def pde_1st_linear_constant_coeff(eq, func, order, match, solvefun):
r"""
Solves a first order linear partial differential equation
with constant coefficients.
The general form of this partial differential equation is
.. math:: a \frac{df(x,y)}{dx} + b \frac{df(x,y)}{dy} + c f(x,y) = G(x,y)
where `a`, `b` and `c` are constants and `G(x, y)` can be an arbitrary
function in `x` and `y`.
The general solution of the PDE is::
>>> from sympy.solvers import pdsolve
>>> from sympy.abc import x, y, a, b, c
>>> from sympy import Function, pprint
>>> f = Function('f')
>>> G = Function('G')
>>> u = f(x,y)
>>> ux = u.diff(x)
>>> uy = u.diff(y)
>>> genform = a*u + b*ux + c*uy - G(x,y)
>>> pprint(genform)
d d
a*f(x, y) + b*--(f(x, y)) + c*--(f(x, y)) - G(x, y)
dx dy
>>> pprint(pdsolve(genform, hint='1st_linear_constant_coeff_Integral'))
// b*x + c*y \
|| / |
|| | |
|| | a*xi |
|| | ------- |
|| | 2 2 |
|| | /b*xi + c*eta -b*eta + c*xi\ b + c |
|| | G|------------, -------------|*e d(xi)|
|| | | 2 2 2 2 | |
|| | \ b + c b + c / |
|| | |
|| / |
|| |
f(x, y) = ||F(eta) + -------------------------------------------------------|*
|| 2 2 |
\\ b + c /
<BLANKLINE>
\|
||
||
||
||
||
||
||
||
-a*xi ||
-------||
2 2||
b + c ||
e ||
||
/|eta=-b*y + c*x, xi=b*x + c*y
Examples
========
>>> from sympy.solvers.pde import pdsolve
>>> from sympy import Function, diff, pprint, exp
>>> from sympy.abc import x,y
>>> f = Function('f')
>>> eq = -2*f(x,y).diff(x) + 4*f(x,y).diff(y) + 5*f(x,y) - exp(x + 3*y)
>>> pdsolve(eq)
f(x, y) == (F(4*x + 2*y) + exp(x/2 + 4*y)/15)*exp(x/2 - y)
References
==========
- Viktor Grigoryan, "Partial Differential Equations"
Math 124A - Fall 2010, pp.7
"""
# TODO : For now homogeneous first order linear PDE's having
# two variables are implemented. Once there is support for
# solving systems of ODE's, this can be extended to n variables.
xi, eta = symbols("xi eta")
f = func.func
x = func.args[0]
y = func.args[1]
b = match[match['b']]
c = match[match['c']]
d = match[match['d']]
e = -match[match['e']]
expterm = exp(-S(d) / (b**2 + c**2) * xi)
functerm = solvefun(eta)
solvedict = solve((b * x + c * y - xi, c * x - b * y - eta), x, y)
# Integral should remain as it is in terms of xi,
# doit() should be done in _handle_Integral.
genterm = (1 / S(b**2 + c**2)) * C.Integral(
(1 / expterm * e).subs(solvedict), (xi, b * x + c * y))
return Eq(
f(x, y),
Subs(expterm * (functerm + genterm), (eta, xi),
(c * x - b * y, b * x + c * y)))
def pde_1st_linear_constant_coeff(eq, func, order, match, solvefun):
r"""
Solves a first order linear partial differential equation
with constant coefficients.
The general form of this partial differential equation is
.. math:: a \frac{\partial f(x,y)}{\partial x}
+ b \frac{\partial f(x,y)}{\partial y}
+ c f(x,y) = G(x,y)
where `a`, `b` and `c` are constants and `G(x, y)` can be an arbitrary
function in `x` and `y`.
