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(-Integer(12) * x / 13 - Integer(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 / Integer(85) - 7 * y / Integer(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_classify():
# When more number of hints are added, add tests for classifying here.
f, g = symbols('f g', cls=Function)
u = f(x, y)
eq1 = a*u + b*u.diff(x) + c*u.diff(y)
eq2 = 3*u + 2*u.diff(x) + u.diff(y)
eq3 = a*u + b*u.diff(x) + 2*u.diff(y)
eq4 = x*u + u.diff(x) + 3*u.diff(y)
eq5 = x**2*u + x*u.diff(x) + x*y*u.diff(y)
eq6 = y*x**2*u + y*u.diff(x) + u.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) == ('1st_linear_variable_coeff',)
# coverage tests
assert classify_pde(eq, u) == ('1st_linear_variable_coeff',)
assert classify_pde(eq, g(x, y)) == ()
assert classify_pde(eq, g(x, y), dict=True) == {'default': None, 'order': 0}
assert classify_pde(u.diff(x) + I*u.diff(y) + y) == ()
assert classify_pde(u.diff(x, y) + x) == ()
assert classify_pde(x*u.diff(x) + u*u.diff(y) + y) == ()
assert classify_pde(x*u.diff(x) + u*u.diff(y) + y,
dict=True) == {'default': None, 'ordered_hints': (),
'order': 1}
assert classify_pde(2*u.diff(x, y, y)) == ()
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) == ('1st_linear_variable_coeff', )
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 / Integer(2) + 4 * y) / Integer(15)) *
exp(x / Integer(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) / Integer(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 / Integer(5) - 3 * y / Integer(5)) -
2 * sin(x) / Integer(5) - cos(x) / Integer(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 / Integer(2) - y / Integer(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/2 + 4*y)/15)*exp(x/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)/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 == Eq(f(x, y),
-x*y + x + y + F(x - y)*exp(-x/2 - y/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_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(-12*x/13 - 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/85 - 7*y/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_classify():
# When more number of hints are added, add tests for classifying here.
f, g = symbols('f g', cls=Function)
u = f(x, y)
eq1 = a*u + b*u.diff(x) + c*u.diff(y)
eq2 = 3*u + 2*u.diff(x) + u.diff(y)
eq3 = a*u + b*u.diff(x) + 2*u.diff(y)
eq4 = x*u + u.diff(x) + 3*u.diff(y)
eq5 = x**2*u + x*u.diff(x) + x*y*u.diff(y)
eq6 = y*x**2*u + y*u.diff(x) + u.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) == ('1st_linear_variable_coeff',)
# coverage tests
assert classify_pde(eq, u) == ('1st_linear_variable_coeff',)
assert classify_pde(eq, g(x, y)) == ()
assert classify_pde(eq, g(x, y), dict=True) == {'default': None, 'order': 0}
assert classify_pde(u.diff(x) + I*u.diff(y) + y) == ()
assert classify_pde(u.diff(x, y) + x) == ()
assert classify_pde(x*u.diff(x) + u*u.diff(y) + y) == ()
assert classify_pde(x*u.diff(x) + u*u.diff(y) + y,
dict=True) == {'default': None, 'ordered_hints': (),
'order': 1}
assert classify_pde(2*u.diff(x, y, y)) == ()
eq = Eq(1 + (2*(u.diff(x)/u)) + (3*(u.diff(y)/u)))
assert classify_pde(eq) == ('1st_linear_constant_coeff_homogeneous',)
