Change of variables (PDE)


Often a partial differential equation can be reduced to a simpler form with a known solution by a suitable change of variables.
The article discusses change of variable for PDEs below in two ways:
  1. by example;
  2. by giving the theory of the method.

Explanation by example

For example, the following simplified form of the Black–Scholes PDE
is reducible to the heat equation
by the change of variables:
in these steps:
Advice on the application of change of variable to PDEs is given by mathematician J. Michael Steele:

Technique in general

Suppose that we have a function and a change of variables such that there exist functions such that
and functions such that
and furthermore such that
and
In other words, it is helpful for there to be a bijection between the old set of variables and the new one, or else one has to
  • Restrict the domain of applicability of the correspondence to a subject of the real plane which is sufficient for a solution of the practical problem at hand, and
  • Enumerate the of exceptions where the otherwise-bijection fails
If a bijection does not exist then the solution to the reduced-form equation will not in general be a solution of the original equation.
We are discussing change of variable for PDEs. A PDE can be expressed as a differential operator applied to a function. Suppose is a differential operator such that
Then it is also the case that
where
and we operate as follows to go from to
  • Apply the chain rule to and expand out giving equation.
  • Substitute for and for in and expand out giving equation.
  • Replace occurrences of by and by to yield, which will be free of and.
In the context of PDEs, Weizhang Huang and Robert D. Russell define and explain the different possible time-dependent transformations in details.

Action-angle coordinates

Often, theory can establish the existence of a change of variables, although the formula itself cannot be explicitly stated. For an integrable Hamiltonian system of dimension, with and, there exist integrals. There exists a change of variables from the coordinates to a set of variables, in which the equations of motion become,, where the functions are unknown, but depend only on. The variables are the action coordinates, the variables are the angle coordinates. The motion of the system can thus be visualized as rotation on torii. As a particular example, consider the simple harmonic oscillator, with and, with Hamiltonian. This system can be rewritten as,, where and are the canonical polar coordinates: and. See V. I. Arnold, `Mathematical Methods of Classical Mechanics', for more details.