The Cole–Hopf transformation is a method of solving parabolic

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

s (PDEs) with a quadratic nonlinearity of the form:

u_ - a\Delta u + b\, \nabla u\, ^ = 0, \quad u(0,x) = g(x)

where

x\in \mathbb^

a,b

are constants,

\Delta

is the

Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is t ...

\nabla

is the

gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...

, and

\, \cdot\, ^

is the

\ell^

norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...

. By assuming that

w = \phi(u)

, where

\phi(\cdot)

is an unknown smooth function, we may calculate:

w_ = \phi'(u)u_, \quad \Delta w = \phi'(u)\Delta u + \phi''(u)\, \nabla u\, ^

Which implies that:

\begin
w_ = \phi'(u)u_ &= \phi'(u)\left( a\Delta u - b\, \nabla u\, ^\right) \\
&= a\Delta w - (a\phi'' + b\phi')\, \nabla u\, ^ \\
&= a\Delta w
\end

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: This is ''the'' ''Cole-Hopf transformation''. With the transformation, the following initial-value problem can now be solved:

w_ - a\Delta w = 0, \quad w(0,x) = e^

The unique, bounded solution of this system is:

w(t,x) =  \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) = -\log \left \int_ e^dy \right

Applications

Aerodynamics Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dyn ...

* Stochastic optimal control * Solving the viscous Burgers' equation

References

{{DEFAULTSORT:Cole-Hopf transformation Partial differential equations Transformation (function) Parabolic partial differential equations