Change of variables (PDE)

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

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

is reducible to the heat equation

:$\backslash frac\{\backslash partial\; u\}\{\backslash partial\; \backslash tau\}\; =\; \backslash frac\{\backslash partial^2\; u\}\{\backslash partial\; x^2\}$

by the change of variables:

:$V(S,t)\; =\; v(x(S),\backslash tau(t))$

:$x(S)\; =\; \backslash ln(S)$

:$\backslash tau(t)\; =\; \backslash frac\{1\}\{2\}\; (T\; -\; t)$

:$v(x,\backslash tau)=\backslash exp(-(1/2)x-(9/4)\backslash tau)\; u(x,\backslash tau)$

in these steps:

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

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

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

::$\backslash frac\{1\}\{2\}\backslash left($

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

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

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

Advice on the application of change of variable to PDEs is given by mathematician J. Michael Steele:J. Michael Steele, Stochastic Calculus and Financial Applications, Springer, New York, 2001

{{quotation|"There is nothing particularly difficult about changing variables and transforming one equation to another, but there is an element of tedium and complexity that slows us down. There is no universal remedy for this molasses effect, but the calculations do seem to go more quickly if one follows a well-defined plan. If we know that $V(S,t)$ satisfies an equation (like the Black–Scholes equation) we are guaranteed that we can make good use of the equation in the derivation of the equation for a new function $v(x,t)$ defined in terms of the old if we write the old V as a function of the new v and write the new $\backslash tau$ and x as functions of the old t and S. This order of things puts everything in the direct line of fire of the chain rule; the partial derivatives $\backslash frac\{\backslash partial\; V\}\{\backslash partial\; t\}$, $\backslash frac\{\backslash partial\; V\}\{\backslash partial\; S\}$ and $\backslash frac\{\backslash partial^2\; V\}\{\backslash partial\; S^2\}$are easy to compute and at the end, the original equation stands ready for immediate use."}}

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

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

Then it is also the case that

:$\backslash mathcal\{L\}v(x\_1,x\_2)=0$

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 $\backslash mathcal\{L\}u(x,t)=0$ to $\backslash mathcal\{L\}v(x\_1,x\_2)=0:$

• Apply the chain rule to $\backslash mathcal\{L\}\; v(x\_1(x,t),x\_2(x,t))=0$ 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 $\backslash mathcal\{L\}v(x\_1,x\_2)=0$, 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.{{cite book |last1=Huang |first1=Weizhang |last2=Russell |first2=Russell |title=Adaptive moving mesh methods |publisher=Springer New York |publication-date=2011 |page=141}}

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 $\backslash dot\{x\}\_i\; =\; \backslash partial\; H/\backslash partial\; p\_j$ and $\backslash dot\{p\}\_j\; =\; -\; \backslash partial\; H/\backslash partial\; x\_j$, there exist $n$ integrals $I\_i$

. There exists a change of variables from the coordinates $\backslash \{\; x\_1,\; \backslash dots,\; x\_n,\; p\_1,\; \backslash dots,\; p\_n\; \backslash \}$ to a set of variables $\backslash \{\; I\_1,\; \backslash dots\; I\_n,\; \backslash varphi\_1,\; \backslash dots,\; \backslash varphi\_n\; \backslash \}$, in which the equations of motion become $\backslash dot\{I\}\_i\; =\; 0$, $\backslash dot\{\backslash varphi\}\_i\; =\; \backslash omega\_i(I\_1,\; \backslash dots,\; I\_n)$, where the functions $\backslash omega\_1,\; \backslash dots,\; \backslash omega\_n$ are unknown, but depend only on $I\_1,\; \backslash dots,\; I\_n$. The variables $I\_1,\; \backslash dots,\; I\_n$ are the action coordinates, the variables $\backslash varphi\_1,\; \backslash dots,\; \backslash varphi\_n$ 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 $\backslash dot\{x\}\; =\; 2p$ and $\backslash dot\{p\}\; =\; -\; 2x$, with Hamiltonian $H(x,p)\; =\; x^2\; +\; p^2$. This system can be rewritten as $\backslash dot\{I\}\; =\; 0$, $\backslash dot\{\backslash varphi\}\; =\; 1$, where $I$ and $\backslash varphi$ are the canonical polar coordinates: $I\; =\; p^2\; +\; q^2$ and $\backslash tan(\backslash varphi)\; =\; p/x$. 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

Category:Multivariable calculus

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