Cole–Hopf transformation explained

The Cole–Hopf transformation is a change of variables that allows to transform a special kind of parabolic partial differential equations (PDEs) with a quadratic nonlinearity into a linear heat equation. In particular, it provides an explicit formula for fairly general solutions of the PDE in terms of the initial datum and the heat kernel.

Consider the following PDE:$$u\_\; -\; a\backslash Delta\; u\; +\; b\backslash |\backslash nabla\; u\backslash |^\; =\; 0,\; \backslash quad\; u(0,x)\; =\; g(x)$$where

x\in R ^{n} , a,b are constants, \Delta is the Laplace operator, \nabla is the gradient, and \| ⋅ \| is the Euclidean norm in R ^{n} . By assuming that w = \phi(u) , where \phi( ⋅ ) is an unknown smooth function, we may calculate:$$w\_\; =\; \backslash phi\text{'}(u)u\_,\; \backslash quad\; \backslash Delta\; w\; =\; \backslash phi\text{'}(u)\backslash Delta\; u\; +\; \backslash phi$$(u)\|\nabla u\|^Which implies that: + b\phi')\|\nabla u\|^ \\&= a\Delta w\endif 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:

This is the Cole-Hopf transformation.^[1] With the transformation, the following initial-value problem can now be solved:$$w\_\; -\; a\backslash Delta\; w\; =\; 0,\; \backslash quad\; w(0,x)\; =\; e^$$The unique, bounded solution of this system is:$$w(t,x)\; =\; \backslash int\_\; e^dy$$Since the Cole–Hopf transformation implies that

u = - (a/b) log w , the solution of the original nonlinear PDE is:$$u(t,x)\; =\; -\backslash log\; \backslash left[\{1\backslash over\{(4\backslash pi\; at)^\{n/2\}\}\}\; \backslash int\_\{\backslash mathbb\{R\}^\{n\}\}\; e^\{-\backslash |x-y\backslash |^\{2\}/4at\; -\; bg(y)/a\}dy\; \backslash right]$$

Applications

• Aerodynamics^[2]
• Stochastic optimal control
• Solving the viscous Burgers' equation^[3]

References

1. Book: Evans, Lawrence C. . Lawrence C. Evans

   . Lawrence C. Evans. Partial Differential Equations . American Mathematical Society . 2010 . 2nd . Graduate Studies in Mathematics . 19 . 206–207.

2. Cole . Julian D. . Julian Cole . 1951 . On a quasi-linear parabolic equation occurring in aerodynamics . Quarterly of Applied Mathematics . en . 9 . 3 . 225–236 . 10.1090/qam/42889 . 0033-569X. free .
3. Hopf . Eberhard . Eberhard Hopf . 1950 . The partial differential equation ut + uux = μxx . Communications on Pure and Applied Mathematics . en . 3 . 3 . 201–230 . 10.1002/cpa.3160030302.