The general solution of the PDE is:
.. math::
f(x, y) = \left. \left[F(\eta) + \frac{1}{a^2 + b^2}
\int\limits^{a x + b y} G\left(\frac{a \xi + b \eta}{a^2 + b^2},
\frac{- a \eta + b \xi}{a^2 + b^2} \right)
e^{\frac{c \xi}{a^2 + b^2}}\, d\xi\right]
e^{- \frac{c \xi}{a^2 + b^2}}
\right|_{\substack{\eta=- a y + b x\\ \xi=a x + b y }}\, ,
where `F(\eta)` is an arbitrary single-valued function. The solution
can be found in SymPy with ``pdsolve``::
>>> from sympy.solvers import pdsolve
>>> from sympy.abc import x, y, a, b, c
>>> from sympy import Function, pprint
>>> f = Function('f')
>>> G = Function('G')
>>> u = f(x,y)
>>> ux = u.diff(x)
>>> uy = u.diff(y)
>>> genform = a*ux + b*uy + c*u - G(x,y)
>>> pprint(genform)
d d
a*--(f(x, y)) + b*--(f(x, y)) + c*f(x, y) - G(x, y)
dx dy
>>> pprint(pdsolve(genform, hint='1st_linear_constant_coeff_Integral'))
// a*x + b*y \
|| / |
|| | |
|| | c*xi |
|| | ------- |
|| | 2 2 |
|| | /a*xi + b*eta -a*eta + b*xi\ a + b |
|| | G|------------, -------------|*e d(xi)|
|| | | 2 2 2 2 | |
|| | \ a + b a + b / |
|| | |
|| / |
|| |
f(x, y) = ||F(eta) + -------------------------------------------------------|*
|| 2 2 |
\\ a + b /
<BLANKLINE>
\|
||
||
||
||
||
||
||
||
-c*xi ||
-------||
2 2||
a + b ||
e ||
||
/|eta=-a*y + b*x, xi=a*x + b*y
Examples
========
>>> from sympy.solvers.pde import pdsolve
>>> from sympy import Function, diff, pprint, exp
>>> from sympy.abc import x,y
>>> f = Function('f')
>>> eq = -2*f(x,y).diff(x) + 4*f(x,y).diff(y) + 5*f(x,y) - exp(x + 3*y)
>>> pdsolve(eq)
Eq(f(x, y), (F(4*x + 2*y) + exp(x/2 + 4*y)/15)*exp(x/2 - y))
References
==========
- Viktor Grigoryan, "Partial Differential Equations"
Math 124A - Fall 2010, pp.7
"""
# TODO : For now homogeneous first order linear PDE's having
# two variables are implemented. Once there is support for
# solving systems of ODE's, this can be extended to n variables.
xi, eta = symbols("xi eta")
f = func.func
x = func.args[0]
y = func.args[1]
b = match[match['b']]
c = match[match['c']]
d = match[match['d']]
e = -match[match['e']]
expterm = exp(-S(d) / (b**2 + c**2) * xi)
functerm = solvefun(eta)
solvedict = solve((b * x + c * y - xi, c * x - b * y - eta), x, y)
# Integral should remain as it is in terms of xi,
# doit() should be done in _handle_Integral.
genterm = (1 / S(b**2 + c**2)) * Integral(
(1 / expterm * e).subs(solvedict), (xi, b * x + c * y))
return Eq(
f(x, y),
Subs(expterm * (functerm + genterm), (eta, xi),
(c * x - b * y, b * x + c * y)))
def test_subs_in_derivative():
expr = sin(x*exp(y))
u = Function('u')
v = Function('v')
assert Derivative(expr, y).subs(expr, y) == Derivative(y, y)
assert Derivative(expr, y).subs(y, x).doit() == \
Derivative(expr, y).doit().subs(y, x)
assert Derivative(f(x, y), y).subs(y, x) == Subs(Derivative(f(x, y), y), y, x)
assert Derivative(f(x, y), y).subs(x, y) == Subs(Derivative(f(x, y), y), x, y)
assert Derivative(f(x, y), y).subs(y, g(x, y)) == Subs(Derivative(f(x, y), y), y, g(x, y)).doit()
assert Derivative(f(x, y), y).subs(x, g(x, y)) == Subs(Derivative(f(x, y), y), x, g(x, y))
assert Derivative(f(x, y), g(y)).subs(x, g(x, y)) == Derivative(f(g(x, y), y), g(y))
assert Derivative(f(u(x), h(y)), h(y)).subs(h(y), g(x, y)) == \
Subs(Derivative(f(u(x), h(y)), h(y)), h(y), g(x, y)).doit()
assert Derivative(f(x, y), y).subs(y, z) == Derivative(f(x, z), z)
assert Derivative(f(x, y), y).subs(y, g(y)) == Derivative(f(x, g(y)), g(y))
assert Derivative(f(g(x), h(y)), h(y)).subs(h(y), u(y)) == \
Derivative(f(g(x), u(y)), u(y))
assert Derivative(f(x, f(x, x)), f(x, x)).subs(
f, Lambda((x, y), x + y)) == Subs(
Derivative(z + x, z), z, 2*x)
assert Subs(Derivative(f(f(x)), x), f, cos).doit() == sin(x)*sin(cos(x))
assert Subs(Derivative(f(f(x)), f(x)), f, cos).doit() == -sin(cos(x))
