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Differential equations can be solved with different methods in Python. Below are examples that show how to solve differential equations with (1) GEKKO Python, (2) Euler's method, (3) the ODEINT function from Scipy.Integrate. Additional information is provided on using APM Python for parameter estimation with dynamic models and scale-up to large-scale problems.

GEKKO Python solves the differential equations with tank overflow conditions. When the first tank overflows, the liquid is lost and does not enter tank 2. The model is composed of variables and equations. The differential variables (h1 and h2) are solved with a mass balance on both tanks.

See [[https://apmonitor.com/pdc/index.php/Main/SolveDifferentialEquations|Introduction to Using ODEINT]] for more information on solving differential equations with SciPy.

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

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!!!! 2. SciPy.Integrate ODEINT Function

!!!! 3. APM Python DAE Integrator and Optimizer

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

 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)

 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()

This 5 minute 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. The tutorial covers the same problem in both MATLAB and Python.

* [[Attach:dynamics.zip|Dynamic Simulation Files (dynamics.zip)]]

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The Python package Scipy offers several solvers to numerically simulate the solution of sets of differential equations. 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.

!!!! Source Code

!!!! DAE Integrator and Optimizer

This same example problem is also demonstrated with [[Main/ExcelEulers|Spreadsheet Programming]] and in the [[Main/MatlabDynamicSim|Matlab programming language]].

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