Boundary-value problems for wave equations with data on the whole boundary

• Main Authors: Makhmud A. Sadybekov, Nurgissa A. Yessirkegenov
• Format: Article
• Language: English
• Published: Texas State University 2016-10-01
• Series: Electronic Journal of Differential Equations
• Subjects: Wave equation; well-posedness of problems; classical solution; strong solution; d'Alembert's formula
• Online Access: http://ejde.math.txstate.edu/Volumes/2016/281/abstr.html

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.