Source code for sympy.solvers.deutils

"""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 Function, Derivative, AppliedUndef
from sympy.core.relational import Equality, Eq
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

[docs]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