[算法] [常微分方程] [欧拉法 改进欧拉法 经典R-K算法]

 

#include<iostream>
#include<cmath>
#include<cstdio>
#include<iomanip>
using namespace std;
double h=0.1;//步差
double xi[11]={0};
double ol_yi[11]={1};
double gol_yi[11]={1};
double rk_yi[11]={1};
double real_yi[11]={1};
double f(double x,double y){
    return 2*x/(3*y*y);
}//f(x,y)
void OLFunction(){
    for(int i=0;i<10;i++){
        ol_yi[i+1]=ol_yi[i]+h*f(xi[i],ol_yi[i]);
    }
}//欧拉方法
void GOLFunction(){
    for(int i=0;i<10;i++){
        gol_yi[i+1]=gol_yi[i]+
            h*(
                f(xi[i],gol_yi[i])
                +f(xi[i+1],gol_yi[i]+h*f(xi[i],gol_yi[i]))
            )/2;
    }
}//改进欧拉方法
void RKFunction(){
    double K1,K2,K3,K4;
    for(int i=0;i<10;i++){
        K1=f(xi[i],rk_yi[i]);
        K2=f(xi[i]+h/2,rk_yi[i]+h*K1/2);
        K3=f(xi[i]+h/2,rk_yi[i]+h*K2/2);
        K4=f(xi[i]+h,rk_yi[i]+h*K3);
        rk_yi[i+1]=rk_yi[i]+h*(K1+2*K2+2*K3+K4)/6;
    }
}//经典龙格贝法
void RFunction(){
    for(int i=1;i<11;i++){
        real_yi[i]=pow(1.0+xi[i]*xi[i],1/3.0);
    }
}//真实解
int main(){
    int i;
    for(i=1;i<11;i++){
        xi[i]=xi[i-1]+h;
    }//求xi[]
    
    OLFunction();//四种计算方法
    GOLFunction();
    RKFunction();
    RFunction();

    printf("-------------------------------------------------\n");
    printf("xi  | 欧拉     | 改进欧拉 | 经典R-K  | 准确解 \n");
    printf("-------------------------------------------------\n");
    for(i=0;i<11;i++){
        printf("%.1lf | %.6lf | %.6lf | %.6lf | %.8lf\n",
            xi[i],ol_yi[i],gol_yi[i],rk_yi[i],real_yi[i]);
    }
    getchar();
    return 0;
}