Example #1
0
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]
Example #2
0
def test_pdsolve_variable_coeff():
    f, F = map(Function, ['f', 'F'])
    u = f(x, y)
    eq = x * (u.diff(x)) - y * (u.diff(y)) + y**2 * u - y**2
    sol = pdsolve(eq, hint="1st_linear_variable_coeff")
    assert sol == Eq(u, F(x * y) * exp(y**2 / 2) + 1)
    assert checkpdesol(eq, sol)[0]

    eq = x**2 * u + x * u.diff(x) + x * y * u.diff(y)
    sol = pdsolve(eq, hint='1st_linear_variable_coeff')
    assert sol == Eq(u, F(y * exp(-x)) * exp(-x**2 / 2))
    assert checkpdesol(eq, sol)[0]

    eq = y * x**2 * u + y * u.diff(x) + u.diff(y)
    sol = pdsolve(eq, hint='1st_linear_variable_coeff')
    assert sol == Eq(u, F(-2 * x + y**2) * exp(-x**3 / 3))
    assert checkpdesol(eq, sol)[0]

    eq = exp(x)**2 * (u.diff(x)) + y
    sol = pdsolve(eq, hint='1st_linear_variable_coeff')
    assert sol == Eq(u, y * exp(-2 * x) / 2 + F(y))
    assert checkpdesol(eq, sol)[0]

    eq = exp(2 * x) * (u.diff(y)) + y * u - u
    sol = pdsolve(eq, hint='1st_linear_variable_coeff')
    assert sol == Eq(u, F(x) * exp(-y * (y - 2) * exp(-2 * x) / 2))
Example #3
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")
Example #4
0
def test_pdsolve_variable_coeff():
    f, F = map(Function, ['f', 'F'])
    u = f(x, y)
    eq = x*(u.diff(x)) - y*(u.diff(y)) + y**2*u - y**2
    sol = pdsolve(eq, hint="1st_linear_variable_coeff")
    assert sol == Eq(u, F(x*y)*exp(y**2/2) + 1)
    assert checkpdesol(eq, sol)[0]

    eq = x**2*u + x*u.diff(x) + x*y*u.diff(y)
    sol = pdsolve(eq, hint='1st_linear_variable_coeff')
    assert sol == Eq(u, F(y*exp(-x))*exp(-x**2/2))
    assert checkpdesol(eq, sol)[0]

    eq = y*x**2*u + y*u.diff(x) + u.diff(y)
    sol = pdsolve(eq, hint='1st_linear_variable_coeff')
    assert sol == Eq(u, F(-2*x + y**2)*exp(-x**3/3))
    assert checkpdesol(eq, sol)[0]

    eq = exp(x)**2*(u.diff(x)) + y
    sol = pdsolve(eq, hint='1st_linear_variable_coeff')
    assert sol == Eq(u, y*exp(-2*x)/2 + F(y))
    assert checkpdesol(eq, sol)[0]

    eq = exp(2*x)*(u.diff(y)) + y*u - u
    sol = pdsolve(eq, hint='1st_linear_variable_coeff')
    assert sol == Eq(u, exp((-y**2 + 2*y + 2*F(x))*exp(-2*x)/2))
Example #5
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",
    )
Example #6
0
def test_checkpdesol():
    f, F = map(Function, ['f', '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)
    for eq in [eq1, eq2, eq3]:
        assert checkpdesol(eq, pdsolve(eq))[0]
    eq4 = x*f(x,y) + f(x,y).diff(x) + 3*f(x,y).diff(y)
    eq5 = 2*f(x,y) + 1*f(x,y).diff(x) + 3*f(x,y).diff(y)
    eq6 = f(x,y) + 1*f(x,y).diff(x) + 3*f(x,y).diff(y)
    assert checkpdesol(eq4, [pdsolve(eq5), pdsolve(eq6)]) == [
        (False, (x - 2)*F(3*x - y)*exp(-x/S(5) - 3*y/S(5))),
         (False, (x - 1)*F(3*x - y)*exp(-x/S(10) - 3*y/S(10)))]
Example #7
0
def test_checkpdesol():
    f, F = map(Function, ['f', '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)
    for eq in [eq1, eq2, eq3]:
        assert checkpdesol(eq, pdsolve(eq))[0]
    eq4 = x * f(x, y) + f(x, y).diff(x) + 3 * f(x, y).diff(y)
    eq5 = 2 * f(x, y) + 1 * f(x, y).diff(x) + 3 * f(x, y).diff(y)
    eq6 = f(x, y) + 1 * f(x, y).diff(x) + 3 * f(x, y).diff(y)
    assert checkpdesol(eq4, [pdsolve(eq5), pdsolve(eq6)]) == [
        (False, (x - 2) * F(3 * x - y) * exp(-x / S(5) - 3 * y / S(5))),
        (False, (x - 1) * F(3 * x - y) * exp(-x / S(10) - 3 * y / S(10)))
    ]
Example #8
0
def test_checkpdesol():
    f, F = map(Function, ['f', '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)
    for eq in [eq1, eq2, eq3]:
        assert checkpdesol(eq, pdsolve(eq))[0]
    eq4 = x*f(x,y) + f(x,y).diff(x) + 3*f(x,y).diff(y)
    eq5 = 2*f(x,y) + 1*f(x,y).diff(x) + 3*f(x,y).diff(y)
    eq6 = f(x,y) + 1*f(x,y).diff(x) + 3*f(x,y).diff(y)
    assert checkpdesol(eq4, [pdsolve(eq5), pdsolve(eq6)]) == [
        (False, (x - 2)*F(3*x - y)*exp(-x/S(5) - 3*y/S(5))),
         (False, (x - 1)*F(3*x - y)*exp(-x/S(10) - 3*y/S(10)))]
    for eq in [eq4, eq5, eq6]:
        assert checkpdesol(eq, pdsolve(eq))[0]
    sol = pdsolve(eq4)
    sol4 = Eq(sol.lhs - sol.rhs, 0)
    raises(NotImplementedError, lambda:
        checkpdesol(eq4, sol4, solve_for_func=False))
Example #9
0
def test_checkpdesol():
    f, F = map(Function, ['f', '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)
    for eq in [eq1, eq2, eq3]:
        assert checkpdesol(eq, pdsolve(eq))[0]
    eq4 = x*f(x,y) + f(x,y).diff(x) + 3*f(x,y).diff(y)
    eq5 = 2*f(x,y) + 1*f(x,y).diff(x) + 3*f(x,y).diff(y)
    eq6 = f(x,y) + 1*f(x,y).diff(x) + 3*f(x,y).diff(y)
    assert checkpdesol(eq4, [pdsolve(eq5), pdsolve(eq6)]) == [
        (False, (x - 2)*F(3*x - y)*exp(-x/S(5) - 3*y/S(5))),
         (False, (x - 1)*F(3*x - y)*exp(-x/S(10) - 3*y/S(10)))]
    for eq in [eq4, eq5, eq6]:
        assert checkpdesol(eq, pdsolve(eq))[0]
    sol = pdsolve(eq4)
    sol4 = Eq(sol.lhs - sol.rhs, 0)
    raises(NotImplementedError, lambda:
        checkpdesol(eq4, sol4, solve_for_func=False))
Example #10
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 / 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')
Example #11
0
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]