Coverage for sympy/solvers/pde.py: 86%

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"""

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

additive or multiplicative separation approach.

These are the helper functions in this module:

- pde_separate_add() - Helper function for searching additive separable solutions.

- 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

more information on each (run help(pde)):

- 1st order linear homogeneous partial differential equations

with constant coefficients.

- 1st order linear general partial differential equations

with constant coefficients.

- 1st order linear partial differential equations with

variable coefficients.

"""

from __future__ import print_function, division

from sympy.simplify import simplify

from sympy.core import Add, S

from sympy.core.compatibility import (reduce, combinations_with_replacement,

is_sequence, range)

from sympy.core.function import Function, expand, AppliedUndef, Subs

from sympy.core.relational import Equality, Eq

from sympy.core.symbol import Symbol, Wild, symbols

from sympy.functions import exp

from sympy.integrals.integrals import Integral

from sympy.utilities.iterables import has_dups

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

from sympy.solvers.solvers import solve

from sympy.simplify.simplify import collect

import operator

allhints = (

"1st_linear_constant_coeff_homogeneous",

"1st_linear_constant_coeff",

"1st_linear_constant_coeff_Integral",

"1st_linear_variable_coeff"

)

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:

- order: The order of the PDE. See also ode_order() in

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.

See also the classify_pde() docstring for more info on hints,

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

- Do help(pde.pde_hintname) to get help more information on a

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)

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 as 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

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 near the beginning of the list will produce

better solutions faster than those near the end, though there are

always exceptions. To make pdsolve use a different classification,

use pdsolve(PDE, func, hint=<classification>). See also the pdsolve()

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

if eq.is_Add:

var = set(combinations_with_replacement((x,y), order))

dummyvar = var.copy()

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:

reduced_eq = collect(reduced_eq, f(x, y))

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

else:

b = Wild('b', exclude=[f(x, y), fx, fy])

c = Wild('c', exclude=[f(x, y), fx, fy])

d = Wild('d', exclude=[f(x, y), fx, fy])

r = reduced_eq.match(b*fx + c*fy + d*f(x,y) + e)

if r:

r.update({'b': b, 'c': c, 'd': d, 'e': e})

matching_hints["1st_linear_variable_coeff"] = 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)

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 = set().union(funcs)

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)([checkpdesol(pde, i, solve_for_func=solve_for_func) for i in 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

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

.. math:: a \frac{df(x,y)}{dx} + b \frac{df(x,y)}{dy} + c f(x,y) = 0

where `a`, `b` and `c` are constants.

The general solution is of the form::

>>> 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))

Eq(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))

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 /

-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)

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 pde_1st_linear_variable_coeff(eq, func, order, match, solvefun):

r"""

Solves a first order linear partial differential equation

with variable coefficients. The general form of this partial differential equation is

.. math:: a(x, y) \frac{df(x, y)}{dx} + a(x, y) \frac{df(x, y)}{dy}

+ c(x, y) f(x, y) - G(x, y)

where `a(x, y)`, `b(x, y)`, `c(x, y)` and `G(x, y)` are arbitrary functions

in `x` and `y`. This PDE is converted into an ODE by making the following transformation.

1] `\xi` as `x`

2] `\eta` as the constant in the solution to the differential equation

`\frac{dy}{dx} = -\frac{b}{a}`

Making the following substitutions reduces it to the linear ODE

.. math:: a(\xi, \eta)\frac{du}{d\xi} + c(\xi, \eta)u - d(\xi, \eta) = 0

which can be solved using dsolve.

