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


 * $$ \frac{\partial V}{\partial t} + \frac{1}{2} S^2\frac{\partial^2 V}{\partial S^2} + S\frac{\partial V}{\partial S} - V = 0. $$

is reducible to the heat equation


 * $$ \frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x^2}$$

by the change of variables:


 * $$ V(S,t) = v(x(S),\tau(t)) $$
 * $$ x(S) = \ln(S) $$
 * $$ \tau(t) = \frac{1}{2} (T - t) $$
 * $$ v(x,\tau)=\exp(-(1/2)x-(9/4)\tau) u(x,\tau) $$

in these steps:


 * Replace $$V(S,t)$$ by $$ v(x(S),\tau(t)) $$ and apply the chain rule to get


 * $$\frac{1}{2}\left(-2v(x(S),\tau)+2 \frac{\partial\tau}{\partial t} \frac{\partial v}{\partial \tau} +S\left(\left(2 \frac{\partial x}{\partial S} + S\frac{\partial^2 x}{\partial S^2}\right)

\frac{\partial v}{\partial x} + S \left(\frac{\partial x}{\partial S}\right)^2 \frac{\partial^2 v}{\partial x^2}\right)\right)=0. $$


 * Replace $$x(S)$$ and $$\tau(t)$$ by $$\ln(S) $$ and $$\frac{1}{2}(T-t)$$ to get


 * $$\frac{1}{2}\left(

-2v(\ln(S),\frac{1}{2}(T-t)) -\frac{\partial v(\ln(S),\frac{1}{2}(T-t))}{\partial\tau} +\frac{\partial v(\ln(S),\frac{1}{2}(T-t))}{\partial x} +\frac{\partial^2 v(\ln(S),\frac{1}{2}(T-t))}{\partial x^2}\right)=0. $$


 * Replace $$\ln(S) $$ and $$\frac{1}{2}(T-t)$$ by $$x(S)$$ and $$\tau(t)$$ and divide both sides by $$\frac{1}{2}$$ to get


 * $$-2 v-\frac{\partial v}{\partial\tau}+\frac{\partial v}{\partial x}+ \frac{\partial^2 v}{\partial x^2}=0.$$


 * Replace $$v(x,\tau)$$ by $$\exp(-(1/2)x-(9/4)\tau) u(x,\tau) $$ and divide through by $$-\exp(-(1/2)x-(9/4)\tau)$$ to yield the heat equation.

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 $$a(x,t), b(x,t)$$ such that


 * $$x_1=a(x,t)$$
 * $$x_2=b(x,t)$$

and functions $$e(x_1,x_2),f(x_1,x_2)$$ such that


 * $$x=e(x_1,x_2)$$
 * $$t=f(x_1,x_2)$$

and furthermore such that


 * $$x_1=a(e(x_1,x_2),f(x_1,x_2))$$
 * $$x_2=b(e(x_1,x_2),f(x_1,x_2))$$

and


 * $$x=e(a(x,t),b(x,t))$$
 * $$t=f(a(x,t),b(x,t))$$

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 $$\mathcal{L}$$ is a differential operator such that


 * $$\mathcal{L}u(x,t)=0$$

Then it is also the case that


 * $$\mathcal{L}v(x_1,x_2)=0$$

where


 * $$v(x_1,x_2)=u(e(x_1,x_2),f(x_1,x_2))$$

and we operate as follows to go from $$\mathcal{L}u(x,t)=0$$ to $$\mathcal{L}v(x_1,x_2)=0:$$
 * Apply the chain rule to $$\mathcal{L} v(x_1(x,t),x_2(x,t))=0$$ and expand out giving equation $$e_1$$.
 * Substitute $$a(x,t)$$ for $$x_1(x,t)$$ and $$b(x,t)$$ for $$x_2(x,t)$$ in $$e_1$$ and expand out giving equation $$e_2$$.
 * Replace occurrences of $$x$$ by $$e(x_1,x_2)$$ and $$t$$ by $$f(x_1,x_2)$$ to yield $$\mathcal{L}v(x_1,x_2)=0$$, which will be free of $$x$$ and $$t$$.

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 $$ \dot{x}_i = \partial H/\partial p_j $$ and $$ \dot{p}_j = - \partial H/\partial x_j $$, there exist $$ n $$ integrals $$ I_i $$. There exists a change of variables from the coordinates $$ \{ x_1, \dots, x_n, p_1, \dots, p_n \} $$ to a set of variables $$ \{ I_1, \dots I_n, \varphi_1, \dots, \varphi_n \} $$, in which the equations of motion become $$ \dot{I}_i = 0 $$, $$ \dot{\varphi}_i = \omega_i(I_1, \dots, I_n) $$, where the functions $$ \omega_1, \dots, \omega_n $$ are unknown, but depend only on $$ I_1, \dots, I_n $$. The variables $$ I_1, \dots, I_n $$ are the action coordinates, the variables $$ \varphi_1, \dots, \varphi_n $$ 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 $$ \dot{x} = 2p $$ and $$ \dot{p} = - 2x $$, with Hamiltonian $$ H(x,p) = x^2 + p^2 $$. This system can be rewritten as $$ \dot{I} = 0 $$, $$ \dot{\varphi} = 1 $$, where $$ I $$ and $$ \varphi $$ are the canonical polar coordinates: $$ I = p^2 + q^2 $$ and $$ \tan(\varphi) = p/x $$. See V. I. Arnold, `Mathematical Methods of Classical Mechanics', for more details.