Izvestiya of Saratov University.
ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

generalized solution

A Mixed Problem for a System of First Order Differential Equations with Continuous Potential

We study a mixed problem for a first order differential system with two independent variables and continuous potential when the initial condition is an arbitrary square summable vector-valued function. The corresponding spectral problem is the Dirac system. It sets the convergence almost everywhere of a formal decision, obtained by the Fourier method. It is shown that the sum of a formal decision is a generalized solution of a mixed problem, understood as the limit of classical solutions for the case of smooth approximation of the initial data of the problem.

Criterion for a Generalized Solution in the Class Lp for the Wave Equation to Be in the Class Wp^1

In this paper we consider the question of whether a generalized solution of the wave equation belongs to different function spaces. Consideration of classical solutions imposes substantial restrictions on the initial data of the problem. But if we proceed not from differential but from integral equations, then the class of solutions and the class of initial boundary value problems can be substantially expanded. To solve the boundary value problem for the wave equation obtained by the wave counting method, it is easy to obtain a sufficient condition for belonging to a particular class.