Cole–Hopf transformation

The Cole–Hopf transformation is a method of solving parabolic partial differential equations (PDEs) with a quadratic nonlinearity of the form:$$u_{t}-a\Delta u+b\|\nabla u\|^{2}=0,\quad u(0,x)=g(x)$$where $x\in \mathbb {R} ^{n}$, $a,b$ are constants, $\Delta$ is the Laplace operator, $\nabla$ is the gradient, and $\|\cdot \|^{2}$ is the $\ell ^{2}$-norm. By assuming that $w=\phi (u)$, where $\phi (\cdot )$ is an unknown smooth function, we may calculate:$$w_{t}=\phi '(u)u_{t},\quad \Delta w=\phi '(u)\Delta u+\phi ''(u)\|\nabla u\|^{2}$$Which implies that:$${\begin{aligned}w_{t}=\phi '(u)u_{t}&=\phi '(u)\left(a\Delta u-b\|\nabla u\|^{2}\right)\\&=a\Delta w-(a\phi ''+b\phi ')\|\nabla u\|^{2}\\&=a\Delta w\end{aligned}}$$if we constrain $\phi$ to satisfy $a\phi ''+b\phi '=0$. Then we may transform the original nonlinear PDE into the canonical heat equation by using the transformation:

$w(u)=e^{-bu/a}$

This is the Cole-Hopf transformation. With the transformation, the following initial-value problem can now be solved:$$w_{t}-a\Delta w=0,\quad w(0,x)=e^{-bg(x)/a}$$The unique, bounded solution of this system is:$$w(t,x)={1 \over {(4\pi at)^{n/2}}}\int _{\mathbb {R} ^{n}}e^{-\|x-y\|^{2}/4at-bg(y)/a}dy$$Since the Cole–Hopf transformation implies that $u=-(a/b)\log w$, the solution of the original nonlinear PDE is:$$u(t,x)=-{a \over {b}}\log \left[{1 \over {(4\pi at)^{n/2}}}\int _{\mathbb {R} ^{n}}e^{-\|x-y\|^{2}/4at-bg(y)/a}dy\right]$$

Applications

• Aerodynamics
• Stochastic optimal control
• Solving the viscous Burgers' equation