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.

Contents

Explanation by exampleEdit

For example, the following simplified form of the Black–Scholes PDE

 

is reducible to the heat equation

 

by the change of variables:[1]

 
 
 
 

in these steps:

  • Replace   by   and apply the chain rule to get
 
  • Replace   and   by   and   to get
 
  • Replace   and   by   and   and divide both sides by   to get
 
  • Replace   by   and divide through by   to yield the heat equation.

Advice on the application of change of variable to PDEs is given by mathematician J. Michael Steele:[2]

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. If we know that   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  ,   and  are easy to compute and at the end, the original equation stands ready for immediate use.

Technique in generalEdit

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 (where again it needs to be a bijection), and
  • Enumerate the (zero or more finite list) of exceptions (poles) where the otherwise-bijection fails (and say why these exceptions don't restrict the applicability of the solution of the reduced equation to the original equation)

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  .

Action-angle coordinatesEdit

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.[3]

ReferencesEdit

  1. ^ Ömür Ugur, An Introduction to Computational Finance, Series in Quantitative Finance, v. 1, Imperial College Press, 298 pp., 2009
  2. ^ J. Michael Steele, Stochastic Calculus and Financial Applications, Springer, New York, 2001
  3. ^ V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, v. 60, Springer-Verlag, New York, 1989