def test_pde_separate_add():
x, y, z, t = symbols("x,y,z,t")
F, T, X, Y, Z, u = map(Function, 'FTXYZu')
eq = Eq(D(u(x, t), x), D(u(x, t), t) * exp(u(x, t)))
res = pde_separate_add(eq, u(x, t), [X(x), T(t)])
assert res == [D(X(x), x) * exp(-X(x)), D(T(t), t) * exp(T(t))]
def test_euler_sineg():
psi = Function('psi')
t = Symbol('t')
x = Symbol('x')
L = D(psi(t, x), t)**2 / 2 - D(psi(t, x), x)**2 / 2 + cos(psi(t, x))
assert euler(L, psi(t, x), [t, x]) == [
Eq(-sin(psi(t, x)) - D(psi(t, x), t, t) + D(psi(t, x), x, x), 0)
]
def test_issue_11726():
x, t = symbols("x t")
f = symbols("f", cls=Function)
X, T = symbols("X T", cls=Function)
u = f(x, t)
eq = u.diff(x, 2) - u.diff(t, 2)
res = pde_separate(eq, u, [T(x), X(t)])
assert res == [D(T(x), x, x) / T(x), D(X(t), t, t) / X(t)]
def test_euler_henonheiles():
x = Function('x')
y = Function('y')
t = Symbol('t')
L = sum(D(z(t), t)**2 / 2 - z(t)**2 / 2 for z in [x, y])
L += -x(t)**2 * y(t) + y(t)**3 / 3
assert euler(L, [x(t), y(t)], t) == [
Eq(-2 * x(t) * y(t) - x(t) - D(x(t), t, t), 0),
Eq(-x(t)**2 + y(t)**2 - y(t) - D(y(t), t, t), 0)
]
def test_euler_interface():
x = Function('x')
y = Symbol('y')
t = Symbol('t')
raises(TypeError, lambda: euler())
raises(TypeError, lambda: euler(D(x(t), t) * y(t), [x(t), y]))
raises(ValueError, lambda: euler(D(x(t), t) * x(y), [x(t), x(y)]))
raises(TypeError, lambda: euler(D(x(t), t)**2, x(0)))
raises(TypeError, lambda: euler(D(x(t), t) * y(t), [t]))
assert euler(D(x(t), t)**2 / 2, {x(t)}) == [Eq(-D(x(t), t, t), 0)]
assert euler(D(x(t), t)**2 / 2, x(t), {t}) == [Eq(-D(x(t), t, t), 0)]
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)
]
def test_pde_separate():
x, y, z, t = symbols("x,y,z,t")
F, T, X, Y, Z, u = map(Function, 'FTXYZu')
eq = Eq(D(u(x, t), x), D(u(x, t), t) * exp(u(x, t)))
raises(ValueError, lambda: pde_separate(eq, u(x, t), [X(x), T(t)], 'div'))
def test_euler_high_order():
# an example from hep-th/0309038
m = Symbol('m')
k = Symbol('k')
x = Function('x')
y = Function('y')
t = Symbol('t')
L = (m * D(x(t), t)**2 / 2 + m * D(y(t), t)**2 / 2 -
k * D(x(t), t) * D(y(t), t, t) + k * D(y(t), t) * D(x(t), t, t))
assert euler(L, [x(t), y(t)]) == [
Eq(2 * k * D(y(t), t, t, t) - m * D(x(t), t, t), 0),
Eq(-2 * k * D(x(t), t, t, t) - m * D(y(t), t, t), 0)
]
w = Symbol('w')
L = D(x(t, w), t, w)**2 / 2
assert euler(L) == [Eq(D(x(t, w), t, t, w, w), 0)]
def test_euler_pendulum():
x = Function('x')
t = Symbol('t')
L = D(x(t), t)**2 / 2 + cos(x(t))
assert euler(L, x(t), t) == [Eq(-sin(x(t)) - D(x(t), t, t), 0)]