Source code for sympy.solvers.pde

"""
Analytical methods for solving Partial Differential Equations
Currently implemented methods:
    - separation of variables - pde_separate

"""

from sympy import Eq, Equality
from sympy.simplify import simplify
from sympy.core.compatibility import reduce
from sympy.utilities.iterables import has_dups

import operator

[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)] See Also ======== pde_separate_add, pde_separate_mul """ do_add = False if strategy == 'add': do_add = True elif strategy == 'mul': do_add = False 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]) if do_add: functions = reduce(operator.add, sep) 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 if not do_add: 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)
[docs]def pde_separate_add(eq, fun, sep): """ 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)] """ return pde_separate(eq, fun, sep, strategy='add')
[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): terms.add(term) continue elif term.is_Derivative and not term.has(*others): terms.add(term) # 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 div.add(ext) # FIXME: Find lcm() of all the divisors and divide with it, instead of # current hack :( # http://code.google.com/p/sympy/issues/detail?id=1498 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 div.add(sep) rhs -= term.expand() # Do the division fulldiv = reduce(operator.add, div) lhs = simplify(lhs/fulldiv).expand() rhs = simplify(rhs/fulldiv).expand() # ...and check whether we were successful :) if lhs.has(*others) or rhs.has(dep): return None return [lhs, rhs]