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.

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

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

is reducible to the heat equation

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

by the change of variables:

${\displaystyle V(S,t)=v(x(S),\tau (t))}$

${\displaystyle x(S)=\ln(S)}$

${\displaystyle \tau (t)={\frac {1}{2}}(T-t)}$

${\displaystyle v(x,\tau )=\exp(-(1/2)x-(9/4)\tau )u(x,\tau )}$

in these steps:

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

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

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

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

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

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

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

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

Suppose that we have a function ${\displaystyle u(x,t)}$and a change of variables ${\displaystyle x_{1},x_{2}}$such that there exist functions ${\displaystyle a(x,t),b(x,t)}$such that

${\displaystyle x_{1}=a(x,t)}$

${\displaystyle x_{2}=b(x,t)}$

and functions ${\displaystyle e(x_{1},x_{2}),f(x_{1},x_{2})}$such that

${\displaystyle x=e(x_{1},x_{2})}$

${\displaystyle t=f(x_{1},x_{2})}$

and furthermore such that

${\displaystyle x_{1}=a(e(x_{1},x_{2}),f(x_{1},x_{2}))}$

${\displaystyle x_{2}=b(e(x_{1},x_{2}),f(x_{1},x_{2}))}$

and

${\displaystyle x=e(a(x,t),b(x,t))}$

${\displaystyle t=f(a(x,t),b(x,t))}$

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

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

Then it is also the case that

${\displaystyle {\mathcal {L}}v(x_{1},x_{2})=0}$

where

${\displaystyle v(x_{1},x_{2})=u(e(x_{1},x_{2}),f(x_{1},x_{2}))}$

and we operate as follows to go from ${\displaystyle {\mathcal {L}}u(x,t)=0}$to ${\displaystyle {\mathcal {L}}v(x_{1},x_{2})=0:}$

• Apply the chain rule to ${\displaystyle {\mathcal {L}}v(x_{1}(x,t),x_{2}(x,t))=0}$and expand out giving equation ${\displaystyle e_{1}}$.
• Substitute ${\displaystyle a(x,t)}$for ${\displaystyle x_{1}(x,t)}$and ${\displaystyle b(x,t)}$for ${\displaystyle x_{2}(x,t)}$in ${\displaystyle e_{1}}$and expand out giving equation ${\displaystyle e_{2}}$.
• Replace occurrences of ${\displaystyle x}$by ${\displaystyle e(x_{1},x_{2})}$and ${\displaystyle t}$by ${\displaystyle f(x_{1},x_{2})}$to yield ${\displaystyle {\mathcal {L}}v(x_{1},x_{2})=0}$, which will be free of ${\displaystyle x}$and ${\displaystyle t}$.

In the context of PDEs, Weizhang Huang and Robert D. Russell define and explain the different possible time-dependent transformations in details.

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 ${\displaystyle n}$, with ${\displaystyle {\dot {x}}_{i}=\partial H/\partial p_{j}}$and ${\displaystyle {\dot {p}}_{j}=-\partial H/\partial x_{j}}$, there exist ${\displaystyle n}$integrals ${\displaystyle I_{i}}$. There exists a change of variables from the coordinates ${\displaystyle \{x_{1},\dots ,x_{n},p_{1},\dots ,p_{n}\}}$to a set of variables ${\displaystyle \{I_{1},\dots I_{n},\varphi _{1},\dots ,\varphi _{n}\}}$, in which the equations of motion become ${\displaystyle {\dot {I}}_{i}=0}$, ${\displaystyle {\dot {\varphi }}_{i}=\omega _{i}(I_{1},\dots ,I_{n})}$, where the functions ${\displaystyle \omega _{1},\dots ,\omega _{n}}$are unknown, but depend only on ${\displaystyle I_{1},\dots ,I_{n}}$. The variables ${\displaystyle I_{1},\dots ,I_{n}}$are the action coordinates, the variables ${\displaystyle \varphi _{1},\dots ,\varphi _{n}}$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 ${\displaystyle {\dot {x}}=2p}$and ${\displaystyle {\dot {p}}=-2x}$, with Hamiltonian ${\displaystyle H(x,p)=x^{2}+p^{2}}$. This system can be rewritten as ${\displaystyle {\dot {I}}=0}$, ${\displaystyle {\dot {\varphi }}=1}$, where ${\displaystyle I}$and ${\displaystyle \varphi }$are the canonical polar coordinates: ${\displaystyle I=p^{2}+q^{2}}$and ${\displaystyle \tan(\varphi )=p/x}$. See V. I. Arnold, `Mathematical Methods of Classical Mechanics', for more details.