# Source code for sympy.solvers.pde

"""
This module contains pdsolve() and different helper functions that it
uses. It is heavily inspired by the ode module and hence the basic
infrastructure remains the same.

**Functions in this module**

These are the user functions in this module:

- pdsolve()     - Solves PDE's
- classify_pde() - Classifies PDEs into possible hints for dsolve().
- pde_separate() - Separate variables in partial differential equation either by

These are the helper functions in this module:

- pde_separate_mul() - Helper function for searching multiplicative
separable solutions.

**Currently implemented solver methods**

The following methods are implemented for solving partial differential
equations.  See the docstrings of the various pde_hint() functions for

- 1st order linear homogeneous partial differential equations
with constant coefficients.

"""
from copy import deepcopy

from sympy.simplify import simplify
from sympy.core import Add, C, S, Mul, Pow, oo
from sympy.core.compatibility import (reduce, combinations_with_replacement,
is_sequence)
from sympy.core.function import (Function, Derivative,
expand, diff, AppliedUndef, Subs)
from sympy.core.numbers import Rational
from sympy.core.relational import Equality, Eq
from sympy.core.symbol import Symbol, Wild, Dummy, symbols
from sympy.functions import exp
from sympy.utilities.iterables import has_dups

from sympy.solvers.deutils import _preprocess, ode_order, _desolve
from sympy.solvers.solvers import solve
import operator

allhints = (
"1st_linear_constant_coeff_homogeneous",
"1st_linear_constant_coeff",
"1st_linear_constant_coeff_Integral"
)

[docs]def pdsolve(eq, func=None, hint='default', dict=False, solvefun=None, **kwargs):
"""
Solves any (supported) kind of partial differential equation.

**Usage**

pdsolve(eq, f(x,y), hint) -> Solve partial differential equation
eq for function f(x,y), using method hint.

**Details**

eq can be any supported partial differential equation (see
the pde docstring for supported methods).  This can either
be an Equality, or an expression, which is assumed to be
equal to 0.

f(x,y) is a function of two variables whose derivatives in that
variable make up the partial differential equation. In many
cases it is not necessary to provide this; it will be autodetected
(and an error raised if it couldn't be detected).

hint is the solving method that you want pdsolve to use.  Use
classify_pde(eq, f(x,y)) to get all of the possible hints for
a PDE.  The default hint, 'default', will use whatever hint
is returned first by classify_pde().  See Hints below for
more options that you can use for hint.

solvefun is the convention used for arbitrary functions returned
by the PDE solver. If not set by the user, it is set by default
to be F.

**Hints**

Aside from the various solving methods, there are also some
meta-hints that you can pass to pdsolve():

"default":
This uses whatever hint is returned first by
classify_pde(). This is the default argument to
pdsolve().

"all":
To make pdsolve apply all relevant classification hints,
use pdsolve(PDE, func, hint="all").  This will return a
dictionary of hint:solution terms.  If a hint causes
pdsolve to raise the NotImplementedError, value of that
hint's key will be the exception object raised.  The
dictionary will also include some special keys:

deutils.py
- default: The solution that would be returned by
default.  This is the one produced by the hint that
appears first in the tuple returned by classify_pde().

"all_Integral":
This is the same as "all", except if a hint also has a
corresponding "_Integral" hint, it only returns the
"_Integral" hint.  This is useful if "all" causes
pdsolve() to hang because of a difficult or impossible
integral.  This meta-hint will also be much faster than
"all", because integrate() is an expensive routine.

and the pde docstring for a list of all supported hints.

**Tips**
- You can declare the derivative of an unknown function this way:
>>> from sympy import Function, Derivative
>>> from sympy.abc import x, y # x and y are the independent variables
>>> f = Function("f")(x, y) # f is a function of x and y
>>> # fx will be the partial derivative of f with respect to x
>>> fx = Derivative(f, x)
>>> # fy will be the partial derivative of f with respect to y
>>> fy = Derivative(f, y)

- See test_pde.py for many tests, which serves also as a set of
examples for how to use pdsolve().
- pdsolve always returns an Equality class (except for the case
when the hint is "all" or "all_Integral"). Note that it is not possible
to get an explicit solution for f(x, y) as in the case of ODE's
specific hint

