Using Partial Differential Algebraic Equations in Modelling

We consider evolutionary systems of partial differential equations depending on a single space variable. It is assumed that the matrices multiplying the derivatives of the desired vector-function are singular in the domain. Such systems are commonly called partial differential algebraic equations (PDAEs). Properties of PDEAs are essentially different to the properties of non-singular systems. In particular, it is impossible to define a type of a system judging by roots of characteristic polynomials. In this paper, we introduce a notion of splittable systems by which we mean systems allowing existence of non-singular transformations that lead to splitting of the original system to the subsystem with a unique solution and the non-singular subsystem of partial differential equations. Such an approach makes it possible to investigate the structure of general solutions to differential algebraic equations and, in some cases, to establish solvability of initial-boundary value problems.