# Issue 13791. No comparison (it's a long formula) but this used to raise an exception.
assert isinstance(v(x, y, u(x, y)).diff(y).diff(x).diff(y), Expr)
# This is also related to issues 13791 and 13795; issue 15190
F = Lambda((x, y), exp(2*x + 3*y))
abstract = f(x, f(x, x)).diff(x, 2)
concrete = F(x, F(x, x)).diff(x, 2)
assert (abstract.subs(f, F).doit() - concrete).simplify() == 0
# don't introduce a new symbol if not necessary
assert x in f(x).diff(x).subs(x, 0).atoms()
# case (4)
assert Derivative(f(x,f(x,y)), x, y).subs(x, g(y)
) == Subs(Derivative(f(x, f(x, y)), x, y), x, g(y))
assert Derivative(f(x, x), x).subs(x, 0
) == Subs(Derivative(f(x, x), x), x, 0)
# issue 15194
assert Derivative(f(y, g(x)), (x, z)).subs(z, x
) == Derivative(f(y, g(x)), (x, x))
df = f(x).diff(x)
assert df.subs(df, 1) is S.One
assert df.diff(df) is S.One
dxy = Derivative(f(x, y), x, y)
dyx = Derivative(f(x, y), y, x)
assert dxy.subs(Derivative(f(x, y), y, x), 1) is S.One
assert dxy.diff(dyx) is S.One
assert Derivative(f(x, y), x, 2, y, 3).subs(
dyx, g(x, y)) == Derivative(g(x, y), x, 1, y, 2)
assert Derivative(f(x, x - y), y).subs(x, x + y) == Subs(
Derivative(f(x, x - y), y), x, x + y)
def test_sympy__core__function__Subs():
from sympy.core.function import Subs
assert _test_args(Subs(x + y, x, 2))
def test_Subs2():
# this reflects a limitation of subs(), probably won't fix
assert Subs(f(x), x**2, x).doit() == f(sqrt(x))
def test_Subs():
assert Subs(1, (), ()) is S.One
# check null subs influence on hashing
assert Subs(x, y, z) != Subs(x, y, 1)
# neutral subs works
assert Subs(x, x, 1).subs(x, y).has(y)
# self mapping var/point
assert Subs(Derivative(f(x), (x, 2)), x, x).doit() == f(x).diff(x, x)
assert Subs(x, x, 0).has(x) # it's a structural answer
assert not Subs(x, x, 0).free_symbols
assert Subs(Subs(x + y, x, 2), y, 1) == Subs(x + y, (x, y), (2, 1))
assert Subs(x, (x,), (0,)) == Subs(x, x, 0)
assert Subs(x, x, 0) == Subs(y, y, 0)
assert Subs(x, x, 0).subs(x, 1) == Subs(x, x, 0)
assert Subs(y, x, 0).subs(y, 1) == Subs(1, x, 0)
assert Subs(f(x), x, 0).doit() == f(0)
assert Subs(f(x**2), x**2, 0).doit() == f(0)
assert Subs(f(x, y, z), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y, z), (x, y, z), (0, 0, 1))
assert Subs(x, y, 2).subs(x, y).doit() == 2
assert Subs(f(x, y), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y) + z, (x, y, z), (0, 1, 0))
assert Subs(f(x, y), (x, y), (0, 1)).doit() == f(0, 1)
assert Subs(Subs(f(x, y), x, 0), y, 1).doit() == f(0, 1)
raises(ValueError, lambda: Subs(f(x, y), (x, y), (0, 0, 1)))
raises(ValueError, lambda: Subs(f(x, y), (x, x, y), (0, 0, 1)))
assert len(Subs(f(x, y), (x, y), (0, 1)).variables) == 2
assert Subs(f(x, y), (x, y), (0, 1)).point == Tuple(0, 1)
assert Subs(f(x), x, 0) == Subs(f(y), y, 0)
assert Subs(f(x, y), (x, y), (0, 1)) == Subs(f(x, y), (y, x), (1, 0))
assert Subs(f(x)*y, (x, y), (0, 1)) == Subs(f(y)*x, (y, x), (0, 1))
assert Subs(f(x)*y, (x, y), (1, 1)) == Subs(f(y)*x, (x, y), (1, 1))
