Makhmud A. Sadybekov, Nurgissa A. Yessirkegenov
Abstract:
In this article we propose a new formulation of boundary-value problem for a
one-dimensional wave equation in a rectangular domain in which boundary
conditions are given on the whole boundary. We prove the well-posedness of
boundary-value problem in the classical and generalized senses.
To substantiate the well-posedness of this problem it is
necessary to have an effective representation of the general
solution of the problem. In this direction we obtain a convenient
representation of the general solution for the wave equation in a
rectangular domain based on d'Alembert classical formula. The
constructed general solution automatically satisfies the boundary
conditions by a spatial variable. Further, by setting different boundary
conditions according to temporary variable, we get some functional
or functional-differential equations. Thus, the proof of the
well-posedness of the formulated problem is reduced to question of the
existence and uniqueness of solutions of the corresponding
functional equations.
Submitted May 12, 2016. Published October 19, 2016.
Math Subject Classifications: 35L05, 35L20, 49K40, 35D35.
Key Words: Wave equation; well-posedness of problems; classical solution;
strong solution; d'Alembert's formula.
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Makhmud A. Sadybekov Institute of Mathematics and Mathematical Modeling 125 Pushkin str., 050010 Almaty, Kazakhstan email: sadybekov@math.kz |
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Nurgissa A. Yessirkegenov Institute of Mathematics and Mathematical Modeling 125 Pushkin str., 050010 Almaty, Kazakhstan email: n.yessirkegenov15@imperial.ac.uk |
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