Solve Differential Equations in Python
Main.PythonDynamicSim History
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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.
to:
Differential equations can be solved with different methods in Python. Below are examples that show how to solve differential equations with (1) [[https://apmonitor.com/pdc/index.php/Main/PythonDifferentialEquations|GEKKO Python]], (2) Euler's method, (3) the [[https://apmonitor.com/pdc/index.php/Main/SolveDifferentialEquations|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.
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----
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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.
to:
See [[https://apmonitor.com/pdc/index.php/Main/PythonDifferentialEquations|Introduction to GEKKO]] for more information on solving differential equations in Python. 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.
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See [[https://apmonitor.com/pdc/index.php/Main/SolveDifferentialEquations|Introduction to Using ODEINT]] for more information on solving differential equations with SciPy.
to:
See [[https://apmonitor.com/pdc/index.php/Main/SolveDifferentialEquations|Introduction to ODEINT]] for more information on solving differential equations with SciPy.
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Attach:tank_height_odeint.png
!!!! 1. GEKKO Python
!!!! 1. GEKKO Python
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Attach:tank_height_odeint.png
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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.
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to:
(:source lang=python:)
import numpy as np
import matplotlib.pyplot as plt
from gekko import GEKKO
m = GEKKO()
# integration time points
m.time = np.linspace(0,10)
# constants
c1 = 0.13
c2 = 0.20
Ac = 2 # m^2
# inflow
qin1 = 0.5 # m^3/hr
# variables
h1 = m.Var(value=0,lb=0,ub=1)
h2 = m.Var(value=0,lb=0,ub=1)
overflow1 = m.Var(value=0,lb=0)
overflow2 = m.Var(value=0,lb=0)
# outflow equations
qin2 = m.Intermediate(c1 * h1**0.5)
qout1 = m.Intermediate(qin2 + overflow1)
qout2 = m.Intermediate(c2 * h2**0.5 + overflow2)
# mass balance equations
m.Equation(Ac*h1.dt()==qin1-qout1)
m.Equation(Ac*h2.dt()==qin2-qout2)
# minimize overflow
m.Obj(overflow1+overflow2)
# set options
m.options.IMODE = 6 # dynamic optimization
# simulate differential equations
m.solve()
# plot results
plt.figure(1)
plt.plot(m.time,h1,'b-')
plt.plot(m.time,h2,'r--')
plt.xlabel('Time (hrs)')
plt.ylabel('Height (m)')
plt.legend(['height 1','height 2'])
plt.show()
(:sourceend:)
import numpy as np
import matplotlib.pyplot as plt
from gekko import GEKKO
m = GEKKO()
# integration time points
m.time = np.linspace(0,10)
# constants
c1 = 0.13
c2 = 0.20
Ac = 2 # m^2
# inflow
qin1 = 0.5 # m^3/hr
# variables
h1 = m.Var(value=0,lb=0,ub=1)
h2 = m.Var(value=0,lb=0,ub=1)
overflow1 = m.Var(value=0,lb=0)
overflow2 = m.Var(value=0,lb=0)
# outflow equations
qin2 = m.Intermediate(c1 * h1**0.5)
qout1 = m.Intermediate(qin2 + overflow1)
qout2 = m.Intermediate(c2 * h2**0.5 + overflow2)
# mass balance equations
m.Equation(Ac*h1.dt()==qin1-qout1)
m.Equation(Ac*h2.dt()==qin2-qout2)
# minimize overflow
m.Obj(overflow1+overflow2)
# set options
m.options.IMODE = 6 # dynamic optimization
# simulate differential equations
m.solve()
# plot results
plt.figure(1)
plt.plot(m.time,h1,'b-')
plt.plot(m.time,h2,'r--')
plt.xlabel('Time (hrs)')
plt.ylabel('Height (m)')
plt.legend(['height 1','height 2'])
plt.show()
(:sourceend:)
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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) Euler's method, (2) the ODEINT function from Scipy.Integrate, and (3) APM Python.
to:
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.
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!!!! 1. Discretize with Euler's Method
to:
!!!! 1. GEKKO Python
!!!! 2. Discretize with Euler's Method
!!!! 2. Discretize with Euler's Method
Changed lines 66-69 from:
(:sourceend:0
!!!! 2. SciPy.Integrate ODEINT Function
to:
(:sourceend:)
!!!! 3. SciPy.Integrate ODEINT Function
!!!! 3. SciPy.Integrate ODEINT Function
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!!!! 3. APM Python DAE Integrator and Optimizer
to:
!!!! APM Python DAE Integrator and Optimizer
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!!!! 4. ODEINT Scale-up for Large Sets of Equations
to:
!!!! Scale-up for Large Sets of Equations
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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)
to:
(:source lang=python:)
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)
(:sourceend:0
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)
(:sourceend:0
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See [[https://apmonitor.com/pdc/index.php/Main/SolveDifferentialEquations|Introduction to Using ODEINT]] for more information on solving differential equations with SciPy.