assert Subs(f(x), x, 0).subs(x, 1).doit() == f(0)
assert Subs(f(x), x, y).subs(y, 0) == Subs(f(x), x, 0)
assert Subs(y*f(x), x, y).subs(y, 2) == Subs(2*f(x), x, 2)
assert (2 * Subs(f(x), x, 0)).subs(Subs(f(x), x, 0), y) == 2*y
assert Subs(f(x), x, 0).free_symbols == set()
assert Subs(f(x, y), x, z).free_symbols == {y, z}
assert Subs(f(x).diff(x), x, 0).doit(), Subs(f(x).diff(x), x, 0)
assert Subs(1 + f(x).diff(x), x, 0).doit(), 1 + Subs(f(x).diff(x), x, 0)
assert Subs(y*f(x, y).diff(x), (x, y), (0, 2)).doit() == \
2*Subs(Derivative(f(x, 2), x), x, 0)
assert Subs(y**2*f(x), x, 0).diff(y) == 2*y*f(0)
e = Subs(y**2*f(x), x, y)
assert e.diff(y) == e.doit().diff(y) == y**2*Derivative(f(y), y) + 2*y*f(y)
assert Subs(f(x), x, 0) + Subs(f(x), x, 0) == 2*Subs(f(x), x, 0)
e1 = Subs(z*f(x), x, 1)
e2 = Subs(z*f(y), y, 1)
assert e1 + e2 == 2*e1
assert e1.__hash__() == e2.__hash__()
assert Subs(z*f(x + 1), x, 1) not in [ e1, e2 ]
assert Derivative(f(x), x).subs(x, g(x)) == Derivative(f(g(x)), g(x))
assert Derivative(f(x), x).subs(x, x + y) == Subs(Derivative(f(x), x),
x, x + y)
assert Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).n(2) == \
Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).evalf(2) == \
z + Rational('1/2').n(2)*f(0)
assert f(x).diff(x).subs(x, 0).subs(x, y) == f(x).diff(x).subs(x, 0)
assert (x*f(x).diff(x).subs(x, 0)).subs(x, y) == y*f(x).diff(x).subs(x, 0)
assert Subs(Derivative(g(x)**2, g(x), x), g(x), exp(x)
).doit() == 2*exp(x)
assert Subs(Derivative(g(x)**2, g(x), x), g(x), exp(x)
).doit(deep=False) == 2*Derivative(exp(x), x)
assert Derivative(f(x, g(x)), x).doit() == Derivative(
f(x, g(x)), g(x))*Derivative(g(x), x) + Subs(Derivative(
f(y, g(x)), y), y, x)
def test_issue_15226():
assert Subs(Derivative(f(y), x, y), y, g(x)).doit() != 0
def test_issue_12005():
e1 = Subs(Derivative(f(x), x), x, x)
assert e1.diff(x) == Derivative(f(x), x, x)
e2 = Subs(Derivative(f(x), x), x, x**2 + 1)
assert e2.diff(x) == 2*x*Subs(Derivative(f(x), x, x), x, x**2 + 1)
e3 = Subs(Derivative(f(x) + y**2 - y, y), y, y**2)
assert e3.diff(y) == 4*y
e4 = Subs(Derivative(f(x + y), y), y, (x**2))
assert e4.diff(y) is S.Zero
e5 = Subs(Derivative(f(x), x), (y, z), (y, z))
assert e5.diff(x) == Derivative(f(x), x, x)
assert f(g(x)).diff(g(x), g(x)) == Derivative(f(g(x)), g(x), g(x))
def test_series_of_Subs():
from sympy.abc import z
subs1 = Subs(sin(x), x, y)
subs2 = Subs(sin(x) * cos(z), x, y)
subs3 = Subs(sin(x * z), (x, z), (y, x))
assert subs1.series(x) == subs1
subs1_series = (Subs(x, x, y) + Subs(-x**3/6, x, y) +
Subs(x**5/120, x, y) + O(y**6))
assert subs1.series() == subs1_series
assert subs1.series(y) == subs1_series
assert subs1.series(z) == subs1
assert subs2.series(z) == (Subs(z**4*sin(x)/24, x, y) +
Subs(-z**2*sin(x)/2, x, y) + Subs(sin(x), x, y) + O(z**6))
assert subs3.series(x).doit() == subs3.doit().series(x)
assert subs3.series(z).doit() == sin(x*y)
raises(ValueError, lambda: Subs(x + 2*y, y, z).series())
assert Subs(x + y, y, z).series(x).doit() == x + z
