def test_pde_1st_linear_constant_coeff_homogeneous():
f, F = map(Function, ['f', 'F'])
u = f(x, y)
eq = 2 * u + u.diff(x) + u.diff(y)
assert classify_pde(eq) == ('1st_linear_constant_coeff_homogeneous', )
sol = pdsolve(eq)
assert sol == Eq(u, F(x - y) * exp(-x - y))
assert checkpdesol(eq, sol)[0]
eq = 4 + (3 * u.diff(x) / u) + (2 * u.diff(y) / u)
assert classify_pde(eq) == ('1st_linear_constant_coeff_homogeneous', )
sol = pdsolve(eq)
assert sol == Eq(u,
F(2 * x - 3 * y) * exp(-S(12) * x / 13 - S(8) * y / 13))
assert checkpdesol(eq, sol)[0]
eq = u + (6 * u.diff(x)) + (7 * u.diff(y))
assert classify_pde(eq) == ('1st_linear_constant_coeff_homogeneous', )
sol = pdsolve(eq)
assert sol == Eq(u, F(7 * x - 6 * y) * exp(-6 * x / S(85) - 7 * y / S(85)))
assert checkpdesol(eq, sol)[0]
eq = a * u + b * u.diff(x) + c * u.diff(y)
sol = pdsolve(eq)
assert checkpdesol(eq, sol)[0]
def test_pde_1st_linear_constant_coeff():
f, F = map(Function, ["f", "F"])
u = f(x, y)
eq = -2 * u.diff(x) + 4 * u.diff(y) + 5 * u - exp(x + 3 * y)
sol = pdsolve(eq)
assert sol == Eq(f(x, y), (F(4 * x + 2 * y) + exp(x / S(2) + 4 * y) / S(15)) * exp(x / S(2) - y))
assert classify_pde(eq) == ("1st_linear_constant_coeff", "1st_linear_constant_coeff_Integral")
assert checkpdesol(eq, sol)[0]
eq = (u.diff(x) / u) + (u.diff(y) / u) + 1 - (exp(x + y) / u)
sol = pdsolve(eq)
assert sol == Eq(f(x, y), F(x - y) * exp(-x / 2 - y / 2) + exp(x + y) / S(3))
assert classify_pde(eq) == ("1st_linear_constant_coeff", "1st_linear_constant_coeff_Integral")
assert checkpdesol(eq, sol)[0]
eq = 2 * u + -u.diff(x) + 3 * u.diff(y) + sin(x)
sol = pdsolve(eq)
assert sol == Eq(f(x, y), F(3 * x + y) * exp(x / S(5) - 3 * y / S(5)) - 2 * sin(x) / S(5) - cos(x) / S(5))
assert classify_pde(eq) == ("1st_linear_constant_coeff", "1st_linear_constant_coeff_Integral")
assert checkpdesol(eq, sol)[0]
eq = u + u.diff(x) + u.diff(y) + x * y
sol = pdsolve(eq)
assert sol == Eq(f(x, y), -x * y + x + y + F(x - y) * exp(-x / S(2) - y / S(2)) - 2)
assert classify_pde(eq) == ("1st_linear_constant_coeff", "1st_linear_constant_coeff_Integral")
assert checkpdesol(eq, sol)[0]
eq = u + u.diff(x) + u.diff(y) + log(x)
assert classify_pde(eq) == ("1st_linear_constant_coeff", "1st_linear_constant_coeff_Integral")
def test_pde_1st_linear_constant_coeff():
f, F = map(Function, ["f", "F"])
u = f(x, y)
eq = -2 * u.diff(x) + 4 * u.diff(y) + 5 * u - exp(x + 3 * y)
sol = pdsolve(eq)
assert sol == Eq(f(x,
y), (F(4 * x + 2 * y) + exp(x / S(2) + 4 * y) / S(15)) *
exp(x / S(2) - y))
assert classify_pde(eq) == (
"1st_linear_constant_coeff",
"1st_linear_constant_coeff_Integral",
)
assert checkpdesol(eq, sol)[0]
eq = (u.diff(x) / u) + (u.diff(y) / u) + 1 - (exp(x + y) / u)
sol = pdsolve(eq)
assert sol == Eq(f(x, y),
F(x - y) * exp(-x / 2 - y / 2) + exp(x + y) / S(3))
assert classify_pde(eq) == (
"1st_linear_constant_coeff",
"1st_linear_constant_coeff_Integral",
)
assert checkpdesol(eq, sol)[0]
eq = 2 * u + -u.diff(x) + 3 * u.diff(y) + sin(x)
sol = pdsolve(eq)
assert sol == Eq(
f(x, y),
F(3 * x + y) * exp(x / S(5) - 3 * y / S(5)) - 2 * sin(x) / S(5) -
cos(x) / S(5),
)
assert classify_pde(eq) == (
"1st_linear_constant_coeff",
"1st_linear_constant_coeff_Integral",
)
assert checkpdesol(eq, sol)[0]
eq = u + u.diff(x) + u.diff(y) + x * y
sol = pdsolve(eq)
assert sol == Eq(f(x, y),
-x * y + x + y + F(x - y) * exp(-x / S(2) - y / S(2)) - 2)
assert classify_pde(eq) == (
"1st_linear_constant_coeff",
"1st_linear_constant_coeff_Integral",
)
assert checkpdesol(eq, sol)[0]
eq = u + u.diff(x) + u.diff(y) + log(x)
assert classify_pde(eq) == (
"1st_linear_constant_coeff",
"1st_linear_constant_coeff_Integral",
)
def test_pde_classify():
