範例EX5-5.c 使用 二階Runge-Kutta 解 ODE y'= -y + t^2 + 1 , 0<=t<=1 , y(0)=1 , 真實解 W(t)= -2e^(-t) + t ^2 - 2t + 3
/* ex5-5.c Second Order Runge-Kutta Method is used
* for solving Ordinary Differential Equation of
* y'=f(y,t) with initial condition of y(t0)=y0.
*/
#include <stdio.h>
#include <math.h>
#define F(y,t) (-y+t*t+1)
#define W(t) (-2*(1.0/exp(t))+pow(t,2)-2*t+3)
void main()
{
int i,n=100;
double h,a=0.0,b=1.0,t0,t,y0=1.0,y,k1,k2;
h=(b-a)/n;
y=y0;
t0=a;
t=t0;
printf("t y(t) w(t) error\n");
printf("=====================================\n");
printf("%.2lf %10.7lf %10.7lf %10.7lf\n",
t,y,W(t),fabs(y-W(t)));
for(i=1;i<=n;i++)
{
k1=h*F(y,t);
k2=h*F((y+k1),(t+h));
y=y+0.5*(k1+k2);
t=t+h;
if(i%10==0)
printf("%.2lf %10.7lf %10.7lf %10.7lf\n",
t,y,W(t),fabs(y-W(t)));
}
return;
}
t y(t) w(t) error
=====================================
0.00 1.0000000 1.0000000 0.0000000
0.10 1.0003269 1.0003252 0.0000017
0.20 1.0025421 1.0025385 0.0000036
0.30 1.0083691 1.0083636 0.0000056
0.40 1.0193675 1.0193599 0.0000076
0.50 1.0369483 1.0369387 0.0000096
0.60 1.0623883 1.0623767 0.0000116
0.70 1.0968430 1.0968294 0.0000136
0.80 1.1413577 1.1413421 0.0000156
0.90 1.1968782 1.1968607 0.0000175
1.00 1.2642605 1.2642411 0.0000194
...Program finished with exit code 38
Press ENTER to exit console.
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