微分方程数值解欧拉法(共10页)

1、精选优质文档-倾情为你奉上1.1、求解初值问题，已知精确解为 当h=0.1时，解为：01100.10.0.9.3616E-030.20.0.1.6057E-020.30.0.2.0637E-020.40.0.2.3555E-020.50.0.2.5182E-020.60.0.2.5823E-020.70.0.2.5723E-020.80.0.2.5080E-020.90.0.2.4053E-021.00.0.2.2768E-02当h=0.05时，解为：01100.050.0.2.4185E-030.100.0.4.4835E-030.150.0.6.2326E-030.200.0.7.6996

2、E-030.250.0.8.9155E-030.300.0.9.9084E-030.350.0.1.0704E-020.400.0.1.1324E-020.450.0.1.1791E-020.500.0.1.2124E-020.550.0.1.2338E-020.600.0.1.2450E-020.650.0.1.2473E-020.700.0.1.2420E-020.750.0.1.2302E-020.800.0.1.2129E-020.850.0.1.1909E-020.900.0.1.1651E-020.950.0.1.1362E-021.000.0.1.1048E-02h=50时，解为

3、：01100.020.0.3.9471E-040.040.0.7.6599E-040.060.0.1.1148E-030.080.0.1.4422E-030.100.0.1.7491E-030.120.0.2.0363E-030.140.0.2.3048E-030.160.0.2.5553E-030.180.0.2.7888E-030.200.0.3.0058E-030.220.0.3.2073E-030.240.0.3.3938E-030.260.0.3.5662E-030.280.0.3.7250E-030.300.0.3.8709E-030.320.0.4.0045E-030.340.0

4、.4.1264E-030.360.0.4.2371E-030.380.0.4.3373E-030.400.0.4.4274E-030.420.0.4.5079E-030.440.0.4.5793E-030.460.0.4.6421E-030.480.0.4.6967E-030.500.0.4.7435E-030.520.0.4.7830E-030.540.0.4.8156E-030.560.0.4.8415E-030.580.0.4.8613E-030.600.0.4.8751E-030.620.0.4.8835E-030.640.0.4.8866E-030.660.0.4.8848E-030

5、.680.0.4.8784E-030.700.0.4.8676E-030.720.0.4.8528E-030.740.0.4.8341E-030.760.0.4.8119E-030.780.0.4.7863E-030.800.0.4.7577E-030.820.0.4.7261E-030.840.0.4.6918E-030.860.0.4.6550E-030.880.0.4.6159E-030.900.0.4.5746E-030.920.0.4.5314E-030.940.0.4.4864E-030.960.0.4.4397E-030.980.0.4.3916E-031.000.0.4.342

6、0E-03有图像看出，当步长越小，计算得到的解越逼近精确解。1.2、求解初值问题，已知精确解为： ，h=0.1欧拉法改进的欧拉法预报-校正格式011110.11.1.1.1.0.21.1.1.1.0.31.1.1.1.0.41.1.1.1.0.51.1.1.1.0.61.1.1.1.0.71.1.1.1.0.81.1.1.1.0.91.1.1.1.1.01.1.1.1.欧拉法图像：改进欧拉法图像：预测-校正法图像：附录：源代码1.1、clearclc X0=0;X1=1;n=10;%更改分点数h=1/n;%步长 y(1)=1;x(1)=X0;for i=1:n x(i+1)=x(i)+h; y

7、(i+1)=y(i)+h*(x(i)*exp(-x(i)-y(i);endx=vpa(x',6)y=vpa(y',6) X=(X0:0.001:X1);n=1/0.001;for(i=1:n+1) Y(i)=0.5*(X(i)2+2)*exp(-X(i);end plot(x,y,'*')hold onplot(X,Y) X=X0:h:X1;Y=0.5.*(X.2+2).*exp(-X);Y=vpa(Y',6)yy=abs(y-Y)1.2、欧拉法：clearclcX0=0;X1=1;n=10;h=1/n;y(1)=1;x(1)=X0;X=X0:h:X1;

8、Y=X+exp(-X);Y=vpa(Y',9);%精确解XX=X0:0.0001:X1;YY=XX+exp(-XX);for i=1:nx(i+1)=x(i)+h;y(i+1)=y(i)+h*(-y(i)+x(i)+1);endy=vpa(y',9)plot(x,y,'*')hold onplot(XX,YY)改进的欧拉法：clearclc X0=0;X1=1;n=10;h=1/n; y(1)=1;x(1)=X0; X=X0:h:X1;Y=X+exp(-X);Y=vpa(Y',9);%精确解 XX=X0:0.0001:X1;YY=XX+exp(-XX);

9、 for i=1:n x(i+1)=x(i)+h; y(i+1)=y(i)+h*(-y(i)+x(i)+1);endfor i=1:n y(i+1)=y(i)+0.5*h*(-y(i)+x(i)+1)+(-y(i+1)+x(i+1)+1);end y=vpa(y',9)plot(x,y,'*')hold onplot(XX,YY)预报-校正格式：clearclc X0=0;X1=1;n=10;h=1/n; y(1)=1;x(1)=X0; X=X0:h:X1;Y=X+exp(-X);Y=vpa(Y',9);%精确解 XX=X0:0.0001:X1;YY=XX+exp(-XX); for i=1:n x(i+1)=x(i)+h; y(i+1)=y(i)+h*(-y(i)+x(i)+1);endfor i=1:n for c