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

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

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

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

• 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

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

• 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: Template:Quotation

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 $ℒ$ is a differential operator such that

$ℒu(x,t)=0$

Then it is also the case that

$ℒv(x_1,x_2)=0$

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 $ℒu(x,t)=0$ to $ℒv(x_1,x_2)=0:$

• Apply the chain rule to $ℒv(x_1(x,t),x_2(x,t))=0$ 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 $ℒv(x_1,x_2)=0$, 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 $\{I_1,\dots I_n,{\phi }_1,\dots ,{\phi }_n\}$, in which the equations of motion become ${\dot{I}}_i=0$, ${\dot{\phi }}_i={\omega }_i(I_1,\dots ,I_n)$, 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 ${\phi }_1,\dots ,{\phi }_n$ 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 $\dot{I}=0$, $\dot{\phi }=1$, where $I$ and $\phi$ are the canonical polar coordinates: $I=p^2+q^2$ and $\tan (\phi )=p/x$. See V. I. Arnold, `Mathematical Methods of Classical Mechanics', for more details.