.1.1)­,(2.1.1)­.,(2.1.1)­(),(2.1.1)­().2.a11(dy)2¡2a12dxdy+a22(dx)2=0:(2.1.2)(2.1.1),(2.1.2)()(2.1.1).a116=0,dydx=a12+p¢a11;dydx=a12¡p¢a11;(2.1.3)(2.1.3)(2.1.1).¢16¢23.¢0.(2.1.3),Á1(x;y)=c1;Á2(x;y)=c2;»=Á1(x;y);´=Á2(x;y);(2.1.1)u»´=A2u»+B2u´+C2u+F2:(2.1.4)®=12(»+´);¯=12(»¡´);(2.1.5)(2.1.1)u®®¡u¯¯=A3u®+B3u¯+C3u+F3:(2.1.6)¢=0.(2.1.3)dydx=a12a11;,Á(x