## 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.

#### 4. ODEINT Scale-up for Large Sets of Equations

#### Additional Material

This same example problem is also demonstrated with Spreadsheet Programming and in the Matlab programming language. Another example problem demonstrates how to calculate the concentration of CO gas buildup in a room.

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