The general form of this PDE is::

>>> from sympy.solvers.pde import pdsolve

>>> from sympy.abc import x, y

>>> from sympy import Function, pprint

>>> a, b, c, G, f= [Function(i) for i in ['a', 'b', 'c', 'G', 'f']]

>>> u = f(x,y)

>>> ux = u.diff(x)

>>> uy = u.diff(y)

>>> genform = a(x, y)*u + b(x, y)*ux + c(x, y)*uy - G(x,y)

>>> pprint(genform)

d d

-G(x, y) + a(x, y)*f(x, y) + b(x, y)*--(f(x, y)) + c(x, y)*--(f(x, y))

dx dy

Examples

========

>>> from sympy.solvers.pde import pdsolve

>>> from sympy import Function, diff, pprint, exp

>>> from sympy.abc import x,y

>>> f = Function('f')

>>> eq = x*(u.diff(x)) - y*(u.diff(y)) + y**2*u - y**2

>>> pdsolve(eq)

Eq(f(x, y), F(x*y)*exp(y**2/2) + 1)

References

==========

- Viktor Grigoryan, "Partial Differential Equations"

Math 124A - Fall 2010, pp.7

"""

from sympy.integrals.integrals import integrate

from sympy.solvers.ode import dsolve

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']]

if not d:

# To deal with cases like b*ux = e or c*uy = e

if not (b and c):

if c:

try:

tsol = integrate(e/c, y)

except NotImplementedError:

raise NotImplementedError("Unable to find a solution"

" due to inability of integrate")

else:

return Eq(f(x,y), solvefun(x) + tsol)

if b:

try:

tsol = integrate(e/b, x)

except NotImplementedError:

raise NotImplementedError("Unable to find a solution"

" due to inability of integrate")

else:

return Eq(f(x,y), solvefun(y) + tsol)

if not c:

# To deal with cases when c is 0, a simpler method is used.

# The PDE reduces to b*(u.diff(x)) + d*u = e, which is a linear ODE in x

plode = f(x).diff(x)*b + d*f(x) - e

sol = dsolve(plode, f(x))

syms = sol.free_symbols - plode.free_symbols - set([x, y])

rhs = _simplify_variable_coeff(sol.rhs, syms, solvefun, y)

return Eq(f(x, y), rhs)

if not b:

# To deal with cases when b is 0, a simpler method is used.

# The PDE reduces to c*(u.diff(y)) + d*u = e, which is a linear ODE in y

plode = f(y).diff(y)*c + d*f(y) - e

sol = dsolve(plode, f(y))

syms = sol.free_symbols - plode.free_symbols - set([x, y])

rhs = _simplify_variable_coeff(sol.rhs, syms, solvefun, x)

return Eq(f(x, y), rhs)

dummy = Function('d')

h = (c/b).subs(y, dummy(x))

sol = dsolve(dummy(x).diff(x) - h, dummy(x))

if isinstance(sol, list):

sol = sol[0]

solsym = sol.free_symbols - h.free_symbols - set([x, y])

if len(solsym) == 1:

solsym = solsym.pop()

etat = (solve(sol, solsym)[0]).subs(dummy(x), y)

ysub = solve(eta - etat, y)[0]

deq = (b*(f(x).diff(x)) + d*f(x) - e).subs(y, ysub)

final = (dsolve(deq, f(x), hint='1st_linear')).rhs

if isinstance(final, list):

final = final[0]

finsyms = final.free_symbols - deq.free_symbols - set([x, y])

rhs = _simplify_variable_coeff(final, finsyms, solvefun, etat)

return Eq(f(x, y), rhs)

else:

raise NotImplementedError("Cannot solve the partial differential equation due"

" to inability of constantsimp")

def _simplify_variable_coeff(sol, syms, func, funcarg):

r"""

Helper function to replace constants by functions in 1st_linear_variable_coeff

"""

eta = Symbol("eta")

if len(syms) == 1:

sym = syms.pop()

final = sol.subs(sym, func(funcarg))

else:

fname = func.__name__

for key, sym in enumerate(syms):

tempfun = Function(fname + str(key))

final = sol.subs(sym, func(funcarg))

return simplify(final.subs(eta, funcarg))

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)]

Coverage for sympy/solvers/pde.py : 86%

333 statements 286 run 47 missing 5 excluded