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Solve Differential Equations in Python

Differential equations can be solved with different methods in Python. Below are examples that show how to solve differential equations with (1) Euler's method, (2) the ODEINT function from Scipy.Integrate, and (3) APM Python.

1. Discretize with Euler's Method

Euler's method is used to solve a set of two differential equations in Excel and Python.

Euler's Method Source Code

 import numpy as np
import matplotlib.pyplot as plt

def tank(c1,c2):
Ac = 2 # m^2
qin = 0.5 # m^3/hr
dt = 0.5 # hr
tf = 10.0 # hr

h1 = 0
h2 = 0
t = 0
ts = np.empty(21)
h1s = np.empty(21)
h2s = np.empty(21)
i = 0
while t<=10.0:
ts[i] = t
h1s[i] = h1
h2s[i] = h2

qout1 = c1 * pow(h1,0.5)
qout2 = c2 * pow(h2,0.5)
h1 = (qin-qout1)*dt/Ac + h1
if h1>1:
h1 = 1
h2 = (qout1-qout2)*dt/Ac + h2
i = i + 1
t = t + dt

# plot data
plt.figure(1)
plt.plot(ts,h1s)
plt.plot(ts,h2s)
plt.xlabel("Time (hrs)")
plt.ylabel("Height (m)")
plt.show()

# call function
tank(0.13,0.20)


2. SciPy.Integrate ODEINT Function

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

def tank(h,t):
# constants
c1 = 0.13
c2 = 0.20
Ac = 2      # m^2
# inflow
qin = 0.5   # m^3/hr
# outflow
qout1 = c1 * h[0]**0.5
qout2 = c2 * h[1]**0.5
# differential equations
dhdt1 = (qin   - qout1) / Ac
dhdt2 = (qout1 - qout2) / Ac
# overflow conditions
if h[0]>=1 and dhdt1>=0:
dhdt1 = 0
if h[1]>=1 and dhdt2>=0:
dhdt2 = 0
dhdt = [dhdt1,dhdt2]
return dhdt

# integrate the equations
t = np.linspace(0,10) # times to report solution
h0 = [0,0]            # initial conditions for height
y = odeint(tank,h0,t) # integrate

# plot results
plt.figure(1)
plt.plot(t,y[:,0],'b-')
plt.plot(t,y[:,1],'r--')
plt.xlabel('Time (hrs)')
plt.ylabel('Height (m)')
plt.legend(['h1','h2'])
plt.show()


3. APM Python DAE Integrator and Optimizer

This tutorial gives step-by-step instructions on how to simulate dynamic systems. Dynamic systems may have differential and algebraic equations (DAEs) or just differential equations (ODEs) that cause a time evolution of the response. Below is an example of solving a first-order decay with the APM solver in Python. The objective is to fit the differential equation solution to data by adjusting unknown parameters until the model and measured values match.