数学实验基础 实验报告(1)常微分方程

实验一 常微分方程

1. 分别用Euler 法和ode45解下列常微分方程并与解析解比较： (1) ,(0)1,13y x y y x '=+=<<

Euler 法：

function [t,y]=euler(Fun,tspan,y0,h) t=tspan(1):h:tspan(2); y(1)=y0;

for i=1:length(t)-1

y(i+1)=y(i)+h.*feval(Fun,t(i),y(i)); end t=t'; y=y';

function f=Fun(x,y) % 常微分方程的右端函数 f=x+y;

>> [x,y]=euler('Fun',[0,3],1,0.1)

>> [x,y] ans =

0 1.0000 0.1000 1.1000 0.2000 1.2200 0.3000 1.3620 0.4000 1.5282 0.5000 1.7210 0.6000 1.9431 0.7000 2.1974 0.8000 2.4872 0.9000 2.8159 1.0000 3.1875 1.1000 3.6062 1.2000 4.0769 1.3000 4.6045 1.4000 5.1950 1.5000 5.8545 1.6000 6.5899 1.7000 7.4089 1.8000 8.3198 1.9000 9.3318 2.0000 10.4550 2.1000 11.7005 2.2000 13.0805 2.3000 14.6086 2.4000 16.2995 2.5000 18.1694 2.6000 20.2364 2.7000 22.5200 2.8000 25.0420 2.9000 27.8262 3.0000 30.8988

ode45：

>> [x,y]=ode45('Fun',[0,3],1) ans =

0 1.0000 0.0502 1.0528 0.1005 1.1109 0.1507 1.1746

0.2010 1.2442 0.2760 1.3596 0.3510 1.4899 0.4260 1.6361

0.5010 1.7996 0.5760 1.9817 0.6510 2.1838 0.7260 2.4074

实验一 常微分方程

0.8010 2.6544 0.8760 2.9264 0.9510 3.2254 1.0260 3.5535

1.1010 3.9131 1.1760 4.3065 1.2510 4.7364 1.3260 5.2056

1.4010 5.7172 1.4760 6.2744 1.5510 6.8810 1.6260 7.5406

1.7010 8.2574 1.7760 9.0359 1.8510 9.8808 1.9260 10.7974

2.0010 11.7912 2.0760 12.8683 2.1510 14.0351 2.2260 15.2986

2.3010 16.6664 2.3760 18.1466 2.4510 19.7478 2.5260 21.4796

2.6010 2

3.3522 2.6760 25.3764 2.7510 27.5641 2.8260 29.9281

2.9010 32.4820 2.9257 3

3.3694 2.9505 3

4.2796 2.9752 3

5.2134

3.0000 36.1711

解析解：>> y=dsolve('Dy=x+y','y(0)=1','x') y =

2*exp(x) - x - 1

(2) 2

0.01()2sin(),(0)0,(0)1,05y y y t y y t ''''-+===<< Euler 法：

实验一常微分方程

function f=Fun(t,y)

% 常微分方程的右端函数

f=[y(2);0.01*y(2)^2-2*y(1)+sin(t)];

>> [t,y]=euler('Fun',[0,5],[0,1],0.2)

ode45：

>> [t,y]=ode45('Fun',[0,5],[0,1])

t =

0 0.0001 0.0001 0.0002 0.0002 0.0005 0.0007 0.0010 0.0012 0.0025

0.0037 0.0050 0.0062 0.0125 0.0188 0.0251 0.0313 0.0627 0.0941 0.1255

0.1569 0.2819 0.4069 0.5319 0.6569 0.7819 0.9069 1.0319 1.1569 1.2819

1.4069 1.5319 1.6569 1.7819 1.9069

2.0319 2.1569 2.2819 2.4069 2.5319

2.6569 2.7819 2.9069

3.0319 3.1569 3.2819 3.4069 3.5319 3.6569 3.7819

3.9069

4.0319 4.1569 4.2819 4.4069 4.5319 4.6569 4.7427 4.8285 4.9142

5.0000

y =

0 1.0000 0.0001 1.0000 0.0001 1.0000 0.0002 1.0000 0.0002 1.0000

0.0005 1.0000 0.0007 1.0000 0.0010 1.0000 0.0012 1.0000 0.0025 1.0000

0.0037 1.0000 0.0050 1.0000 0.0062 1.0000 0.0125 1.0000 0.0188 1.0000

0.0251 0.9999 0.0313 0.9998 0.0627 0.9987 0.0941 0.9965 0.1253 0.9934

0.1564 0.9893 0.2786 0.9632 0.3966 0.9220 0.5085 0.8662 0.6126 0.7967

0.7072 0.7146 0.7908 0.6210 0.8620 0.5176 0.9198 0.4058 0.9632 0.2876

0.9915 0.1647 1.0043 0.0392 1.0013 -0.0869 0.9826 -0.2117 0.9485 -0.3331

0.8996 -0.4490 0.8365 -0.5578 0.7605 -0.6577 0.6725 -0.7471 0.5742 -0.8246