solving system of ode using matlab

问题

I have 9 equations with a time dependent coefficient g

% MY M file
function dy =tarak(t,y)
G= 3.16;
g =  0.1*exp(-((t-200)/90).^2);
dy=zeros(9,1);
dy(1)=-2*2*y(1)+2*G*y(5)+2*g*y(7);
dy(2)=2*y(1)-2*G*y(5);
dy(3)=2*y(1)-2*g*y(7);
dy(4)=-2*y(4)+g*y(9);
dy(5)=-2*y(5)+G*(y(2)-y(1))+g*y(8);
dy(6)=-2*y(6)-G*y(9);
dy(7)=-2*y(7)+g*(y(3)-y(1))+G*y(8);
dy(8)=-G*y(7)-g*y(5);
dy(9)=G*y(6)-g*y(4);

then in command window:

[T,Y] = ode45(@tarak,[0 ,500],[0 0 1 0 0 0 0 0 0])

where coefficient G = 3.16 and g = 0.1*exp(-((t-200)/90).^2) is a time dependent coefficient and time t = 0:500; Initial condition [0 0 1 0 0 0 0 0 0].

I'm getting WRONG negative values for output y(1), y(2). Can someone pls try to solve above eqns with ode45 so that i can compare the results.

回答1:

With a simple application of RK4 I get this picture

nicely positive, with one strange initial jump in the y(1) component. But note the scale, on the whole y(1) is rather small. It seems that the system is stiff at this point, so rk45 might have problems, an implicit Runge-Kutta method would be better.

And a zoom of the initial oscillations

Python code

import numpy as np
import matplotlib.pyplot as plt
from math import exp

def dydt(t,y):
    dy = np.array(y);

    G = 3.16;
    g = 0.1*exp(-((t-200)/90)**2);

    dy[0]=-2*2*y[0]+2*G*y[4]+2*g*y[6];
    dy[1]=   2*y[0]-2*G*y[4];
    dy[2]=   2*y[0]-2*g*y[6];
    dy[3]=  -2*y[3]+  g*y[8];
    dy[4]=  -2*y[4]+  G*(y[1]-y[0])+g*y[7];
    dy[5]=  -2*y[5]-  G*y[8];
    dy[6]=  -2*y[6]+  g*(y[2]-y[0])+G*y[7];
    dy[7]=  -G*y[6]-  g*y[4];
    dy[8]=   G*y[5]-  g*y[3];
    return dy;

def RK4Step(f,x,y,h):
    k1=f(x      , y         )
    k2=f(x+0.5*h, y+0.5*h*k1)
    k3=f(x+0.5*h, y+0.5*h*k2)
    k4=f(x+    h, y+    h*k3)
    return (k1+2*(k2+k3)+k4)/6.0


t= np.linspace(0,500,200+1);
dt = t[1]-t[0];
y0=np.array([0, 0, 1, 0, 0, 0, 0, 0, 0]);

y = [y0]

for t0 in t[0:-1]:
    N=200;
    h = dt/N;
    for i in range(N):
        y0 = y0 + h*RK4Step(dydt,t0+i*h,y0,h);
    y.append(y0);

y = np.array(y);

plt.subplot(121);
plt.title("y(1)")
plt.plot(t,y[:,0],"b.--")
plt.subplot(122);
plt.title("y(2)")
plt.plot(t,y[:,1],"b-..")
plt.show()

回答2:

And in Matlab:

options = odeset('AbsTol', 1e-12);
[T,Y] = ode45(@tarak, [0, 500], [0 0 1 0 0 0 0 0 0], options);

来源：https://stackoverflow.com/questions/29384358/solving-system-of-ode-using-matlab

标签

matlab

ode

differential-equations