When , Burgers' equation becomes the inviscid Burgers' equation:
which is a prototype for equations for which the solution can develop discontinuities (shock waves).
The inviscid Burgers' equation is a first order partial differential equation. Its solution can be constructed by the method of characteristics. This method yields that if is a solution of the ordinary differential equation
then is constant as a function of . Hence is a solution of the system of ordinary equations
The solutions of this system are given in terms of the initial values by
Substitute , then . Now the system becomes
This is an implicit relation that determines the solution of the inviscid Burgers' equation.
The viscous Burgers equation can be linearized by the Cole-Hopf substitution
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