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. 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 Stochastic optimal control Solving the viscous Burgers' equation References Partial differential equations Transformation (function) Parabolic partial differential equations