Solving system of coupled differential equations using Runge-Kutta in python

Question

This python code can solve one non- coupled differential equation:

import numpy as np
import matplotlib.pyplot as plt 
import numba
import time
start_time = time.clock()


@numba.jit()
# A sample differential equation "dy / dx = (x - y**2)/2" 
def dydx(x, y): 
    return ((x - y**2)/2) 

# Finds value of y for a given x using step size h 
# and initial value y0 at x0. 
def rungeKutta(x0, y0, x, h): 
    # Count number of iterations using step size or 
    # step height h 
    n = (int)((x - x0)/h)  
    # Iterate for number of iterations 
    y = y0 
    for i in range(1, n + 1): 
        "Apply Runge Kutta Formulas to find next value of y"
        k1 = h * dydx(x0, y) 
        k2 = h * dydx(x0 + 0.5 * h, y + 0.5 * k1) 
        k3 = h * dydx(x0 + 0.5 * h, y + 0.5 * k2) 
        k4 = h * dydx(x0 + h, y + k3) 
  
        # Update next value of y 
        y = y + (1.0 / 6.0)*(k1 + 2 * k2 + 2 * k3 + k4) 
  
        # Update next value of x 
        x0 = x0 + h 
    return y 

def dplot(start,end,steps):
    Y=list()
    for x in np.linspace(start,end,steps):
        Y.append(rungeKutta(x0, y, x , h))
    plt.plot(np.linspace(start,end,steps),Y)
    print("Execution time:",time.clock() - start_time, "seconds")
    plt.show()

start,end = 0, 10
steps = end* 100
x0 = 0
y = 1
h = 0.002

dplot(start,end,steps)

This code can solve this differential equation:

    dydx= (x - y**2)/2

Now I have a system of coupled differential equations:

    dydt= (x - y**2)/2
    dxdt= x*3 + 3y

How can I implement these two as a system of coupled differential equations in the above code? Is there any more generalized way for system of n-number of coupled differential equations?

Your code should work just fine for n-coupled differential equations, just make sure your initial value is a np.array and dxdy similarly returns a np.array — Peter Meisrimel
I didn't understand your answer, can you edit the code for the second example mentioned above? (For initial values just assume a number). — Amirhosein Rezaee
def dxdy(x, y): return np.array([..., ...]) and y = np.array([.., ...]) are the only changes you should need. Plotting will need to be adjusted though. See also: stackoverflow.com/questions/63606503/… — Peter Meisrimel
Also, you should only call rungeKutta once and store the solution inside the function, rather than calling it steps times. — Peter Meisrimel
Your (last) system does not make sense. Did you mean dy/dt = ... and dx/dt = ? In an ODE system you only have one independent variable. — Lutz Lehmann

Amirhosein Rezaee Amirhosein Rezaee · Accepted Answer · 2020-09-10T20:23:17

With the help of others, I got to this:

import numpy as np
from math import sqrt
import matplotlib.pyplot as plt
import numba
import time
start_time = time.clock()

a=1
b=1
c=1
d=1

# Equations:
@numba.jit()

#du/dt=V(u,t)
def V(u,t):
    x, y, vx, vy = u
    return np.array([vy,vx,a*x+b*y,c*x+d*y])

def rk4(f, u0, t0, tf , n):
    t = np.linspace(t0, tf, n+1)
    u = np.array((n+1)*[u0])
    h = t[1]-t[0]
    for i in range(n):
        k1 = h * f(u[i], t[i])    
        k2 = h * f(u[i] + 0.5 * k1, t[i] + 0.5*h)
        k3 = h * f(u[i] + 0.5 * k2, t[i] + 0.5*h)
        k4 = h * f(u[i] + k3, t[i] + h)
        u[i+1] = u[i] + (k1 + 2*(k2 + k3 ) + k4) / 6
    return u, t

u, t  = rk4(V, np.array([1., 0., 1. , 0.]) , 0. , 10. , 100000)
x,y, vx,vy  = u.T
# plt.plot(t, x, t,y)
plt.semilogy(t, x, t,y)
plt.grid('on')
print("Execution time:",time.clock() - start_time, "seconds")
plt.show()

Solving system of coupled differential equations using Runge-Kutta in python

1 Answers