[FIXED] Can Sympy Compute Definite Integral By Cases?

Issue

This is a heat equation integration problem.

So we have u(x,0), the initial condition, that we want to integrate, that is:

1. x, when 0 <= x <= 10
2. 20-x, when 10 <= x <= 20

Now, the problem is, if I want to calculate the integral of u(x,0) from 0 to 20, if we use hand, it should be: int_{0}^{10} x dx + int_{10}^{20} (20-x) dx

with Python' SymPy code this is what I can create, very manual (the integral of u(x,0) from 0 to 20 is represented by function g):

import numpy as np
import sympy as sm
from sympy import *
from spb import *

x = sm.symbols("x")
t = sm.symbols("t")
n = sm.symbols("n", integer=True)

L = 20
f1 = (2/L)*x*sin(n*np.pi*x/20)
f2 = (2/L)*(20-x)*sin(n*np.pi*x/20)
fint1 = sm.integrate(f1,(x,0,10))
fint2 = sm.integrate(f2,(x,10,20))

D = 0.475
g = (fint1+fint2)*sin(n*np.pi*x/20)*exp(-(n**2)*(np.pi**2)*D*t/400).nsimplify()

is there a SymPy' technique of integration for this case, so I do not need to write manually fint1, fint2, ... ?

Solution

You can use Piecewise for this:

In [8]: f = (S(2)/L)*sin(n*pi*x/20)*Piecewise((x, (0 <= x) & (x <= 10)), (20-x, (10 < x) & (x <= 20)))

In [9]: f
Out[9]: 
⎛⎧  x     for x ≥ 0 ∧ x ≤ 10 ⎞    ⎛π⋅n⋅x⎞
⎜⎨                           ⎟⋅sin⎜─────⎟
⎝⎩20 - x  for x ≤ 20 ∧ x > 10⎠    ⎝ 20  ⎠
─────────────────────────────────────────
                   10                    

In [10]: piecewise_fold(f)
Out[10]: 
⎧        ⎛π⋅n⋅x⎞                         
⎪   x⋅sin⎜─────⎟                         
⎪        ⎝ 20  ⎠                         
⎪   ────────────      for x ≥ 0 ∧ x ≤ 10 
⎪        10                              
⎨                                        
⎪            ⎛π⋅n⋅x⎞                     
⎪(20 - x)⋅sin⎜─────⎟                     
⎪            ⎝ 20  ⎠                     
⎪───────────────────  for x ≤ 20 ∧ x > 10
⎩        10                              

Now you can integrate the Piecewise expression once:

In [11]: integrate(f, (x, 0, 20))
Out[11]: 
⎧      ⎛π⋅n⎞           
⎪80⋅sin⎜───⎟           
⎪      ⎝ 2 ⎠           
⎪───────────  for n ≠ 0
⎨    2  2              
⎪   π ⋅n               
⎪                      
⎪     0       otherwise
⎩  

Answered By - Oscar Benjamin