# When more number of hints are added, add tests for classifying here.
f = Function('f')
eq1 = a * f(x, y) + b * f(x, y).diff(x) + c * f(x, y).diff(y)
eq2 = 3 * f(x, y) + 2 * f(x, y).diff(x) + f(x, y).diff(y)
eq3 = a * f(x, y) + b * f(x, y).diff(x) + 2 * f(x, y).diff(y)
eq4 = x * f(x, y) + f(x, y).diff(x) + 3 * f(x, y).diff(y)
eq5 = x**2 * f(x, y) + x * f(x, y).diff(x) + x * y * f(x, y).diff(y)
eq6 = y * x**2 * f(x, y) + y * f(x, y).diff(x) + f(x, y).diff(y)
for eq in [eq1, eq2, eq3]:
assert classify_pde(eq) == ('1st_linear_constant_coeff_homogeneous', )
for eq in [eq4, eq5, eq6]:
assert classify_pde(eq) == ()
def test_pde_classify():
# When more number of hints are added, add tests for classifying here.
f = Function("f")
eq1 = a * f(x, y) + b * f(x, y).diff(x) + c * f(x, y).diff(y)
eq2 = 3 * f(x, y) + 2 * f(x, y).diff(x) + f(x, y).diff(y)
eq3 = a * f(x, y) + b * f(x, y).diff(x) + 2 * f(x, y).diff(y)
eq4 = x * f(x, y) + f(x, y).diff(x) + 3 * f(x, y).diff(y)
eq5 = x ** 2 * f(x, y) + x * f(x, y).diff(x) + x * y * f(x, y).diff(y)
eq6 = y * x ** 2 * f(x, y) + y * f(x, y).diff(x) + f(x, y).diff(y)
for eq in [eq1, eq2, eq3]:
assert classify_pde(eq) == ("1st_linear_constant_coeff_homogeneous",)
for eq in [eq4, eq5, eq6]:
assert classify_pde(eq) == ()
def test_pde_1st_linear_constant_coeff():
f, F = map(Function, ['f', 'F'])
u = f(x, y)
eq = -2 * u.diff(x) + 4 * u.diff(y) + 5 * u - exp(x + 3 * y)
sol = pdsolve(eq)
assert sol == Eq(f(
x, y), (F(4 * x + 2 * y) * exp(x / 2) + exp(x + 4 * y) / 15) * exp(-y))
assert classify_pde(eq) == ('1st_linear_constant_coeff',
'1st_linear_constant_coeff_Integral')
assert checkpdesol(eq, sol)[0]
eq = (u.diff(x) / u) + (u.diff(y) / u) + 1 - (exp(x + y) / u)
sol = pdsolve(eq)
assert sol == Eq(f(x, y), F(x - y) * exp(-x / 2 - y / 2) + exp(x + y) / 3)
assert classify_pde(eq) == ('1st_linear_constant_coeff',
'1st_linear_constant_coeff_Integral')
assert checkpdesol(eq, sol)[0]
eq = 2 * u + -u.diff(x) + 3 * u.diff(y) + sin(x)
sol = pdsolve(eq)
assert sol == Eq(
f(x, y),
F(3 * x + y) * exp(x / 5 - 3 * y / 5) - 2 * sin(x) / 5 - cos(x) / 5)
assert classify_pde(eq) == ('1st_linear_constant_coeff',
'1st_linear_constant_coeff_Integral')
assert checkpdesol(eq, sol)[0]
eq = u + u.diff(x) + u.diff(y) + x * y
sol = pdsolve(eq)
assert sol.expand() == Eq(
f(x, y), x + y + (x - y)**2 / 4 - (x + y)**2 / 4 +
F(x - y) * exp(-x / 2 - y / 2) - 2).expand()
assert classify_pde(eq) == ('1st_linear_constant_coeff',
'1st_linear_constant_coeff_Integral')
assert checkpdesol(eq, sol)[0]
eq = u + u.diff(x) + u.diff(y) + log(x)
assert classify_pde(eq) == ('1st_linear_constant_coeff',
'1st_linear_constant_coeff_Integral')
def test_pde_1st_linear_constant_coeff_homogeneous():
f, F = map(Function, ["f", "F"])
u = f(x, y)
eq = 2 * u + u.diff(x) + u.diff(y)
assert classify_pde(eq) == ("1st_linear_constant_coeff_homogeneous",)
sol = pdsolve(eq)
assert sol == Eq(u, F(x - y) * exp(-x - y))
assert checkpdesol(eq, sol)[0]
eq = 4 + (3 * u.diff(x) / u) + (2 * u.diff(y) / u)
assert classify_pde(eq) == ("1st_linear_constant_coeff_homogeneous",)
sol = pdsolve(eq)
assert sol == Eq(u, F(2 * x - 3 * y) * exp(-S(12) * x / 13 - S(8) * y / 13))
assert checkpdesol(eq, sol)[0]
eq = u + (6 * u.diff(x)) + (7 * u.diff(y))
assert classify_pde(eq) == ("1st_linear_constant_coeff_homogeneous",)
sol = pdsolve(eq)
assert sol == Eq(u, F(7 * x - 6 * y) * exp(-6 * x / S(85) - 7 * y / S(85)))
assert checkpdesol(eq, sol)[0]
eq = a * u + b * u.diff(x) + c * u.diff(y)
sol = pdsolve(eq)
assert checkpdesol(eq, sol)[0]