The Structure theory of Second Order Scalar Partial Differential Equation

Abstract

The construction of the structure theory of second order scalar differential equations in the plane is considered. Because of this we introduce the two main classes of partial differential equations which are the subject of this dissertation. The classes are the determined first order systems of partial differential equations for two unknown functions of two variables and second order scalar partial differential equations in the plane. These two classes of equation have similar geometric structure, and this is precisely the reason that we can develop the theory for these systems in parallel. Therefore, we investigate a formulation of the structure theory of second order scalar differential equations in the plane in terms of contact distribution. In this research we consider the work of vessiot that explain the geometric characterization of these equations which summarized by Stomark and Duistermaat, to discuss a new formulation and extend the structure theory given by this authers, and then we construct a framing on the equation manifold of first 4‫‪order‬‬ ‫‪systems‬‬ ‫‪that‬‬ ‫‪are‬‬ ‫‪invariant‬‬ ‫‪under‬‬ ‫‪general‬‬ ‫‪contact‬‬ ‫‪transformation‬‬ ‫‪as‬‬ ‫‪an‬‬ ‫‪application‬‬ ‫‪of‬‬ ‫‪the‬‬ ‫‪theory‬‬ ‫‪.‬‬