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:
#by example;
#by giving the theory of the method.

Explanation by example

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

:$\frac + \frac S^2\frac + S\frac - V = 0.$

is reducible to the heat equation

:$\frac = \frac$

by the change of variables:

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

in these steps:

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

::$\frac\left(-2v(x(S),\tau)+2 \frac \frac +S\left(\left(2 \frac + S\frac\right) \frac + S \left(\frac\right)^2 \frac\right)\right)=0.$

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

::$\frac\left( -2v(\ln(S),\frac(T-t)) -\frac +\frac +\frac\right)=0.$

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

::$-2 v-\frac+\frac+ \frac=0.$

* 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 $\mathcal$ is a differential operator such that

:$\mathcalu(x,t)=0$

Then it is also the case that

:$\mathcalv(x_1,x_2)=0$

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 $\mathcalu(x,t)=0$ to $\mathcalv(x_1,x_2)=0:$
* Apply the chain rule to $\mathcal v(x_1(x,t),x_2(x,t))=0$ 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 $\mathcalv(x_1,x_2)=0$, 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 $\dot_i = \partial H/\partial p_j$ and $\dot_j = - \partial H/\partial x_j$, 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 $\dot_i = 0$, $\dot_i = \omega_i(I_1, \dots, I_n)$, 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 $\dot = 2p$ and $\dot = - 2x$, with Hamiltonian $H(x,p) = x^2 + p^2$. This system can be rewritten as $\dot = 0$, $\dot{\varphi} = 1$, 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. V. I. Arnold, ''Mathematical Methods of Classical Mechanics'', Graduate Texts in Mathematics, v. 60, Springer-Verlag, New York, 1989

References

Multivariable calculus
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