四阶龙格库塔法

一：各位大神帮帮忙吧！要交作业，求解啊，用Matlab编程，龙格库塔法四阶

h=1/128;tf=3 ;t = 1:h:tf;x = zeros(1,length(t));x(1) = 2; F_tx = @(t, x)(t.*x-x.^2)./t.^2; for i=1:(length(t)-1) k_1 = F_tx(t(i),x(i)); k_2 = F_tx(t(i)+0.5*h,x(i)+0.5*h*k_1); k_3 = F_tx((t(i)+0.5*h),(x(i)+0.5*h*k_2)); k_4 = F_tx((t(i)+h),(x(i)+k_3*h)); x(i+1) = x(i) + (1/6)*(k_1+2*k_2+2*k_3+k_4)*h;endt0 = t(1); x0 = x(1);xspan = [t0 tf];[x_ode45, y_ode45] = ode45(F_tx,xspan,x0);subplot(211)plot(x_ode45,y_ode45,'--'); xlabel('t');ylabel('x');legend('Exact');subplot(212)plot(t,x,'-');xlabel('t');ylabel('x');legend('Approximation');

二：如何用四阶龙格库塔法解2阶的微分方程

给你一个例子。如何用四阶龙格库塔法解2阶的微分方程。

三：四阶龙格库塔 步长

假设求解初值问题：

y'=y-2x/y (0

y(0)=1

设步长h=0.2,从x=0直到x=1用四阶龙格库塔法：

Private Sub Form_click()

Dim x As Single, y As Single

Dim k1 As Single, k2 As Single, k3 As Single, k4 As Single

Dim y1 As Single, y2 As Single, y3 As Single, y4 As Single

x = 0

y = 1

h = 0.2

For i = 0 To 1 Step 0.2

k1 = f(x, y)

y1 = y + h / 2 * k1

x = x + h / 2

k2 = f(x, y1)

y2 = y + h / 2 * k2

k3 = f(x, y2)

y3 = y + h * k3

x = x + h / 2

k4 = f(x, y3)

y = y + h / 6 * (k1 + 2 * k2 + 2 * k3 + k4)

Print "x="; x; " y="; y

Next

End Sub

Private Function f(a As Single, b As Single) As Single

f = b - 2 * a / b

End Function

四：如何在Matlab中用四阶龙格库塔法解微分方程

建立.m文件 --------------------------------------------- function theta=danbai(t,X) x=X(1); dx=X(2); ddx=-sin(x); theta=[dx;ddx]; ---------------------------------------------- 命令窗口输入 >> [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') 自编龙格库塔 ------------------------------------------------ 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') %θ-t图 hold on plot(f1(x,y,z),'g') % dθ-t图 hold on plot(f2(x,y,z),'b-')%d2θ-t图 %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)

五：用matlab编程实现四阶龙格库塔解二元二阶微分方程组

求解二阶微分方程，初始条件还需要给出y1'(0)和y2'(0)。这里暂时按照0处理。

function zd530003514

a=0.1;

b=0.1;

Y0 = [b-1; 0; b; 0];

% 解方程

[t,Y]= ode45(@ode,[0 10],Y0);

y1=Y(:,1);

y2=Y(:,3);

% 绘图

subplot 211

plot(t,y1);

subplot 212

plot(t,y2);

% 微分方程定义

function dY = ode(t, Y)

L1=5;

L2=0.01;

a0=2;

b0=2;

c0=2;

y1=Y(1);y2=Y(3);

dY = [

Y(2);

-(a0*y2+b0*y2^2+c0*y2^3) - L1^2*L2*y1 - L1^2*贰1;

Y(4);

-(a0*y2+b0*y2^2+c0*y2^3) - L1^2*L2*y1;

];

六：四阶龙格库塔法的JAVA编程

Java语言的二三四阶龙格库塔程序如下:

public class D {

static double[] x=new double[6];

static double[] y=new double[6];

public static double function(double a,double b)

{

return b-2*a/b;

}

/*n表示几等分，n+1表示他输出的个数*/

public static void RungeKutta(double y0,double a,double b,int n,int style)

{

double h=(b-a)/n,k1,k2,k3,k4;

int i;

x[0]=a;

y[0]=y0;

switch(style)

{

case 2:

for(i=0;i

{

x[i+1]=x[i]+h;

k1=function(x[i],y[i]);

k2=function(x[i]+h/2,y[i]+h*k1/2);

y[i+1]=y[i]+h*k2;

}

break;

case 3:

for(i=0;i

{

x[i+1]=x[i]+h;

k1=function(x[i],y[i]);

k2=function(x[i]+h/2,y[i]+h*k1/2);

k3=function(x[i]+h,y[i]-h*k1+2*h*k2);

y[i+1]=y[i]+h*(k1+4*k2+k3)/6;

}

break;

case 4:

for(i=0;i

{

x[i+1]=x[i]+h;

k1=function(x[i],y[i]);

k2=function(x[i]+h/2,y[i]+h*k1/2);

k3=function(x[i]+h/2,y[i]+h*k2/2);

k4=function(x[i]+h,y[i]+h*k3);

y[i+1]=y[i]+h*(k1+2*k2+2*k3+k4)/6;

}

break;

}

}

public static void main(String[] args) {

//例子求y'=y-2*x/y(0

System.out.println("用二阶龙格-库塔方法");

RungeKutta(1,0,1,5,2);

for(int i=0;i<6;i++)

System.out.printf("x[%d]=%f,y[%d]=%f\n",i,x[i],i,y[i]);

System.out.println("用三阶龙格-库塔方法");

RungeKutta(1,0,1,5,3);

for(int i=0;i<6;i++)

System.out.printf("x[%d]=%f,y[%d]=%f\n",i......余下全文>>

七：用四级四阶龙格-库塔公式求解初值问题

主程序

clc;clear

x0=0;y0=1;

h=0.1;

Nstep=floor(1/h);

[x2,y2]=Rung_Kutta(@myfun,x0,y0,h,Nstep);

plot(x2,y2)

xlabel('x')

ylabel('y')

box off

grid on

函数程序1

function y1=myfun(x,y)

%% a self-defined function

y1=sqrt(x+y);

end

函数程序2

function [x,y]=Rung_Kutta(fxy,x0,y0,h,Nstep)

%% Rung_Kutta法求解微分方程

% y'=fxy y(x0)=y0

% h: 步长 Nstep: 步数

x=zeros(Nstep+1,1);

y=x;

x(1)=x0;

y(1)=y0;

for i=1:Nstep

x(i+1)=x(i)+h;

k1=fxy(x(i),y(i));

k2=fxy(x(i)+h/2,y(i)+h/2*k1);

k3=fxy(x(i)+h/2,y(i)+h/2*k2);

k4=fxy(x(i)+h,y(i)+h*k3);

y(i+1)=y(i)+h/6*(k1+2*k2+2*k3+k4);

end

end

最后结果

八：MATLAB 用四阶龙格库塔解微分方程组

可以用ode45求解。

需定义mass函数，用以计算质量矩阵，即化为下述形式求解：

M(t,y) * dy/dt = f(t,y)。

请给出ABCDEF的具体值。