Examples
========

>>> from sympy.solvers.pde import pdsolve
>>> from sympy import Function, diff, Eq
>>> from sympy.abc import x, y
>>> f = Function('f')
>>> u = f(x, y)
>>> ux = u.diff(x)
>>> uy = u.diff(y)
>>> eq = Eq(1 + (2*(ux/u)) + (3*(uy/u)))
>>> pdsolve(eq)
f(x, y) == F(3*x - 2*y)*exp(-2*x/13 - 3*y/13)

"""

given_hint = hint  # hint given by the user.

if not solvefun:
solvefun = Function('F')

# See the docstring of _desolve for more details.
hints = _desolve(eq, func=func,
hint=hint, simplify=True, type='pde', **kwargs)
eq = hints.pop('eq', False)
all_ = hints.pop('all', False)

if all_:
# TODO : 'best' hint should be implemented when adequate
# number of hints are added.
pdedict = {}
failed_hints = {}
gethints = classify_pde(eq, dict=True)
pdedict.update({'order': gethints['order'],
'default': gethints['default']})
for hint in hints:
try:
rv = _helper_simplify(eq, hint, hints[hint]['func'],
hints[hint]['order'], hints[hint][hint], solvefun)
except NotImplementedError, detail:
failed_hints[hint] = detail
else:
pdedict[hint] = rv
pdedict.update(failed_hints)
return pdedict

else:
return _helper_simplify(eq, hints['hint'],
hints['func'], hints['order'], hints[hints['hint']], solvefun)

def _helper_simplify(eq, hint, func, order, match, solvefun):
"""Helper function of pdsolve that calls the respective
pde functions to solve for the partial differential
equations. This minimises the computation in
calling _desolve multiple times.
"""

if hint.endswith("_Integral"):
solvefunc = globals()[
"pde_" + hint[:-len("_Integral")]]
else:
solvefunc = globals()["pde_" + hint]
return _handle_Integral(solvefunc(eq, func, order,
match, solvefun), func, order, hint)

def _handle_Integral(expr, func, order, hint):
r"""
Converts a solution with integrals in it into an actual solution.

Simplifies the integral mainly using doit()
"""
if hint.endswith("_Integral"):
return expr

elif hint == "1st_linear_constant_coeff":
return simplify(expr.doit())

else:
return expr

[docs]def classify_pde(eq, func=None, dict=False, **kwargs):
"""
Returns a tuple of possible pdsolve() classifications for a PDE.

The tuple is ordered so that first item is the classification that
pdsolve() uses to solve the PDE by default.  In general,
classifications at the near the beginning of the list will produce
better solutions faster than those near the end, thought there are
always exceptions.  To make pdsolve use a different classification,
docstring for different meta-hints you can use.

If dict is true, classify_pde() will return a dictionary of
hint:match expression terms. This is intended for internal use by
pdsolve().  Note that because dictionaries are ordered arbitrarily,
this will most likely not be in the same order as the tuple.

You can get help on different hints by doing help(pde.pde_hintname),
where hintname is the name of the hint without "_Integral".

See sympy.pde.allhints or the sympy.pde docstring for a list of all
supported hints that can be returned from classify_pde.

Examples
========
>>> from sympy.solvers.pde import classify_pde
>>> from sympy import Function, diff, Eq
>>> from sympy.abc import x, y
>>> f = Function('f')
>>> u = f(x, y)
>>> ux = u.diff(x)
>>> uy = u.diff(y)
>>> eq = Eq(1 + (2*(ux/u)) + (3*(uy/u)))
>>> classify_pde(eq)
('1st_linear_constant_coeff_homogeneous',)
"""

prep = kwargs.pop('prep', True)

if func and len(func.args) != 2:
raise NotImplementedError("Right now only partial "
"differential equations of two variables are supported")

if prep or func is None:
prep, func_ = _preprocess(eq, func)
if func is None:
func = func_

if isinstance(eq, Equality):
if eq.rhs != 0:
return classify_pde(eq.lhs - eq.rhs, func)
eq = eq.lhs

f = func.func
x = func.args[0]
y = func.args[1]
fx = f(x,y).diff(x)
fy = f(x,y).diff(y)

# TODO : For now pde.py uses support offered by the ode_order function
# to find the order with respect to a multi-variable function. An
# improvement could be to classify the order of the PDE on the basis of
# individual variables.
order = ode_order(eq, f(x,y))

