DOI: https://doi.org/10.36719/2789-6919/45/224-233
Oruj Huseynov
Ganja State University
PhD in Mathematics
https://orcid.org/0009-0008-4144-8975
oruc.huseynov@gdu.edu.az
Tahir Mammadov
Ganja State University
PhD in Mathematics
https://orcid.org/0009-0002-9546-3683
tahir.mammadov@gdu.edu.az
Elchin Javadov
Ganja State University
PhD in Engineering
https://orcid.org/0009-0002-8868-9617
elcin.cavadov@gdu.edu.az
Nazir İsmayilov
Ganja State University
https://orcid.org/0009-0009-9100-9302
nazir.ismayilov@gdu.edu.az
Fundamental Solution of Multipoint Nonlocal Boundary Value Problems for Third-Order Hyperbolic Equations and Integral Representation of the Solution in S.L.Sobolev Type Spaces
Abstract
In the literature, the issues of correct solvability and construction of fundamental solutions of local and nonlocal problems for linear hyperbolic equations with discontinuous coefficients or coefficients from Lp have been little studied. Therefore, it is of theoretical importance to study these issues together with the question of finding Isomorphisms associated with local and nonlocal problems for such equations.
In this paper, we consider a multipoint nonlocal problem associated with finding the solution to a third-order hyperbolic equation with a dominant mixed derivative that satisfies multipoint boundary conditions. These multipoint nonlocal conditions generalize all previously known classes of such conditions in the case of smooth and continuous coefficients. Using one version of the perturbation method, we find sufficient conditions under which the operator of the problem generates an isomorphism together with its adjoint operator between the pores of S.L. Sobolev-type spaces. Under these same conditions, by constructing special conjugate systems of the type of “integro-algebraic” equations, the concept of a fundamental solution to the problem under consideration was also introduced and an integral representation of the solution to the inhomogeneous problem was obtained.
Keywords: Spaces of S.L.Sobolev type, generalized derivatives in the sense of S.L.Sobolev, third-order hyperbolic equations, nonlocal problem, coupled system, fundamental solution, integral representation of the solution