Coverage for sympy/solvers/deutils.py : 87%

"""Utility functions for classifying and solving 

ordinary and partial differential equations. 

Contains 

======== 

_preprocess 

ode_order 

_desolve 

""" 

from __future__ import print_function, division 

from sympy.core.function import Derivative, AppliedUndef 

from sympy.core.relational import Equality 

from sympy.core.symbol import Wild 

def _preprocess(expr, func=None, hint='_Integral'): 

"""Prepare expr for solving by making sure that differentiation 

is done so that only func remains in unevaluated derivatives and 

(if hint doesn't end with _Integral) that doit is applied to all 

other derivatives. If hint is None, don't do any differentiation. 

(Currently this may cause some simple differential equations to 

fail.) 

In case func is None, an attempt will be made to autodetect the 

function to be solved for. 

>>> from sympy.solvers.deutils import _preprocess 

>>> from sympy import Derivative, Function, Integral, sin 

>>> from sympy.abc import x, y, z 

>>> f, g = map(Function, 'fg') 

Apply doit to derivatives that contain more than the function 

of interest: 

>>> _preprocess(Derivative(f(x) + x, x)) 

(Derivative(f(x), x) + 1, f(x)) 

Do others if the differentiation variable(s) intersect with those 

of the function of interest or contain the function of interest: 

>>> _preprocess(Derivative(g(x), y, z), f(y)) 

(0, f(y)) 

>>> _preprocess(Derivative(f(y), z), f(y)) 

(0, f(y)) 

Do others if the hint doesn't end in '_Integral' (the default 

assumes that it does): 

>>> _preprocess(Derivative(g(x), y), f(x)) 

(Derivative(g(x), y), f(x)) 

>>> _preprocess(Derivative(f(x), y), f(x), hint='') 

(0, f(x)) 

Don't do any derivatives if hint is None: 

>>> eq = Derivative(f(x) + 1, x) + Derivative(f(x), y) 

>>> _preprocess(eq, f(x), hint=None) 

(Derivative(f(x) + 1, x) + Derivative(f(x), y), f(x)) 

If it's not clear what the function of interest is, it must be given: 

>>> eq = Derivative(f(x) + g(x), x) 

>>> _preprocess(eq, g(x)) 

(Derivative(f(x), x) + Derivative(g(x), x), g(x)) 

>>> try: _preprocess(eq) 

... except ValueError: print("A ValueError was raised.") 

A ValueError was raised. 

""" 

derivs = expr.atoms(Derivative) 

if not func: 

funcs = set().union(*[d.atoms(AppliedUndef) for d in derivs]) 

if len(funcs) != 1: 

raise ValueError('The function cannot be ' 

'automatically detected for %s.' % expr) 

func = funcs.pop() 

fvars = set(func.args) 

if hint is None: 

return expr, func 

reps = [(d, d.doit()) for d in derivs if not hint.endswith('_Integral') or 

d.has(func) or set(d.variables) & fvars] 

eq = expr.subs(reps) 

return eq, func 

def ode_order(expr, func): 

""" 

Returns the order of a given differential 

equation with respect to func. 

This function is implemented recursively. 

Examples 

======== 

>>> from sympy import Function 

>>> from sympy.solvers.deutils import ode_order 

>>> from sympy.abc import x 

>>> f, g = map(Function, ['f', 'g']) 

>>> ode_order(f(x).diff(x, 2) + f(x).diff(x)**2 + 

... f(x).diff(x), f(x)) 

2 

>>> ode_order(f(x).diff(x, 2) + g(x).diff(x, 3), f(x)) 

2 

>>> ode_order(f(x).diff(x, 2) + g(x).diff(x, 3), g(x)) 

3 

""" 

a = Wild('a', exclude=[func]) 

if expr.match(a): 

return 0 

if isinstance(expr, Derivative): 

if expr.args[0] == func: 

return len(expr.variables) 

else: 

order = 0 

for arg in expr.args[0].args: 

order = max(order, ode_order(arg, func) + len(expr.variables)) 

return order 

else: 

order = 0 

for arg in expr.args: 

order = max(order, ode_order(arg, func)) 

return order 

def _desolve(eq, func=None, hint="default", ics=None, simplify=True, **kwargs): 

"""This is a helper function to dsolve and pdsolve in the ode 

and pde modules. 

If the hint provided to the function is "default", then a dict with 

the following keys are returned 

'func' - It provides the function for which the differential equation 

has to be solved. This is useful when the expression has 

more than one function in it. 

'default' - The default key as returned by classifier functions in ode 

and pde.py 

'hint' - The hint given by the user for which the differential equation 

is to be solved. If the hint given by the user is 'default', 

then the value of 'hint' and 'default' is the same. 