# hint:matchdict or hint:(tuple of matchdicts)
# Also will contain "default":<default hint> and "order":order items.
matching_hints = {'order': order}

if not order:
if dict:
matching_hints["default"] = None
return matching_hints
else:
return ()

eq = expand(eq)

a = Wild('a', exclude = [f(x,y)])
b = Wild('b', exclude = [f(x,y), fx, fy, x, y])
c = Wild('c', exclude = [f(x,y), fx, fy, x, y])
d = Wild('d', exclude = [f(x,y), fx, fy, x, y])
e = Wild('e', exclude = [f(x,y), fx, fy])
n = Wild('n', exclude = [x, y])
# Try removing the smallest power of f(x,y)
# from the highest partial derivatives of f(x,y)
reduced_eq = None
var = set(combinations_with_replacement((x,y), order))
dummyvar = deepcopy(var)
power = None
for i in var:
coeff = eq.coeff(f(x,y).diff(*i))
if coeff != 1:
match = coeff.match(a*f(x,y)**n)
if match and match[a]:
power = match[n]
dummyvar.remove(i)
break
dummyvar.remove(i)
for i in dummyvar:
coeff = eq.coeff(f(x,y).diff(*i))
if coeff != 1:
match = coeff.match(a*f(x,y)**n)
if match and match[a] and match[n] < power:
power = match[n]
if power:
den = f(x,y)**power
reduced_eq = Add(*[arg/den for arg in eq.args])
if not reduced_eq:
reduced_eq = eq

if order == 1:
r = reduced_eq.match(b*fx + c*fy + d*f(x,y) + e)
if r:
if not r[e]:
## Linear first-order homogeneous partial-differential
## equation with constant coefficients
r.update({'b': b, 'c': c, 'd': d})
matching_hints["1st_linear_constant_coeff_homogeneous"] = r
else:
if r[b]**2 + r[c]**2 != 0:
## Linear first-order general partial-differential
## equation with constant coefficients
r.update({'b': b, 'c': c, 'd': d, 'e': e})
matching_hints["1st_linear_constant_coeff"] = r
matching_hints[
"1st_linear_constant_coeff_Integral"] = r

# Order keys based on allhints.
retlist = []
for i in allhints:
if i in matching_hints:
retlist.append(i)

if dict:
# Dictionaries are ordered arbitrarily, so make note of which
# hint would come first for pdsolve().  Use an ordered dict in Py 3.
matching_hints["default"] = None
matching_hints["ordered_hints"] = tuple(retlist)
for i in allhints:
if i in matching_hints:
matching_hints["default"] = i
break
return matching_hints
else:
return tuple(retlist)

[docs]def checkpdesol(pde, sol, func=None, solve_for_func=True):
"""
Checks if the given solution satisfies the partial differential
equation.

pde is the partial differential equation which can be given in the
form of an equation or an expression. sol is the solution for which
the pde is to be checked. This can also be given in an equation or
an expression form. If the function is not provided, the helper
function _preprocess from deutils is used to identify the function.

If a sequence of solutions is passed, the same sort of container will be
used to return the result for each solution.

The following methods are currently being implemented to check if the
solution satisfies the PDE:

1. Directly substitute the solution in the PDE and check. If the
solution hasn't been solved for f, then it will solve for f
provided solve_for_func hasn't been set to False.

If the solution satisfies the PDE, then a tuple (True, 0) is returned.
Otherwise a tuple (False, expr) where expr is the value obtained
after substituting the solution in the PDE. However if a known solution
returns False, it may be due to the inability of doit() to simplify it to zero.

Examples
========
>>> from sympy import Function, symbols, diff
>>> from sympy.solvers.pde import checkpdesol, pdsolve
>>> x, y = symbols('x y')
>>> f = Function('f')
>>> eq = 2*f(x,y) + 3*f(x,y).diff(x) + 4*f(x,y).diff(y)
>>> sol = pdsolve(eq)
>>> assert checkpdesol(eq, sol)[0]
>>> eq = x*f(x,y) + f(x,y).diff(x)
>>> checkpdesol(eq, sol)
(False, (x*F(4*x - 3*y) - 6*F(4*x - 3*y)/25 + 4*Subs(Derivative(F(_xi_1), _xi_1), (_xi_1,), (4*x - 3*y,)))*exp(-6*x/25 - 8*y/25))
"""

