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.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 ) {displaystyle 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 ) {displaystyle 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 τ {displaystyle au } 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 ∂ V ∂ t {displaystyle {frac {partial V}{partial t}}} , ∂ V ∂ S {displaystyle {frac {partial V}{partial S}}} and ∂ 2 V ∂ S 2 {displaystyle {frac {partial ^{2}V}{partial S^{2}}}} are easy to compute and at the end, the original equation stands ready for immediate use. Often a partial differential equation can be reduced to a simpler form with a known solution by a suitable change of variables.