'order' - The order of the function as returned by ode_order 

'match' - It returns the match as given by the classifier functions, for 

the default hint. 

If the hint provided to the function is not "default" and is not in 

('all', 'all_Integral', 'best'), then a dict with the above mentioned keys 

is returned along with the keys which are returned when dict in 

classify_ode or classify_pde is set True 

If the hint given is in ('all', 'all_Integral', 'best'), then this function 

returns a nested dict, with the keys, being the set of classified hints 

returned by classifier functions, and the values being the dict of form 

as mentioned above. 

Key 'eq' is a common key to all the above mentioned hints which returns an 

expression if eq given by user is an Equality. 

See Also 

======== 

classify_ode(ode.py) 

classify_pde(pde.py) 

""" 

prep = kwargs.pop('prep', True) 

if isinstance(eq, Equality): 

eq = eq.lhs - eq.rhs 

# preprocess the equation and find func if not given 

if prep or func is None: 

eq, func = _preprocess(eq, func) 

prep = False 

# type is an argument passed by the solve functions in ode and pde.py 

# that identifies whether the function caller is an ordinary 

# or partial differential equation. Accordingly corresponding 

# changes are made in the function. 

type = kwargs.get('type', None) 

xi = kwargs.get('xi') 

eta = kwargs.get('eta') 

x0 = kwargs.get('x0', 0) 

terms = kwargs.get('n') 

if type == 'ode': 

from sympy.solvers.ode import classify_ode, allhints 

classifier = classify_ode 

string = 'ODE ' 

dummy = '' 

elif type == 'pde': 

from sympy.solvers.pde import classify_pde, allhints 

classifier = classify_pde 

string = 'PDE ' 

dummy = 'p' 

# Magic that should only be used internally. Prevents classify_ode from 

# being called more than it needs to be by passing its results through 

# recursive calls. 

if kwargs.get('classify', True): 

hints = classifier(eq, func, dict=True, ics=ics, xi=xi, eta=eta, 

n=terms, x0=x0, prep=prep) 

else: 

# Here is what all this means: 

# 

# hint: The hint method given to _desolve() by the user. 

# hints: The dictionary of hints that match the DE, along with other 

# information (including the internal pass-through magic). 

# default: The default hint to return, the first hint from allhints 

# that matches the hint; obtained from classify_ode(). 

# match: Dictionary containing the match dictionary for each hint 

# (the parts of the DE for solving). When going through the 

# hints in "all", this holds the match string for the current 

# hint. 

# order: The order of the DE, as determined by ode_order(). 

hints = kwargs.get('hint', 

{'default': hint, 

hint: kwargs['match'], 

'order': kwargs['order']}) 

if hints['order'] == 0: 

raise ValueError( 

str(eq) + " is not a differential equation in " + str(func)) 

if not hints['default']: 

# classify_ode will set hints['default'] to None if no hints match. 

if hint not in allhints and hint != 'default': 

raise ValueError("Hint not recognized: " + hint) 

elif hint not in hints['ordered_hints'] and hint != 'default': 

raise ValueError(string + str(eq) + " does not match hint " + hint) 

else: 

raise NotImplementedError(dummy + "solve" + ": Cannot solve " + str(eq)) 

if hint == 'default': 

return _desolve(eq, func, ics=ics, hint=hints['default'], simplify=simplify, 

prep=prep, x0=x0, classify=False, order=hints['order'], 

match=hints[hints['default']], xi=xi, eta=eta, n=terms, type=type) 

elif hint in ('all', 'all_Integral', 'best'): 

retdict = {} 

failedhints = {} 

gethints = set(hints) - set(['order', 'default', 'ordered_hints']) 

if hint == 'all_Integral': 

for i in hints: 

if i.endswith('_Integral'): 

gethints.remove(i[:-len('_Integral')]) 

# special cases 

for k in ["1st_homogeneous_coeff_best", "1st_power_series", 

"lie_group", "2nd_power_series_ordinary", "2nd_power_series_regular"]: 

if k in gethints: 

gethints.remove(k) 

for i in gethints: 

sol = _desolve(eq, func, ics=ics, hint=i, x0=x0, simplify=simplify, prep=prep, 

classify=False, n=terms, order=hints['order'], match=hints[i], type=type) 

retdict[i] = sol 

retdict['all'] = True 

retdict['eq'] = eq 

return retdict 

elif hint not in allhints: # and hint not in ('default', 'ordered_hints'): 

raise ValueError("Hint not recognized: " + hint) 

elif hint not in hints: 

raise ValueError(string + str(eq) + " does not match hint " + hint) 

else: 

# Key added to identify the hint needed to solve the equation 

hints['hint'] = hint 

hints.update({'func': func, 'eq': eq}) 

return hints