Cole–Hopf transformation
Appearance
The Cole–Hopf transformation is a method of solving parabolic partial differential equations (PDEs) with a quadratic nonlinearity of the form:where , are constants, is the Laplace operator, is the gradient, and is the -norm. By assuming that , where is an unknown smooth function, we may calculate:Which implies that:if we constrain to satisfy . 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:The unique, bounded solution of this system is:Since the Cole–Hopf transformation implies that , the solution of the original nonlinear PDE is:
Applications
- Aerodynamics[2]
- Stochastic optimal control
- Solving the viscous Burgers' equation[3]
References
- ^ Evans, Lawrence C. (2010). Partial Differential Equations. Graduate Studies in Mathematics. Vol. 19 (2nd ed.). American Mathematical Society. pp. 206–207.
- ^ Cole, Julian D. (1951). "On a quasi-linear parabolic equation occurring in aerodynamics". Quarterly of Applied Mathematics. 9 (3): 225–236. doi:10.1090/qam/42889. ISSN 0033-569X.
- ^ Hopf, Eberhard (1950). "The partial differential equation ut + uux = μxx". Communications on Pure and Applied Mathematics. 3 (3): 201–230. doi:10.1002/cpa.3160030302.