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1、一：题目重述-d2uxdx2+sinxux=cosx+sinxcosx定义区间为：0,π2u0=1,uπ2=0,当N=1000时，求ui和u(xi)二：解题过程2.1求出差分方程差分方程为：Au=fA=b1cab2cab3⋱0⋱⋱0b998cab999,u=u1u2u3⋮⋮u998u999,f=f1+u0h2f2f3⋮f998f999+u1000h2其中：xi=π2000ii=1,2,3……999,h=π2000,a=c=-1h2,qi=sinxi,fi=cosxi+sinxicosxi,bi=2h2+qi2.2用追赶法编程求解差分
2、方程functionx=chase()tic;%开始计时n=999;%对a,b,c,f进行赋值a=zeros(n-1,1);a=a-4000000/pi^2;c=zeros(n-1,1);c=c-4000000/pi^2;b=zeros(n,1);f=zeros(n,1);qx=@(x)sin(x);fx=@(x)cos(x)+sin(x)*cos(x);x2=zeros(n,1);%记录u（x）的真实值error=zeros(n,1);%记录误差fori=1:nb(i)=8000000/pi^2+qx(pi*i/2000);f(i
3、)=fx(pi*i/2000);x2(i)=cos(pi*i/2000);endf(1)=f(1)+4000000/pi^2;%对A进行分解,追赶法u(1)=b(1);fori=2:nif(u(i-1)~=0)l(i-1)=a(i-1)/u(i-1);u(i)=b(i)-l(i-1)*c(i-1);elseendendL=eye(n)+diag(l,-1);U=diag(u)+diag(c,1);x=zeros(n,1);y=x;%求解Ly=b,追赶法第一步，消去最下面的斜行y(1)=f(1);fori=2:ny(i)=f(i)-l
4、(i-1)*y(i-1);end%求解Ux=y，追赶法第二步if(u(n)~=0)x(n)=y(n)/u(n);endfori=n-1:-1:1x(i)=(y(i)-c(i)*x(i+1))/u(i);endtime=toc;%计时结束，并赋值disp('运算时间为：');disp(time);%列出计算结果result=[xx2error];disp('数值解真实解误差项');disp(result);三：计算结果3.1Matlab运行的结果（图1）图13.2整理后的部分结果（表1）x=i*π/2000数值解真实值误差i=10.9
5、999987664882190.9999987662997040.00000000018851520.9999950655783810.9999950652018580.00000000037652330.9999888972796190.9999888967155960.00000000056402340.9999802616071530.9999802608561370.00000000075101650.9999691585822910.9999691576447900.00000000093750160.9999555882
6、324300.9999555871089500.00000000112348070.9999395505910520.9999395492821010.00000000130895180.9999210456977310.9999210442038160.00000000149391590.9999000735981260.9999000719197540.0000000016783724930.7148389611275530.7148389245461190.0000000365814344940.713739638835332
7、0.7137396022764210.0000000365589114950.7126385554612950.7126385189252050.0000000365360904960.7115357137222550.7115356772092850.0000000365129704970.7104311163393660.7104310798498130.0000000364895534980.7093247660381120.7093247295722740.0000000364658384990.70821666554830
8、30.7082166291064760.0000000364418275000.7071068176040670.7071067811865480.0000000364175195010.7059952249438450.705995
