International Journal of Differential Equations
Volume 2011 (2011), Article ID 268465, 10 pages
http://dx.doi.org/10.1155/2011/268465
Research Article

Boundary Value Problems with Integral Gluing Conditions for Fractional-Order Mixed-Type Equation

A. S. Berdyshev,1 E. T. Karimov,2 and N. Akhtaeva1

1Kazakh National Pedagogical University named after Abai, 050010 Almaty, Kazakhstan
2University of Santiago de Compostela, 15782 Santiago de Compostela, Spain

Received 18 May 2011; Accepted 11 October 2011

Academic Editor: Shaher M. Momani

Copyright © 2011 A. S. Berdyshev 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

Analogs of the Tricomi and the Gellerstedt problems with integral gluing conditions for mixed parabolic-hyperbolic equation with parameter have been considered. The considered mixed-type equation consists of fractional diffusion and telegraph equation. The Tricomi problem is equivalently reduced to the second-kind Volterra integral equation, which is uniquely solvable. The uniqueness of the Gellerstedt problem is proven by energy integrals' method and the existence by reducing it to the ordinary differential equations. The method of Green functions and properties of integral-differential operators have been used.