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:

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 u(x,t) and a change of variables x_1,x_2 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. Suppose \mathcal{L} is a differential operator such that Then it is also the case that where and we operate as follows to go from to 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 n, with and, there exist n integrals I_i. 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 \dot{I} = 0,, where I and \varphi are the canonical polar coordinates: and. See V. I. Arnold, `Mathematical Methods of Classical Mechanics', for more details.