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1、偏微分方程数值解法（带程序）例1求解初边值问题要求采用树脂格式，，完成下列计算：（1）取分别计算时刻的数值解。（2）取分别计算时刻的数值解。（3）取分别计算时刻的数值解。并与解析解进行比较。解：程序functionA=zhongxinchafen(x,y,la)U=zeros(length(x),length(y));fori=1:size(x,2)ifx(i)>0&x(i)<=0.5U(i,1)=2*x(i);elseifx(i)>0.5&x(i)<1U(i,1)=2*(1-x(i));endendforj=1:length(y)
2、-1fori=1:length(x)-2U(i+1,j+1)=U(i+1,j)+la*(U(i+2,j)-2*U(i+1,j)+U(i,j));endendA=U(:,size(U,2))functionu=jiexijie1(x,t)fori=1:size(x,2)k=3;a1=(1/(1^2)*sin(1*pi/2)*sin(1*pi*x(i))*exp(-1^2*pi^2*t));a2=a1+(1/(2^2)*sin(2*pi/2)*sin(2*pi*x(i))*exp(-2^2*pi^2*t));whileabs(a2-a1
3、)>0.00001a1=a2;a2=a1+(1/(k^2)*sin(k*pi/2)*sin(k*pi*x(i))*exp(-k^2*pi^2*t));k=k+1;endu(i)=8/(pi^2)*a2;endclc;%第1题第1问clear;t1=0.01;t2=0.02;t3=0.1;x=[0:0.1:1];y1=[0:0.001:t1];y2=[0:0.001:t2];y3=[0:0.001:t3];la=0.1;subplot(131)A1=zhongxinchafen(x,y1,la);u1=jiexijie1(x,t1)l
4、ine(x,A1,'color','r','linestyle',':','linewidth',1.5);holdonline(x,u1,'color','b','linewidth',1);A2=zhongxinchafen(x,y2,la);u2=jiexijie1(x,t2)line(x,A2,'color','r','linestyle',':','linewidth',1.5);line(x,u2,'color','b','linewidth',1);A3=zhongxinchafen(x,y3,la);u3=jiexi
5、jie1(x,t3)line(x,A3,'color','r','linestyle',':','linewidth',1.5);line(x,u3,'color','b','linewidth',1);title('例1(1)');subplot(132);line(x,u1,'color','b','linewidth',1);line(x,u2,'color','b','linewidth',1);line(x,u3,'color','b','linewidth',1);title('解析解');subplot(133);li
6、ne(x,A1,'color','r','linestyle',':','linewidth',1.5);line(x,A2,'color','r','linestyle',':','linewidth',1.5);line(x,A3,'color','r','linestyle',':','linewidth',1.5);title('数值解');clc;%第1题第2问clear;t1=0.01;t2=0.02;t3=0.1;x=[0:0.1:1];y1=[0:0.005:t1];y2=[0:0.005:t2];y3=[0:0.0
7、05:t3];la=0.5;subplot(131);A1=zhongxinchafen(x,y1,la);u1=jiexijie1(x,t1)line(x,A1,'color','r','linestyle',':','linewidth',1.5);holdonline(x,u1,'color','b','linewidth',1);A2=zhongxinchafen(x,y2,la);u2=jiexijie1(x,t2)line(x,A2,'color','r','linestyle',':','linewidth',1.5)
8、;line(x,u2,'color','b','linewidth',1);A3=zhongxinchafen(x,y3,la);u3=jiexijie1(x,t3)line(x,A3,'color','r','linestyle',
