Change of variables (PDE) explained

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

(\partial V)/(\partial t) + (1)/(2) S ^{2 (\partial ^{2} V)/(\partial S ^{2})} + S (\partial V)/(\partial S) - V = 0 .

is reducible to the heat equation

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

• Replace

x(S) and \tau(t) by ln (S) and (1)/(2) (T - t) to get(1)/(2) \left( -2 v( ln (S), (1)/(2) (T - t)) - (\partial v( ln (S), (1)/(2) (T - t)))/(+ \partial\tau) (\partial v( ln (S), (1)/(2) (T - t)))/(+ \partial x) (\partial ^{2} v( ln (S), (1)/(2) (T - t)))/(\partial x ^{2}) \right) =0 .

• Replace

ln (S) and (1)/(2) (T - t) by x(S) and \tau(t) and divide both sides by (1)/(2) to get-2 v - (\partial v)/(+ \partial\tau) (\partial v)/(\partial x) + (\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:^[1]

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 thatx _{1=a(x,t)} x _{2=b(x,t)}

and functions

e(x _{1,x} _{2),f(x} _{1,x} _{2)} such thatx = 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{L} is a differential operator such thatl{L}u(x,t)=0

Then it is also the case that

l{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

l{L}u(x,t)=0 to l{L}v(x _{1,x} _{2)=0:}

• Apply the chain rule to

l{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 l{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.^[2]

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 • x _{i} = \partial H/\partial p _{j} and • p _{j} = - \partial H/\partial x _{j} , there exist n integrals I _{i} . There exists a change of variables from the coordinates \{ x _{1,} ... , x _{n,} p _{1,} ... , p _{n} \} to a set of variables \{ I _{1,} ... I _{n,} \varphi _{1,} ... , \varphi _{n} \} , in which the equations of motion become • I _{i} = 0 , • \varphi _{i} = \omega _{i(I} _{1,} ... , I _{n)} , where the functions \omega _{1,} ... , \omega _{n} are unknown, but depend only on I _{1,} ... , I _{n} . The variables I _{1,} ... , I _{n} are the action coordinates, the variables \varphi _{1,} ... , \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 • x = 2 p and • p = - 2 x , with Hamiltonian H(x,p) = x ^{2} + p ^{2} . This system can be rewritten as • I = 0 , • \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.^[3]

References

1. [J. Michael Steele]
2. Book: Huang . Weizhang . Russell . Russell . . Springer New York . 2011 . 141.
3. [Vladimir Arnold|V. I. Arnold]