 Cole–Hopf transformation

The Cole-Hopf transformation $u=-2\nu {\varphi }_x/\varphi$ converts the viscous Burgers equation into the heat equation ${\varphi }_t=\nu {\varphi }_{xx}$. Initial data $u(x,0)=g(x)$ correspond to $\varphi (x,0)=C\exp [-(2\nu {)}^{-1}{\int }^{x}g(s)ds]$.

The Cole–Hopf transformation is a mathematical technique used to transform certain nonlinear partial differential equations into linear ones. It is primarily applied to the Burgers' equation, a fundamental equation in fluid dynamics and various fields of applied mathematics.