代码
构造函数
function [yn,xn] = RKorder4(f,a,b,h,y0)
%a,b为上下界;h为步长;y0为初值。
%返回yn、xn向量,并绘制曲线
yn=zeros(1,((b-a)/h)+1);
yn(1)=y0;
xn=a:h:b;
for i=1:((b-a)/h)
k1=h*f(xn(i),yn(i));
k2=h*f(xn(i)+h/2,yn(i)+k1/2);
k3=h*f(xn(i)+h/2,yn(i)+k2/2);
k4=h*f(xn(i)+h,yn(i)+k3);
yn(i+1)=yn(i)+(1/6)*(k1+2*k2+2*k3+k4);
end
end
实例
对于某暂态电路的初值问题:
{didt=622sin314t−20ii(0)=0
\left\{\begin{matrix}\frac{\mathrm{d} i}{\mathrm{d} t}=622sin314 t-20i
\\
i(0)=0
\end{matrix}\right.
{dtdi=622sin314t−20ii(0)=0
f(tn,in)=622sin314t−20i\ f(t_n,i_n)=622sin314 t-20i f(tn,in)=622sin314t−20i
取h=0.001,在[0,0.01]的时间区间内求解微分方程,得出电流值。
在命令窗口输入
a=0;b=0.01;h=0.001;
y0=0;
f=@(t,i) 622*sin(314*t)-20*i;
[yn,xn] = RKorder4(f,a,b,h,y0);
plot(xn,yn);grid on
又由暂态电路理论分析得,此电路的解析解为:
i(t)=4882724749(e−20t+10157sin314t−cos314t)
i(t)=\frac{48827}{24749}\left( e^{-20t} +\frac{10}{157}sin314t-cos314t\right)
i(t)=2474948827(e−20t+15710sin314t−cos314t)
输入命令,将解析解和数值解的曲线绘制在同一图内:
i=@(t) (48827/24749).*(exp(-20.*t)+(10/157)*sin(314.*t)-cos(314.*t));
plot(linspace(0,0.01,1000),i(linspace(0,0.01,1000)),'r',xn,yn,'b');grid on