Английская Википедия:Change of variables (PDE)

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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

<math> \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. </math>

is reducible to the heat equation

<math> \frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x^2}</math>

by the change of variables:

<math> V(S,t) = v(x(S),\tau(t)) </math>

<math> x(S) = \ln(S) </math>

<math> \tau(t) = \frac{1}{2} (T - t) </math>

<math> v(x,\tau)=\exp(-(1/2)x-(9/4)\tau) u(x,\tau) </math>

in these steps:

• Replace <math>V(S,t)</math> by <math> v(x(S),\tau(t)) </math> and apply the chain rule to get

<math>\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. </math>

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

<math>\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.

</math>

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

<math>-2 v-\frac{\partial v}{\partial\tau}+\frac{\partial v}{\partial x}+ \frac{\partial^2 v}{\partial x^2}=0.</math>

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

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

Technique in general

Suppose that we have a function <math>u(x,t)</math> and a change of variables <math>x_1,x_2</math> such that there exist functions <math>a(x,t), b(x,t)</math> such that

<math>x_1=a(x,t)</math>

<math>x_2=b(x,t)</math>

and functions <math>e(x_1,x_2),f(x_1,x_2)</math> such that

<math>x=e(x_1,x_2)</math>

<math>t=f(x_1,x_2)</math>

and furthermore such that

<math>x_1=a(e(x_1,x_2),f(x_1,x_2))</math>

<math>x_2=b(e(x_1,x_2),f(x_1,x_2))</math>

and

<math>x=e(a(x,t),b(x,t))</math>

<math>t=f(a(x,t),b(x,t))</math>

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 <math>\mathcal{L}</math> is a differential operator such that

<math>\mathcal{L}u(x,t)=0</math>

Then it is also the case that

<math>\mathcal{L}v(x_1,x_2)=0</math>

where

<math>v(x_1,x_2)=u(e(x_1,x_2),f(x_1,x_2))</math>

and we operate as follows to go from <math>\mathcal{L}u(x,t)=0</math> to <math>\mathcal{L}v(x_1,x_2)=0:</math>

• Apply the chain rule to <math>\mathcal{L} v(x_1(x,t),x_2(x,t))=0</math> and expand out giving equation <math>e_1</math>.
• Substitute <math>a(x,t)</math> for <math>x_1(x,t)</math> and <math>b(x,t)</math> for <math>x_2(x,t)</math> in <math>e_1</math> and expand out giving equation <math>e_2</math>.
• Replace occurrences of <math>x</math> by <math>e(x_1,x_2)</math> and <math>t</math> by <math>f(x_1,x_2)</math> to yield <math>\mathcal{L}v(x_1,x_2)=0</math>, which will be free of <math>x</math> and <math>t</math>.

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