Equivalence Conditions and Invariants for the General Form of Burgers’ Equations

Equivalence of differential equations is one of the most important concepts in the theory of differential equations. In this paper, the moving coframe method is applied to solve the local equivalence problem for the general form of Burgers’ equation, which has two independent variables under action of a pseudo-group of contact transformations. Using this method, we found the structure equations and invariants of these equations, as a result some conditions for equivalence of them will be given.

On the Equivalence of Differential Equations in the Sense of Coincidence Reflecting Functions

We use a new method for constructing some differential equations which are equivalent to a given equation in the sense of having the same reflecting function. We completely solve the problem: when is a polynomial differential equation equivalent to a given polynomial differential equation? Many sufficient conditions have been established for one differential equation to be equivalent to a given differential equation. We apply the obtained results to study the boundary value problem of two equivalent differential equations.