在 python 中用 ODEINT 求解微分方程

Question

I have a coupled system of differential equations that I've already solved with Euler in Excel. Now I want to make it more precise with an ODE-solver in python. However, there must be a mistake in my code because the curves look different than in Excel. I don't expect the curves to reach 1 and 0 in the end.

import numpy as np
from scipy.integrate import odeint
import matplotlib.pyplot as plt

# define reactor
def reactor(x,z):
    n_a = x[0]
    n_b = x[1]
    n_c = x[2]

    dn_adz = A * (-1) * B * (n_a/(n_a + n_b + n_c)) / (1 + C * (n_c/(n_a + n_b + n_c)))
    dn_bdz = A * (1) * B * (n_a/(n_a + n_b + n_c)) / (1 + C * (n_c/(n_a + n_b + n_c)))
    dn_cdz = A * (1) * B * (n_a/(n_a + n_b + n_c)) / (1 + C * (n_c/(n_a + n_b + n_c)))
    dxdz = [dn_adz,dn_bdz,dn_cdz]
    return dxdz

# initial conditions
n_a0 = 0.5775
n_b0 = 0.0
n_c0 = 0.0
x0 = [n_a0, n_b0, n_c0]

# parameters
A = 0.12
B = 3.1e-9
C = 4.02e15

# number of steps
n = 100

# z step interval (m)
z = np.linspace(0,0.0274,n)

# solve ODEs
x = odeint(reactor,x0,z)


# Plot the results
plt.plot(z,x[:,0],'b-')
plt.plot(z,x[:,1],'r--')
plt.plot(z,x[:,2],'k:')
plt.show()

Is is a problem with the initial condition that stays constant and does not change from step to step? Should it be like in Excel with Euler, where the next step uses the conditions/values of the precious step?

Answer 1

From the structure of the right sides you get constant combinations of the state variables, n_a+n_b=n_a0+n_b0 and n_a+n_c=n_a0+n_c0 . This means that the dynamic reduces to the one-dimensional dynamic of n_a .

By the first equation, the derivative of n_a is negative for positive n_a , so that the solution is falling towards n_a=0 . By the constants of the dynamics, n_b converges to n_a0+n_b0 and n_c converges to n_a0+n_c0 .

It is unclear how you get convergence towards 1 in some components, as that is not supported by the initial conditions. Apart from that, the described odeint result fits this qualitative behavior.