274 lines
10 KiB
Python
274 lines
10 KiB
Python
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"""Utility functions for classifying and solving
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ordinary and partial differential equations.
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Contains
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========
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_preprocess
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ode_order
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_desolve
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"""
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from sympy.core import Pow
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from sympy.core.function import Derivative, AppliedUndef
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from sympy.core.relational import Equality
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from sympy.core.symbol import Wild
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def _preprocess(expr, func=None, hint='_Integral'):
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"""Prepare expr for solving by making sure that differentiation
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is done so that only func remains in unevaluated derivatives and
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(if hint doesn't end with _Integral) that doit is applied to all
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other derivatives. If hint is None, don't do any differentiation.
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(Currently this may cause some simple differential equations to
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fail.)
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In case func is None, an attempt will be made to autodetect the
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function to be solved for.
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>>> from sympy.solvers.deutils import _preprocess
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>>> from sympy import Derivative, Function
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>>> from sympy.abc import x, y, z
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>>> f, g = map(Function, 'fg')
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If f(x)**p == 0 and p>0 then we can solve for f(x)=0
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>>> _preprocess((f(x).diff(x)-4)**5, f(x))
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(Derivative(f(x), x) - 4, f(x))
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Apply doit to derivatives that contain more than the function
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of interest:
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>>> _preprocess(Derivative(f(x) + x, x))
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(Derivative(f(x), x) + 1, f(x))
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Do others if the differentiation variable(s) intersect with those
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of the function of interest or contain the function of interest:
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>>> _preprocess(Derivative(g(x), y, z), f(y))
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(0, f(y))
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>>> _preprocess(Derivative(f(y), z), f(y))
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(0, f(y))
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Do others if the hint doesn't end in '_Integral' (the default
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assumes that it does):
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>>> _preprocess(Derivative(g(x), y), f(x))
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(Derivative(g(x), y), f(x))
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>>> _preprocess(Derivative(f(x), y), f(x), hint='')
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(0, f(x))
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Don't do any derivatives if hint is None:
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>>> eq = Derivative(f(x) + 1, x) + Derivative(f(x), y)
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>>> _preprocess(eq, f(x), hint=None)
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(Derivative(f(x) + 1, x) + Derivative(f(x), y), f(x))
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If it's not clear what the function of interest is, it must be given:
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>>> eq = Derivative(f(x) + g(x), x)
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>>> _preprocess(eq, g(x))
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(Derivative(f(x), x) + Derivative(g(x), x), g(x))
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>>> try: _preprocess(eq)
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... except ValueError: print("A ValueError was raised.")
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A ValueError was raised.
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"""
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if isinstance(expr, Pow):
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# if f(x)**p=0 then f(x)=0 (p>0)
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if (expr.exp).is_positive:
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expr = expr.base
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derivs = expr.atoms(Derivative)
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if not func:
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funcs = set().union(*[d.atoms(AppliedUndef) for d in derivs])
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if len(funcs) != 1:
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raise ValueError('The function cannot be '
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'automatically detected for %s.' % expr)
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func = funcs.pop()
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fvars = set(func.args)
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if hint is None:
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return expr, func
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reps = [(d, d.doit()) for d in derivs if not hint.endswith('_Integral') or
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d.has(func) or set(d.variables) & fvars]
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eq = expr.subs(reps)
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return eq, func
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def ode_order(expr, func):
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"""
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Returns the order of a given differential
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equation with respect to func.
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This function is implemented recursively.
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Examples
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========
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>>> from sympy import Function
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>>> from sympy.solvers.deutils import ode_order
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>>> from sympy.abc import x
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>>> f, g = map(Function, ['f', 'g'])
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>>> ode_order(f(x).diff(x, 2) + f(x).diff(x)**2 +
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... f(x).diff(x), f(x))
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2
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>>> ode_order(f(x).diff(x, 2) + g(x).diff(x, 3), f(x))
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2
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>>> ode_order(f(x).diff(x, 2) + g(x).diff(x, 3), g(x))
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3
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"""
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a = Wild('a', exclude=[func])
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if expr.match(a):
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return 0
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if isinstance(expr, Derivative):
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if expr.args[0] == func:
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return len(expr.variables)
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else:
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order = 0
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for arg in expr.args[0].args:
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order = max(order, ode_order(arg, func) + len(expr.variables))
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return order
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else:
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order = 0
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for arg in expr.args:
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order = max(order, ode_order(arg, func))
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return order
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def _desolve(eq, func=None, hint="default", ics=None, simplify=True, *, prep=True, **kwargs):
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"""This is a helper function to dsolve and pdsolve in the ode
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and pde modules.
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If the hint provided to the function is "default", then a dict with
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the following keys are returned
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'func' - It provides the function for which the differential equation
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has to be solved. This is useful when the expression has
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more than one function in it.