# Converting the pde into an equation
if not isinstance(pde, Equality):
pde = Eq(pde, 0)

# If no function is given, try finding the function present.
if func is None:
try:
_, func = _preprocess(pde.lhs)
except ValueError:
funcs = [s.atoms(AppliedUndef) for s in (
sol if is_sequence(sol, set) else [sol])]
funcs = reduce(set.union, funcs, set())
if len(funcs) != 1:
raise ValueError(
'must pass func arg to checkpdesol for this case.')
func = funcs.pop()

# If the given solution is in the form of a list or a set
# then return a list or set of tuples.
if is_sequence(sol, set):
return type(sol)(map(lambda i: checkpdesol(pde, i,
solve_for_func=solve_for_func), sol))

# Convert solution into an equation
if not isinstance(sol, Equality):
sol = Eq(func, sol)

# Try solving for the function
if solve_for_func and not (sol.lhs == func and not sol.rhs.has(func)) and not \
(sol.rhs == func and not sol.lhs.has(func)):
try:
solved = solve(sol, func)
if not solved:
raise NotImplementedError
except NotImplementedError:
pass
else:
if len(solved) == 1:
result = checkpdesol(pde, Eq(func, solved[0]),
order=order, solve_for_func=False)
else:
result = checkpdesol(pde, [Eq(func, t) for t in solved],
order=order, solve_for_func=False)

# The first method includes direct substitution of the solution in
# the PDE and simplifying.
pde = pde.lhs - pde.rhs
if sol.lhs == func:
s = pde.subs(func, sol.rhs).doit()
elif sol.rhs == func:
s = pde.subs(func, sol.lhs).doit()
if s:
ss = simplify(s)
if ss:
return False, ss
else:
return True, 0
else:
return True, 0

[docs]def pde_1st_linear_constant_coeff_homogeneous(eq, func, order, match, solvefun):
r"""
Solves a first order linear homogeneous
partial differential equation with constant coefficients.

The general form of this partial differential equation is
a*f(x,y).diff(x) + b*f(x,y).diff(y) + c*f(x,y) = 0
where a, b and c are constants.

The general solution of the differential equation, can be found
by the method of characteristics. It is given by
f(x,y) = F(b*x - a*y)*exp(-c/(a**2 + b**2)*(a*x + b*y))

>>> from sympy.solvers import pdsolve
>>> from sympy.abc import x, y, a, b, c
>>> from sympy import Function, pprint
>>> f = Function('f')
>>> u = f(x,y)
>>> ux = u.diff(x)
>>> uy = u.diff(y)
>>> genform = a*ux + b*uy + c*u
>>> pprint(genform)
d               d
a*--(f(x, y)) + b*--(f(x, y)) + c*f(x, y)
dx              dy

>>> pprint(pdsolve(genform))
-c*(a*x + b*y)
---------------
2    2
a  + b
f(x, y) = F(-a*y + b*x)*e

Examples
========

>>> from sympy.solvers.pde import (
... pde_1st_linear_constant_coeff_homogeneous)
>>> from sympy import pdsolve
>>> from sympy import Function, diff, pprint
>>> from sympy.abc import x,y
>>> f = Function('f')
>>> pdsolve(f(x,y) + f(x,y).diff(x) + f(x,y).diff(y))
f(x, y) == F(x - y)*exp(-x/2 - y/2)
>>> pprint(pdsolve(f(x,y) + f(x,y).diff(x) + f(x,y).diff(y)))
x   y
- - - -
2   2
f(x, y) = F(x - y)*e

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.

f = func.func
x = func.args[0]
y = func.args[1]
b = match[match['b']]
c = match[match['c']]
d = match[match['d']]
return Eq(f(x,y), exp(-S(d)/(b**2 + c**2)*(b*x + c*y))*solvefun(c*x - b*y))

[docs]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
a*f(x,y).diff(x) + b*f(x,y).diff(y) + c*f(x,y) = G(x,y)
where a, b and c are constants and G 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)))

[docs]def pde_separate(eq, fun, sep, strategy='mul'):
"""Separate variables in partial differential equation either by additive
or multiplicative separation approach. It tries to rewrite an equation so
that one of the specified variables occurs on a different side of the
equation than the others.

:param eq: Partial differential equation

:param fun: Original function F(x, y, z)

:param sep: List of separated functions [X(x), u(y, z)]

:param strategy: Separation strategy. You can choose between additive
separation ('add') and multiplicative separation ('mul') which is
default.