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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()
to:
(:source lang=python:)
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()
(:sourceend:)
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()
(:sourceend:)
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(:htmlend:)
!!!! 4. ODEINT Scale-up for Large Sets of Equations
(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/8kx6vC9gTLo" frameborder="0" allowfullscreen></iframe>
!!!! 4. ODEINT Scale-up for Large Sets of Equations
(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/8kx6vC9gTLo" frameborder="0" allowfullscreen></iframe>
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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)]]
(:html:)
<iframe width="560" height="315" src="//www.youtube.com/embed/-IDTagajoyA?rel=0" frameborder="0" allowfullscreen></iframe>
(:htmlend:)
----
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.
* [[Attach:dynamics.zip|Dynamic Simulation Files (dynamics.zip)]]
(:html:)
<iframe width="560" height="315" src="//www.youtube.com/embed/-IDTagajoyA?rel=0" frameborder="0" allowfullscreen></iframe>
(:htmlend:)
----
The Python package Scipy offers several solvers to numerically simulate the solution of sets of differential equations
to:
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.
* [[Attach:dynamic_estimation.zip|Dynamic Estimation Files (dynamic_estimation.zip)]]
* [[Attach:dynamic_estimation.zip|Dynamic Estimation Files (dynamic_estimation.zip)]]
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(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/U7uyj9BaNKg" frameborder="0" allowfullscreen></iframe>
(:htmlend:)
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* [[Attach:gravity_tanks.pdf|Gravity Drained Tank Problem]]
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to:
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()
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()
Changed line 18 from:
!!!! Source Code
to:
!!!! Euler's Method Source Code
Added lines 5-8:
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
!!!! 1. Discretize with Euler's Method
Changed lines 61-62 from:
to:
!!!! 2. SciPy.Integrate ODEINT Function
!!!! 3. APM Python DAE Integrator and Optimizer
!!!! 3. APM Python DAE Integrator and Optimizer
Added lines 13-56:
!!!! 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)
Added lines 4-14:
Euler's method is used to solve a set of two differential equations in Excel and Python.
* [[Attach:gravity_tanks.pdf|Gravity Drained Tank Problem]]
* [[Attach:gravity_tank_files.zip|Gravity Drained Tank Files]]
(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/ygoohjN_Lww" frameborder="0" allowfullscreen></iframe>
(:htmlend:)
!!!! DAE Integrator and Optimizer
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to:
!!!! Additional Material
This same example problem is also demonstrated with [[Main/ExcelEulers|Spreadsheet Programming]] and in the [[Main/MatlabDynamicSim|Matlab programming language]]. Another example problem demonstrates how to calculate the concentration of CO gas buildup in a room.
* %list list-page% [[https://apmonitor.com/wiki/index.php/Apps/CarbonMonoxide | Case Study on CO Buildup in a Room]]
This same example problem is also demonstrated with [[Main/ExcelEulers|Spreadsheet Programming]] and in the [[Main/MatlabDynamicSim|Matlab programming language]]. Another example problem demonstrates how to calculate the concentration of CO gas buildup in a room.
* %list list-page% [[https://apmonitor.com/wiki/index.php/Apps/CarbonMonoxide | Case Study on CO Buildup in a Room]]
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<iframe width="560" height="315" src="//www.youtube.com/embed/YvjG2LRNtKU" frameborder="0" allowfullscreen></iframe>
to:
<iframe width="560" height="315" src="//www.youtube.com/embed/-IDTagajoyA?rel=0" frameborder="0" allowfullscreen></iframe>
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* [[Attach:dynamics.zip|Dynamic Simulation Files (dynamics.zip)]]
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* [[Attach:dynamic_estimation.zip|Dynamic Estimation Files (dynamic_estimation.zip)]]
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<iframe width="560" height="315" src="//www.youtube.com/embed/YvjG2LRNtKU" frameborder="0" allowfullscreen></iframe>
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(:title Solve Differential Equations in Python:)
(:keywords introduction, Python, programming language, differential equations, nonlinear, university course:)
(:description Solve Differential Equations in Python - Problem-Solving Techniques for Chemical Engineers at Brigham Young University:)
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.
(:html:)
<iframe width="560" height="315" src="//www.youtube.com/embed/YvjG2LRNtKU" frameborder="0" allowfullscreen></iframe>
(:htmlend:)
----
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.
(:html:)
(:htmlend:)
This same example problem is also demonstrated with [[Main/ExcelEulers|Spreadsheet Programming]] and in the [[Main/MatlabDynamicSim|Matlab programming language]].
----
(:html:)
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(:htmlend:)
(:keywords introduction, Python, programming language, differential equations, nonlinear, university course:)
(:description Solve Differential Equations in Python - Problem-Solving Techniques for Chemical Engineers at Brigham Young University:)
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.
(:html:)
<iframe width="560" height="315" src="//www.youtube.com/embed/YvjG2LRNtKU" frameborder="0" allowfullscreen></iframe>
(:htmlend:)
----
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.
(:html:)
(:htmlend:)
This same example problem is also demonstrated with [[Main/ExcelEulers|Spreadsheet Programming]] and in the [[Main/MatlabDynamicSim|Matlab programming language]].
----
(:html:)
<div id="disqus_thread"></div>
<script type="text/javascript">
/* * * CONFIGURATION VARIABLES: EDIT BEFORE PASTING INTO YOUR WEBPAGE * * */
var disqus_shortname = 'apmonitor'; // required: replace example with your forum shortname
/* * * DON'T EDIT BELOW THIS LINE * * */
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