Abstract and Applied Analysis
Volume 2012 (2012), Article ID 219040, 27 pages
http://dx.doi.org/10.1155/2012/219040
Research Article

Solution of the First Boundary-Value Problem for a System of Autonomous Second-Order Linear Partial Differential Equations of Parabolic Type with a Single Delay

Josef Diblík,1,2 Denis Khusainov,3 Oleksandra Kukharenko,3 and Zdeněk Svoboda2

1Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 60200 Brno, Czech Republic
2Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 61600 Brno, Czech Republic
3Department of Cybernetics, Taras Shevchenko National University of Kyiv, Kyiv 01601, Ukraine

Received 29 March 2012; Accepted 9 May 2012

Academic Editor: Miroslava Růžičková

Copyright © 2012 Josef Diblík et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The first boundary-value problem for an autonomous second-order system of linear partial differential equations of parabolic type with a single delay is considered. Assuming that a decomposition of the given system into a system of independent scalar second-order linear partial differential equations of parabolic type with a single delay is possible, an analytical solution to the problem is given in the form of formal series and the character of their convergence is discussed. A delayed exponential function is used in order to analytically solve auxiliary initial problems (arising when Fourier method is applied) for ordinary linear differential equations of the first order with a single delay.