Examples
========

>>> from sympy import E, Eq, Function, pde_separate, Derivative as D
>>> from sympy.abc import x, t
>>> u, X, T = map(Function, 'uXT')

>>> eq = Eq(D(u(x, t), x), E**(u(x, t))*D(u(x, t), t))
>>> pde_separate(eq, u(x, t), [X(x), T(t)], strategy='add')
[exp(-X(x))*Derivative(X(x), x), exp(T(t))*Derivative(T(t), t)]

>>> eq = Eq(D(u(x, t), x, 2), D(u(x, t), t, 2))
>>> pde_separate(eq, u(x, t), [X(x), T(t)], strategy='mul')
[Derivative(X(x), x, x)/X(x), Derivative(T(t), t, t)/T(t)]

========
"""

elif strategy == 'mul':
else:
assert ValueError('Unknown strategy: %s' % strategy)

if isinstance(eq, Equality):
if eq.rhs != 0:
return pde_separate(Eq(eq.lhs - eq.rhs), fun, sep, strategy)
assert eq.rhs == 0

# Handle arguments
orig_args = list(fun.args)
subs_args = []
for s in sep:
for j in range(0, len(s.args)):
subs_args.append(s.args[j])

else:
functions = reduce(operator.mul, sep)

# Check whether variables match
if len(subs_args) != len(orig_args):
raise ValueError("Variable counts do not match")
# Check for duplicate arguments like  [X(x), u(x, y)]
if has_dups(subs_args):
raise ValueError("Duplicate substitution arguments detected")
# Check whether the variables match
if set(orig_args) != set(subs_args):
raise ValueError("Arguments do not match")

# Substitute original function with separated...
result = eq.lhs.subs(fun, functions).doit()

# Divide by terms when doing multiplicative separation
eq = 0
for i in result.args:
eq += i/functions
result = eq

svar = subs_args[0]
dvar = subs_args[1:]
return _separate(result, svar, dvar)

"""
Helper function for searching additive separable solutions.

Consider an equation of two independent variables x, y and a dependent
variable w, we look for the product of two functions depending on different
arguments:

w(x, y, z) = X(x) + y(y, z)

Examples
========

>>> from sympy import E, Eq, Function, pde_separate_add, Derivative as D
>>> from sympy.abc import x, t
>>> u, X, T = map(Function, 'uXT')

>>> eq = Eq(D(u(x, t), x), E**(u(x, t))*D(u(x, t), t))
>>> pde_separate_add(eq, u(x, t), [X(x), T(t)])
[exp(-X(x))*Derivative(X(x), x), exp(T(t))*Derivative(T(t), t)]

"""

[docs]def pde_separate_mul(eq, fun, sep):
"""
Helper function for searching multiplicative separable solutions.

Consider an equation of two independent variables x, y and a dependent
variable w, we look for the product of two functions depending on different
arguments:

w(x, y, z) = X(x)*u(y, z)

Examples
========

>>> from sympy import Function, Eq, pde_separate_mul, Derivative as D
>>> from sympy.abc import x, y
>>> u, X, Y = map(Function, 'uXY')

>>> eq = Eq(D(u(x, y), x, 2), D(u(x, y), y, 2))
>>> pde_separate_mul(eq, u(x, y), [X(x), Y(y)])
[Derivative(X(x), x, x)/X(x), Derivative(Y(y), y, y)/Y(y)]

"""
return pde_separate(eq, fun, sep, strategy='mul')

def _separate(eq, dep, others):
"""Separate expression into two parts based on dependencies of variables."""

# FIRST PASS
# Extract derivatives depending our separable variable...
terms = set()
for term in eq.args:
if term.is_Mul:
for i in term.args:
if i.is_Derivative and not i.has(*others):
continue
elif term.is_Derivative and not term.has(*others):
# Find the factor that we need to divide by
div = set()
for term in terms:
ext, sep = term.expand().as_independent(dep)
# Failed?
if sep.has(*others):
return None
# FIXME: Find lcm() of all the divisors and divide with it, instead of
# current hack :(
if len(div) > 0:
final = 0
for term in eq.args:
eqn = 0
for i in div:
eqn += term / i
final += simplify(eqn)
eq = final

# SECOND PASS - separate the derivatives
div = set()
lhs = rhs = 0
for term in eq.args:
# Check, whether we have already term with independent variable...
if not term.has(*others):
lhs += term
continue
# ...otherwise, try to separate
temp, sep = term.expand().as_independent(dep)
# Failed?
if sep.has(*others):
return None
# Extract the divisors