Hopf-Cole transformation

The Hopf-Cole transformation is a mathematical transformation that allows the non-linear (viscous) Burgers equation to be traced back to the linear heat conduction equation and thus to be solved. The transformation was discovered in 1950 and 1951 by Eberhard Hopf and Julian Cole, independently of one another.

Details

The one-dimensional viscous Burgers equation

${\displaystyle {\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}=\nu {\frac {\partial ^{2}u}{\partial x^{2}}}}$

is going through the transformation

${\displaystyle u=-2\nu {\frac {\partial }{\partial x}}\log(\phi )}$

into the heat conduction equation

${\displaystyle {\frac {\partial \phi }{\partial t}}=\nu {\frac {\partial ^{2}\phi }{\partial x^{2}}}}$

convicted. This results in the following formula for the solution of the Cauchy problem of the original equation:

${\displaystyle u(x,t)=-2\nu {\frac {\partial }{\partial x}}\log \left((4\pi \nu t)^{-1/2}\int _{-\infty }^{\infty }\exp {\Bigl [}-{\frac {(x-x')^{2}}{4\nu t}}-{\frac {1}{2\nu }}\int _{0}^{x'}u(x'',0)dx''{\Bigr ]}dx'\right).}$

generalization

The quasi-linear parabolic differential equation becomes somewhat more general

${\displaystyle {\frac {\partial u}{\partial t}}+-a\Delta u+b|\mathrm {grad} \,u|^{2}=0}$

through the transformation

${\displaystyle u=-{\frac {a}{b}}\log(\phi )}$

into the heat conduction equation

${\displaystyle {\frac {\partial \phi }{\partial t}}-a\Delta \phi =0}$

convicted.

swell

• JD Cole: On a quasi-linear parabolic equation occurring in aerodynamics. In: Quart. Appl. Math. 9, 1951, pp. 225-236.
• L. Debnath: Nonlinear partial differential equations for scientists and engineers. Birkhäuser, 1997, ISBN 0-8176-3902-0 , pp. 289-293.
• LC Evans: Partial Differential Equations. American Mathematical Society, 1999, ISBN 0-8218-0772-2 , pp. 194-195.
• E. Hopf: The partial differential equation ut + uux = μuxx. In: Commun. Pure Appl. Math. 3, 1950, pp. 201-230.