热传导方程热传导方程的导出及其定解条件

u=u(t;x1;x2;¢¢¢;xn)ut=k4ukn=1n=3n=3Cauchy()FourierLi-YauHanarcktCauchyF.JohnPartialDi®erentialEquations,Springer-Verlag,1982.x1:1.1Du(t;x;y;z)D(x;y;z)tFourierdtndSdQdS@u@ndQ=¡k(x;y;z)@u@ndSdt;(1.1)k(x;y;z)(x;y;z)(1.1)dQ@u@n1DS­(1.1)t1t2Q=Zt2t1½ZZSk(x;y;z)@u@ndS¾dt;(1.2)@u@nuSn(t1;t2)u(t1;x;y;z)u(t2;x;y;z)ZZZ­º(x;y;z)½(x;y;z)[u(t2;x;y;z)¡u(t1;x;y;z)]dxdydz;º½Zt2t1ZZSk@u@ndSdt=ZZZ­º½[u(t2;x;y;z)¡u(t1;x;y;z)]dxdydz:(1.3)ux;y;ztGreen(1.3)Zt2t1ZZZ­·@@xµk@u@x¶+@@yµk@u@y¶+@@zµk@u@z¶¸dxdydzdt=ZZZ­º½µZt2t1@u@tdt¶dxdydz;Zt2t1ZZZ­·º½@u@t¡@@xµk@u@x¶¡@@yµk@u@y¶¡@@zµk@u@z¶¸dxdydzdt=0:(1.4)t1t2­º½@u@t=@@xµk@u@x¶+@@yµk@u@y¶+@@zµk@u@z¶:(1.5)(1.5)::::::::::::::kº½k=º½=c2@u@t=c2µ@2u@x2+@2u@y2+@2u@z2¶:(1.6)()(1.5)2F(t;x;y;z)Zt2t1ZZSk@u@ndSdt+Zt2t1ZZZ­F(t;x;y;z)dxdydzdt=ZZZ­º½[u(t2;x;y;z)¡u(t1;x;y;z)]dxdydz:(1.5)º½@u@t=@@xµk@u@x¶+@@yµk@u@y¶+@@zµk@u@z¶+F(t;x;y;z):(1.7)(1.6)@u@t=c2µ@2u@x2+@2u@y2+@2u@z2¶+f(t;x;y;z);(1.8)f(t;x;y;z)=F(t;x;y;z)º½:(1.9)(1.6)::::::::::::::::::(1.8):::::::::::::::::::::Fourier(1.1)(1.3)dm=¡°(x;y;z)@U@ndSdt;(1.10)Zt2t1ZZS°@U@ndSdt=ZZZ­[U(t2;x;y;z)¡U(t1;x;y;z)]dxdydz;(1.11)UdmdtndS°(x;y;z):::::::::::(1.1)(1.3)3(1.10)(1.11)(1.1)(1.3)QukmU°(1.3)º½1@U@t=@@xµ°@U@x¶+@@yµ°@U@y¶+@@zµ°@U@z¶:(1.12)°°=c2(1.12)(1.6)1.2()u(0;x;y;z)='(x;y;z);(1.13)'(x;y;z)t=0u(t;x;y;z)j(x;y;z)2S=g(t;x;y;z);(1.14)Sg(t;x;y;z)[0;T]£ST::::::::::::::::::::::::::::::::Dirichlet::::::::QFourierdQ=¡k@u@ndSdtu@u@n¯¯¯¯(x;y;z)2S=g(t;x;y;z);(1.15)4@u@nuSng(t;x;y;z)[0;T]£S::::::::::::::::::::::::::::::::Neumann::::::::11u1uu11dQ=°(u¡u1)dSdt;(1.16)°:::::::::::::SFourier1¡k@u@ndSdt=°(u¡u1)dSdt;°u+k@u@n=°u1:°kµ@u@n+¾u¶¯¯¯¯(x;y;z)2S=g(t;x;y;z);(1.17)@u@nuSng(t;x;y;z)[0;T]£S¾:::::::::::::::::::::::::::::Cauchy:::u(0;x;y;z)='(x;y;z)(¡1x;y;z1):(1.18)5uxt::::::::::::::::::@u@t=c2@2u@x2:(1.19):::::::::::::::::::@u@t=c2µ@2u@x2+@2u@y2¶:(1.20)Cauchy1.LdQ=°(u¡u1)dSdt:½ºku2.3.Q(t)Q0dQdt=¡¯Q¯º½ku6