Partial differential equation
Change of variables

Often a PDE can be reduced to a simpler form with a known solution by a suitable change of variables. For example, the Black–Scholes equation ∂ V ∂ t + 1 2 σ 2 S 2 ∂ 2 V ∂ S 2 + r S ∂ V ∂ S − r V = 0 {\displaystyle {\frac {\partial V}{\partial t}}+{\tfrac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}-rV=0} is reducible to the heat equation ∂ u ∂ τ = ∂ 2 u ∂ x 2 {\displaystyle {\frac {\partial u}{\partial \tau }}={\frac {\partial ^{2}u}{\partial x^{2}}}} by the change of variables V (S, t) = v (x, τ), x = ln (S), τ = 1 2 σ 2 (T − t), v (x, τ) = e − α x − β τ u (x, τ). {\displaystyle {\begin{aligned}V(S,t)&=v(x,\tau),\\x&=\ln \left(S\right),\\\tau &={\tfrac {1}{2}}\sigma ^{2}(T-t),\\v(x,\tau)&=e^{-\alpha x-\beta \tau }u(x,\tau).\end{aligned}}}
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