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'default' - The default key as returned by classifier functions in ode
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and pde.py
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'hint' - The hint given by the user for which the differential equation
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is to be solved. If the hint given by the user is 'default',
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then the value of 'hint' and 'default' is the same.
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'order' - The order of the function as returned by ode_order
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'match' - It returns the match as given by the classifier functions, for
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the default hint.
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If the hint provided to the function is not "default" and is not in
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('all', 'all_Integral', 'best'), then a dict with the above mentioned keys
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is returned along with the keys which are returned when dict in
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classify_ode or classify_pde is set True
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If the hint given is in ('all', 'all_Integral', 'best'), then this function
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returns a nested dict, with the keys, being the set of classified hints
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returned by classifier functions, and the values being the dict of form
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as mentioned above.
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Key 'eq' is a common key to all the above mentioned hints which returns an
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expression if eq given by user is an Equality.
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See Also
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========
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classify_ode(ode.py)
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classify_pde(pde.py)
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"""
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if isinstance(eq, Equality):
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eq = eq.lhs - eq.rhs
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# preprocess the equation and find func if not given
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if prep or func is None:
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eq, func = _preprocess(eq, func)
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prep = False
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# type is an argument passed by the solve functions in ode and pde.py
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# that identifies whether the function caller is an ordinary
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# or partial differential equation. Accordingly corresponding
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# changes are made in the function.
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type = kwargs.get('type', None)
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xi = kwargs.get('xi')
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eta = kwargs.get('eta')
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x0 = kwargs.get('x0', 0)
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terms = kwargs.get('n')
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if type == 'ode':
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from sympy.solvers.ode import classify_ode, allhints
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classifier = classify_ode
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string = 'ODE '
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dummy = ''
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elif type == 'pde':
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from sympy.solvers.pde import classify_pde, allhints
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classifier = classify_pde
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string = 'PDE '
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dummy = 'p'
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# Magic that should only be used internally. Prevents classify_ode from
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# being called more than it needs to be by passing its results through
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# recursive calls.
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if kwargs.get('classify', True):
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hints = classifier(eq, func, dict=True, ics=ics, xi=xi, eta=eta,
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n=terms, x0=x0, hint=hint, prep=prep)
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else:
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# Here is what all this means:
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#
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# hint: The hint method given to _desolve() by the user.
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# hints: The dictionary of hints that match the DE, along with other
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# information (including the internal pass-through magic).
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# default: The default hint to return, the first hint from allhints
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# that matches the hint; obtained from classify_ode().
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# match: Dictionary containing the match dictionary for each hint
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# (the parts of the DE for solving). When going through the
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# hints in "all", this holds the match string for the current
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# hint.
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# order: The order of the DE, as determined by ode_order().
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hints = kwargs.get('hint',
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{'default': hint,
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hint: kwargs['match'],
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'order': kwargs['order']})
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if not hints['default']:
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# classify_ode will set hints['default'] to None if no hints match.
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if hint not in allhints and hint != 'default':
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raise ValueError("Hint not recognized: " + hint)
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elif hint not in hints['ordered_hints'] and hint != 'default':
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raise ValueError(string + str(eq) + " does not match hint " + hint)
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# If dsolve can't solve the purely algebraic equation then dsolve will raise
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# ValueError
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elif hints['order'] == 0:
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raise ValueError(
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str(eq) + " is not a solvable differential equation in " + str(func))
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else:
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raise NotImplementedError(dummy + "solve" + ": Cannot solve " + str(eq))
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if hint == 'default':
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return _desolve(eq, func, ics=ics, hint=hints['default'], simplify=simplify,
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prep=prep, x0=x0, classify=False, order=hints['order'],
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match=hints[hints['default']], xi=xi, eta=eta, n=terms, type=type)
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elif hint in ('all', 'all_Integral', 'best'):
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retdict = {}
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gethints = set(hints) - {'order', 'default', 'ordered_hints'}
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if hint == 'all_Integral':
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for i in hints:
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if i.endswith('_Integral'):
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gethints.remove(i[:-len('_Integral')])
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# special cases
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for k in ["1st_homogeneous_coeff_best", "1st_power_series",
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"lie_group", "2nd_power_series_ordinary", "2nd_power_series_regular"]:
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if k in gethints:
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gethints.remove(k)
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for i in gethints:
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sol = _desolve(eq, func, ics=ics, hint=i, x0=x0, simplify=simplify, prep=prep,
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classify=False, n=terms, order=hints['order'], match=hints[i], type=type)
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retdict[i] = sol
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retdict['all'] = True
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retdict['eq'] = eq
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return retdict
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elif hint not in allhints: # and hint not in ('default', 'ordered_hints'):
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raise ValueError("Hint not recognized: " + hint)
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elif hint not in hints:
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raise ValueError(string + str(eq) + " does not match hint " + hint)
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else:
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# Key added to identify the hint needed to solve the equation
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hints['hint'] = hint
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hints.update({'func': func, 'eq': eq})
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return hints
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