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: 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:[1] "There is nothing particularly difficult about changing variables and transforming one equation to another, but there is an element of tedium and complexity that slows us down.
There is no universal remedy for this molasses effect, but the calculations do seem to go more quickly if one follows a well-defined plan.
satisfies an equation (like the Black–Scholes equation) we are guaranteed that we can make good use of the equation in the derivation of the equation for a new function
defined in terms of the old if we write the old V as a function of the new v and write the new
and x as functions of the old t and S. This order of things puts everything in the direct line of fire of the chain rule; the partial derivatives
are easy to compute and at the end, the original equation stands ready for immediate use.
"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 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.
is a differential operator such that Then it is also the case that where and we operate as follows to go from
In the context of PDEs, Weizhang Huang and Robert D. Russell define and explain the different possible time-dependent transformations in details.
[2] 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
There exists a change of variables from the coordinates
to a set of variables
, in which the equations of motion become
are unknown, but depend only on
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
This system can be rewritten as
are the canonical polar coordinates:
tan ( φ ) = p
See V. I. Arnold, `Mathematical Methods of Classical Mechanics', for more details.