1数学物理方程 第三版 谷超豪 习题解答 齐海涛

,2, . . .ntu(x,t)=Asinωlasinωxasinωt+2Aωal∞Xn=1(?1)nω2/a2?n2π2/l2sinnπaltsinnπlx.~~~3.3|u???§utt?a2uxx=0,0<x<l,t>0÷v±e?)^??):(1)u|x=0=ux|x=l=0,u|t=0=sin32lπx,ut|t=0=sin52lπx.(2)ux|x=0=ux|x=l=0,u|t=0=x,ut|t=0=0.à°7htqi2008@95ê??n?§6SK?ù?ìà??%°??): (1)÷v>.^??)?u(x,t)=∞Xk=0??????Akcos(k+12)πalt+Bksin(k+12)πalt??????sin(k+12)πlx.dD?^?:∞Xk=0Aksin(k+12)πlx=sin3π2lx,∞Xk=0Bk(k+12)πalsin(k+12)πlx=sin5π2lx.?A1=1,Ak=0,k,1;B2=2l5πa,Bk=0,k,2.u(x,t)=cos3πa2ltsin3π2lx+2l5πasin5πa2ltsin5π2lx.(2)÷v>.^??)?u(x,t)=12A0+12B0t+∞Xk=1Akcoskπalt+Bksinkπalt!coskπlx.dD?^?:12A0+∞Xk=1Akcoskπlx=x,12B0+∞Xk=1Bkkπalcoskπlx=0.?A0=l,Ak=2l(?1)k?1k2π2;Bk=0.u(x,t)=l2?4lπ2∞Xk=11(2k?1)2cos(2k?1)πaltcos(2k?1)πlx.~~~3.4^?lCt{|)D>?ˉK:?????????????utt?a2uxx=g,0<x<l,t>0,u|x=0=ux|x=l=0,u|t=0=0,ut|t=0=sinπx2l,ù￥g?~ê.à°7htqi2008@10