def test_pde_separate_mul():
x, y, z, t = symbols("x,y,z,t")
c = Symbol("C", real=True)
Phi = Function("Phi")
F, R, T, X, Y, Z, u = map(Function, "FRTXYZu")
r, theta, z = symbols("r,theta,z")
# Something simple :)
eq = Eq(D(F(x, y, z), x) + D(F(x, y, z), y) + D(F(x, y, z), z))
# Duplicate arguments in functions
raises(ValueError, lambda: pde_separate_mul(eq, F(x, y, z), [X(x), u(z, z)]))
# Wrong number of arguments
raises(ValueError, lambda: pde_separate_mul(eq, F(x, y, z), [X(x), Y(y)]))
# Wrong variables: [x, y] -> [x, z]
raises(ValueError, lambda: pde_separate_mul(eq, F(x, y, z), [X(t), Y(x, y)]))
assert pde_separate_mul(eq, F(x, y, z), [Y(y), u(x, z)]) == [
D(Y(y), y) / Y(y),
-D(u(x, z), x) / u(x, z) - D(u(x, z), z) / u(x, z),
]
assert pde_separate_mul(eq, F(x, y, z), [X(x), Y(y), Z(z)]) == [
D(X(x), x) / X(x),
-D(Z(z), z) / Z(z) - D(Y(y), y) / Y(y),
]
# wave equation
wave = Eq(D(u(x, t), t, t), c ** 2 * D(u(x, t), x, x))
res = pde_separate_mul(wave, u(x, t), [X(x), T(t)])
assert res == [D(X(x), x, x) / X(x), D(T(t), t, t) / (c ** 2 * T(t))]
# Laplace equation in cylindrical coords
eq = Eq(
1 / r * D(Phi(r, theta, z), r)
+ D(Phi(r, theta, z), r, 2)
+ 1 / r ** 2 * D(Phi(r, theta, z), theta, 2)
+ D(Phi(r, theta, z), z, 2)
)
# Separate z
res = pde_separate_mul(eq, Phi(r, theta, z), [Z(z), u(theta, r)])
assert res == [
D(Z(z), z, z) / Z(z),
-D(u(theta, r), r, r) / u(theta, r)
- D(u(theta, r), r) / (r * u(theta, r))
- D(u(theta, r), theta, theta) / (r ** 2 * u(theta, r)),
]
# Lets use the result to create a new equation...
eq = Eq(res[1], c)
# ...and separate theta...
res = pde_separate_mul(eq, u(theta, r), [T(theta), R(r)])
assert res == [
D(T(theta), theta, theta) / T(theta),
-r * D(R(r), r) / R(r) - r ** 2 * D(R(r), r, r) / R(r) - c * r ** 2,
]
# ...or r...
res = pde_separate_mul(eq, u(theta, r), [R(r), T(theta)])
assert res == [
r * D(R(r), r) / R(r) + r ** 2 * D(R(r), r, r) / R(r) + c * r ** 2,
-D(T(theta), theta, theta) / T(theta),
]
def test_pde_separate_mul():
x, y, z, t = symbols("x,y,z,t")
c = Symbol("C", real=True)
Phi = Function('Phi')
F, R, T, X, Y, Z, u = map(Function, 'FRTXYZu')
r, theta, z = symbols('r,theta,z')
# Something simple :)
eq = Eq(D(F(x, y, z), x) + D(F(x, y, z), y) + D(F(x, y, z), z), 0)
# Duplicate arguments in functions
raises(ValueError,
lambda: pde_separate_mul(eq, F(x, y, z), [X(x), u(z, z)]))
# Wrong number of arguments
raises(ValueError, lambda: pde_separate_mul(eq, F(x, y, z), [X(x), Y(y)]))
# Wrong variables: [x, y] -> [x, z]
raises(ValueError,
lambda: pde_separate_mul(eq, F(x, y, z), [X(t), Y(x, y)]))
assert pde_separate_mul(eq, F(x, y, z), [Y(y), u(x, z)]) == \
[D(Y(y), y)/Y(y), -D(u(x, z), x)/u(x, z) - D(u(x, z), z)/u(x, z)]
assert pde_separate_mul(eq, F(x, y, z), [X(x), Y(y), Z(z)]) == \
[D(X(x), x)/X(x), -D(Z(z), z)/Z(z) - D(Y(y), y)/Y(y)]
# wave equation
wave = Eq(D(u(x, t), t, t), c**2 * D(u(x, t), x, x))
res = pde_separate_mul(wave, u(x, t), [X(x), T(t)])
assert res == [D(X(x), x, x) / X(x), D(T(t), t, t) / (c**2 * T(t))]
# Laplace equation in cylindrical coords
eq = Eq(
1 / r * D(Phi(r, theta, z), r) + D(Phi(r, theta, z), r, 2) +
1 / r**2 * D(Phi(r, theta, z), theta, 2) + D(Phi(r, theta, z), z, 2),
0)
# Separate z
res = pde_separate_mul(eq, Phi(r, theta, z), [Z(z), u(theta, r)])
assert res == [
D(Z(z), z, z) / Z(z),
-D(u(theta, r), r, r) / u(theta, r) - D(u(theta, r), r) /
(r * u(theta, r)) - D(u(theta, r), theta, theta) / (r**2 * u(theta, r))
]
# Lets use the result to create a new equation...
eq = Eq(res[1], c)
# ...and separate theta...
res = pde_separate_mul(eq, u(theta, r), [T(theta), R(r)])
assert res == [
D(T(theta), theta, theta) / T(theta),
-r * D(R(r), r) / R(r) - r**2 * D(R(r), r, r) / R(r) - c * r**2
]
# ...or r...
res = pde_separate_mul(eq, u(theta, r), [R(r), T(theta)])
assert res == [
r * D(R(r), r) / R(r) + r**2 * D(R(r), r, r) / R(r) + c * r**2,
-D(T(theta), theta, theta) / T(theta)
]