How to: Solve an ODE in Python (Boundary Value Problem)

Learn out to numerically solve an ordinary differential equation (ODE) in Python using a built in solver for boundary value problems: "scipy.integrate.solve_bvp()".
Here I discuss a 2nd order ODE with 2 boundary conditions (i.e. boundary value problem - BVP). A 1D steady-state heat transfer problem is used as an example, with various boundary conditions (fixed temperature, radiation) and with the option for heat generation. In order to solve the BPV, the 2nd order ODE must be converted into a system of 1st order ODEs in the format required by "scipy.integrate.solve_bvp()".
The Python script and PDF of the notes can be found here: www.hageslab.com/Resources.ht...
Here we are using "Spyder" IDE with the numpy, scipy, and matplotlib libraries
Script (for the radiation example):
import numpy as np #For basic math functions
import scipy.integrate as intg #For advanced math functions
import matplotlib.pyplot as plt #For plotting
#Define Constants
sigma = 5.67e-8 #[W/m^2/K^4]
#Set-up Paramters
qgen = 0 #[W/m**3]
k = 40 #[W/m/K]
T1 = 273.15 #[K]
epsilon = 0.8
Tsurr = 500+273.15 #[K]
#Set-up Grid
L = 1 #[m]
nodes = 100
x = np.linspace(0,L,nodes)
#Define Equations & Boundary Conditions
def f(x,y):
return np.vstack((y[1],np.full_like(x,-qgen/k)))
def bc(ya,yb):
return np.array([ya[0]-T1,k*yb[1]+sigma*epsilon*(yb[0]**4-Tsurr**4)])
#Inital Guess
y0 = np.zeros((2,x.size))
#Solve
sol = intg.solve_bvp(f,bc,x,y0)
T = sol.y[0]
dTdX = sol.y[1]
#Surface Temps
TS1 = T[0]
TS2 = T[-1]
#Compute Flux
q = -k*dTdX
#Plot
plt.figure(1,dpi=120)
plt.yscale('linear')
plt.xscale('linear')
#plt.xlim(0,1)
plt.ylim(273.15,500+273.15)
plt.title("Temperature Profile")
plt.xlabel("Distance / m")
plt.ylabel("T / K")
plt.plot(x,T)
plt.figure(2,dpi=120)
plt.yscale('linear')
plt.xscale('linear')
#plt.xlim(0,0.1)
#plt.ylim(0,100)
plt.title("Heat Flux")
plt.xlabel("Distance / m")
plt.ylabel("Flux / kW m$^{-1}$ K$^{-1}$")
plt.plot(x,q/1000)

[@andrewsawyer5003]

This is my favorite Minecraft lets play series, keep up the good content

[@Just_Jabi]

Incredible video, nice and clear explanation. 10/10

[@zzzzzz3690]

thank you for video. it was so easy to understand how to use this function. really perfect you know, I had a problem with code, and I didn't know what I need to do to solve. the problem was in boundary conditions. It really easy to solve my problem)

[@scottfinefrock1096]

Thanks for excellent video. Flux is in W/m2.

[@user-lt1jk4cm1k]

how do you solve a 3 order ODE, is it the same as this since we have 3 dimensions for y? and the boundary conditions also double.

[@bellgrooversmeetup5570]

is there a way to solve these type of problems when you also have boundary conditions for the first derivative? e.g. a function that starts at (T, T') = (T0, 0) and ends at (T, T') = (0, 0) . That would mean 4 boundary conditions for a 1D problem, is it possible?

[@andrewl9797]

I LOVE YOU SO MUCH THANK YOU THANK YOU THANK YOU THANK YOU THANK YOU