matlab用龙格库塔法求解变系数常微分方程 急！！！求matlab 用四阶龙格-库塔法求解常微分方程
Matlab\u56db\u9636\u9f99\u683c\u5e93\u5854\u6cd5\u6c42\u89e3\u5e38\u5fae\u5206\u65b9\u7a0b\u7528Matlab\u56db\u9636\u9f99\u683c\u5e93\u5854\u6cd5\u6c42\u5e38\u5fae\u5206\u65b9\u7a0b\u53ef\u4ee5\u6309\u7167\u4ee5\u4e0b\u65b9\u6cd5\u53bb\u5b9e\u73b0\u3002
1\u3001\u9996\u5148\u5efa\u7acb\u81ea\u5b9a\u4e49\u5fae\u5206\u65b9\u7a0b\u51fd\u6570
function f = ode_fun(x,y)
f=y+2*x/y^2;
end
2\u3001\u7136\u540e\u7528\u56db\u9636\u9f99\u683c\u5e93\u5854\u6cd5\u6c42\u5176\u6570\u503c\u89e3
figure(2)
y0=[1]; %\u521d\u503cy\uff080\uff09=1
h=0.1;
a=0;
b=5;
[x,y] = runge_kutta(@(x,y)ode_fun(x,y),y0,h,a,b);
disp(' x y')
A=[x',y']
plot(x,y,'LineWidth',1.5),grid on
xlabel('x'),ylabel('y(x)');
3\u3001\u7ed8\u5236y-x\u7684\u66f2\u7ebf\u56fe
\u5efa\u7acb.m\u6587\u4ef6
---------------------------------------------
function theta=danbai(t,X)
x=X(1);
dx=X(2);
ddx=-sin(x);
theta=[dx;ddx];
----------------------------------------------
\u547d\u4ee4\u7a97\u53e3\u8f93\u5165
>> [t,Y]=ode45(@danbai,[0 6],[pi/3 -1/2]);
>> plot(t,Y(:,1),'ro-',t,Y(:,2),'bv-');
>> legend('\theta-t','d\theta-t')
\u81ea\u7f16\u9f99\u683c\u5e93\u5854
------------------------------------------------
function [y,z]=Runge_kutta(a,b,y0,z0,h)
x=a:h:b;
y(1)=y0;
z(1)=z0;
n=(b-a)/h+1;
for i=2:n
K(1,1)=f1(x(i-1),y(i-1),z(i-1));
K(2,1)=f2(x(i-1),y(i-1),z(i-1));
K(1,2)=f1(x(i-1)+h/2,y(i-1)+K(1,1)*h/2,z(i-1)+K(2,1)*h/2);
K(2,2)=f2(x(i-1)+h/2,y(i-1)+K(1,1)*h/2,z(i-1)+K(2,1)*h/2);
K(1,3)=f1(x(i-1)+h/2,y(i-1)+K(1,2)*h/2,z(i-1)+K(2,2)*h/2);
K(2,3)=f2(x(i-1)+h/2,y(i-1)+K(1,2)*h/2,z(i-1)+K(2,2)*h/2);
K(1,4)=f1(x(i-1)+h,y(i-1)+K(1,3)*h,z(i-1)+K(2,3)*h);
K(2,4)=f2(x(i-1)+h,y(i-1)+K(1,3)*h,z(i-1)+K(2,3)*h);
y(i)=y(i-1)+h/6*(K(1,1)+2*K(1,2)+2*K(1,3)+K(1,4));
z(i)=z(i-1)+h/6*(K(2,1)+2*K(2,2)+2*K(2,3)+K(2,4));
end
y(2)
plot(y,'r') %\u03b8-t\u56fe
hold on
plot(f1(x,y,z),'g') % d\u03b8-t\u56fe
hold on
plot(f2(x,y,z),'b-')%d2\u03b8-t\u56fe
%f1.m
function f1=f1(x,y,z)
f1=z;
%f2.m
function f2=f2(x,y,z)
f2=-sin(y);
-----------------------------------------
Runge_kutta(0,6,pi/3,-1/2,0.02)
该二阶微分方程用龙格库塔法可以这样来求解。
第一步，根据该二阶微分方程，自定义微分方程函数，func(t,y)
第二步，根据初始条件，确定y和y'的初值，即y0=[0,0]
第三步，使用ode45函数求解【t,y】的数值解，即
[t,y] = ode45(@func,[0 0.0005],y0);
第四步，根据t、y、y'值，绘制t—y(t)，t—y'(t)曲线图
下图为求解结果
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