Non-local boundary value problems in the cylindrical domain fir the multidimenstional wave equation

Author(s):  S.A. Aldashev, Dr., Prof., Kazakh National Pedagogical University named after Abai, Almaty, Kazakhstan , аldash51@mail.ru

Issue:  Volume 50, №1

Rubric:  Mathematics

Annotation:  Hadamard showed that one of the fundamental problems of the mathematical physics - the study of the oscillating string - is ill-posed when the boundary value conditions are defined on the entire boundary of the domain. A.V. Bitsadze and A.M. Nakhushev noted that the Dirichlet problem is ill-posed (in terms of unique solvability) not only for the wave equation but for the general hyperbolic equations. The Di-richlet and Poincare problems, and their related local boundary value problems for multidimensional hyperbolic equations have been studied, and it has been shown that the unique solvability of these prob-lems crucially depends on the height of the cylindrical domain under study. Non-local boundary value problems for these equations have not yet been analyzed. Using the method proposed by the author ear-lier, this paper shows the unique solvability and obtains the explicit form of the classical solution for the multidimensional wave equation in the cylindrical domain, which are the generalization of the mixed problem and the Dirichlet and Poincare problems. We also obtain the criterion of uniqueness of the reg-ular solution of these problems

Keywords:  non-local problem, multidimensional equation, uniqueness criterion, unique solvability, Bessel functions

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