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#!/usr/bin/env python
"""Partial Differential Equations example
Demonstrates various ways to solve partial differential equations
"""
from sympy import symbols, Eq, Function, pde_separate, pprint, sin, cos, latex
from sympy import Derivative as D
def main():
r, phi, theta = symbols("r,phi,theta")
Xi = Function('Xi')
R, Phi, Theta, u = map(Function, ['R', 'Phi', 'Theta', 'u'])
C1, C2 = symbols('C1,C2')
pprint("Separation of variables in Laplace equation in spherical coordinates")
pprint("Laplace equation in spherical coordinates:")
eq = Eq(D(Xi(r, phi, theta), r, 2) + 2/r * D(Xi(r, phi, theta), r) +
1/(r**2 * sin(phi)**2) * D(Xi(r, phi, theta), theta, 2) +
cos(phi)/(r**2 * sin(phi)) * D(Xi(r, phi, theta), phi) +
1/r**2 * D(Xi(r, phi, theta), phi, 2))
pprint(eq)
pprint("We can either separate this equation in regards with variable r:")
res_r = pde_separate(eq, Xi(r, phi, theta), [R(r), u(phi, theta)])
pprint(res_r)
pprint("Or separate it in regards of theta:")
res_theta = pde_separate(eq, Xi(r, phi, theta), [Theta(theta), u(r, phi)])
pprint(res_theta)
res_phi = pde_separate(eq, Xi(r, phi, theta), [Phi(phi), u(r, theta)])
pprint("But we cannot separate it in regards of variable phi: ")
pprint("Result: %s" % res_phi)
pprint("\n\nSo let's make theta dependent part equal with -C1:")
eq_theta = Eq(res_theta[0], -C1)
pprint(eq_theta)
pprint("\nThis also means that second part is also equal to -C1:")
eq_left = Eq(res_theta[1], -C1)
pprint(eq_left)
pprint("\nLets try to separate phi again :)")
res_theta = pde_separate(eq_left, u(r, phi), [Phi(phi), R(r)])
pprint("\nThis time it is successful:")
pprint(res_theta)
pprint("\n\nSo our final equations with separated variables are:")
pprint(eq_theta)
pprint(Eq(res_theta[0], C2))
pprint(Eq(res_theta[1], C2))
if __name__ == "__main__":
main()
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