Substituting a differential equation into an expression

Say I have a (heat) partial differential equation of $f(t,x)$ $$\frac{\partial f}{\partial t}=\frac12\frac{\partial^2 f}{\partial x^2}.$$ I would like to express an algebraic function $g$ of the variable $\Big(\frac{\partial f}{\partial t}, \frac{\partial^2 f}{\partial t\partial x}\Big)$ all in the partial derivatives of $x$. That is $$\frac{\partial^2 f}{\partial t\partial x}=\frac12\frac{\partial^3 f}{\partial x^3}.$$ The result should be $$g\Big(\frac{\partial f}{\partial t}, \frac{\partial^2 f}{\partial t\partial x}\Big)=g\Big(\frac12\frac{\partial^2 f}{\partial x^2}, \frac12\frac{\partial^3 f}{\partial x^3}\Big).$$

How can I accomplish this with SageMath?

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This is tricky, but it can be done. The idea is to replace (in the expression tree) derivatives of $f$ (with at least one derivative with respect to $t$) by the right-hand side of the equation, substituted into itself as many times as there are derivatives with respect to $t$ minus one, and finally replacing $f$ by the derivative of $f$ with respect to the remaining variables (if any).

My attempt to implement this (based on SubstituteFunction) is below, with an example (your example) in the docstring.

from sage.symbolic.expression_conversions import ExpressionTreeWalker
from sage.symbolic.operators import FDerivativeOperator

class SubstituteEvolutionaryPDE(ExpressionTreeWalker):
def __init__(self, pde, t):
"""
A class that walks the tree and replaces derivatives of f with
respect to t by the right-hand side of an evolutionary PDE.
EXAMPLES::
sage: var('x,t'); f = function('f'); g = function('g')
sage: pde = diff(f(t,x),t) == 1/2*diff(f(t,x),x,x)
sage: s = SubstituteEvolutionaryPDE(pde, t)
sage: h = g(diff(f(t,x),t), diff(f(t,x),t,x))
sage: s(h)
g(1/2*diff(f(t, x), x, x), 1/2*diff(f(t, x), x, x, x))

ASSUMPTION::

pde is of the form diff(f(t,...),t) == ... and the first
argument of f is always t.
"""
f_expr = integrate(pde.lhs(), t)
self.f = f_expr.operator()
self.args = f_expr.operands()
self.rhs = pde.rhs()

def derivative(self, ex, operator):
if operator.function() == self.f and 0 in operator.parameter_set(): # f is differentiated with respect to t
t_derivatives = [p for p in operator.parameter_set() if p == 0] # (assumes t is the first argument)
result = self.rhs
for _ in range(len(t_derivatives)-1):
result = result.substitute_function(self.f, self.rhs.function(*self.args))
if len(t_derivatives) < len(operator.parameter_set()): # derivatives w.r.t variables other than t
other_derivatives = [p for p in operator.parameter_set() if not p == 0] # (assumes t is the first argument)
new_operator = FDerivativeOperator(operator.function(), other_derivatives)
result = result.substitute_function(self.f, new_operator(*[self(_) for _ in ex.operands()]).function(*self.args))
return result
else:
return operator(*[self(_) for _ in ex.operands()])


Edit: updated to use substitute_function correctly (the second argument should also be a function).

more

Impressive! I am digesting your answer. What does the * in 'new_operator(*[self(_) for _ in ex.operands()])' do?

( 2018-11-01 14:57:53 -0500 )edit

This "unpacks" the list so e.g. f(*[t,x]) == f(t,x); it's a way to insert positional arguments from a (possibly dynamically generated, e.g. of variable length) list.

( 2018-11-01 15:21:12 -0500 )edit