Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 579645, 10 pages
http://dx.doi.org/10.1155/2011/579645
Research Article

On the General Solution of the Ultrahyperbolic Bessel Operator

Rattapan Damkengpan1 and Kamsing Nonlaopon1,2

1Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand
2Centre of Excellence in Mathematics, Commission on Higher Education (CHE), Si Ayuthaya road, Bangkok 10400, Thailand

Received 6 April 2011; Accepted 21 June 2011

Academic Editor: Alexei Mailybaev

Copyright © 2011 Rattapan Damkengpan and Kamsing Nonlaopon. 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

We study the general solution of equation □ 𝑘 𝐵 , 𝑐 𝑢 ( 𝑥 ) = 𝑓 ( 𝑥 ) , where □ 𝑘 𝐵 , 𝑐 is the ultrahyperbolic Bessel operator iterated 𝑘 -times and is defined by □ 𝑘 𝐵 , 𝑐 = [ ( 1 / 𝑐 2 ) ( 𝐵 𝑥 1 + 𝐵 𝑥 2 + ⋯ + 𝐵 𝑥 𝑝 ) − ( 𝐵 𝑥 𝑝 + 1 + ⋯ + 𝐵 𝑥 𝑝 + 𝑞 ) ] 𝑘 , 𝑝 + 𝑞 = 𝑛 , 𝑛 is the dimension of ℝ + 𝑛 = { 𝑥 ∶ 𝑥 = ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) , 𝑥 > 0 , … , 𝑥 𝑛 > 0 } , 𝐵 𝑥 𝑖 = 𝜕 2 / 𝜕 𝑥 2 𝑖 + ( 2 𝑣 𝑖 / 𝑥 𝑖 ) ( 𝜕 / 𝜕 𝑥 𝑖 ) , 2 𝑣 𝑖 = 2 𝛽 𝑖 + 1 , 𝛽 𝑖 > − 1 / 2 , 𝑥 𝑖 > 0 ( 𝑖 = 1 , 2 , … , 𝑛 ) , 𝑓 ( 𝑥 ) is a given generalized function, 𝑢 ( 𝑥 ) is an unknown generalized function, 𝑘 is a nonnegative integer, 𝑐 is a positive constant, and 𝑥 ∈ ℝ + 𝑛 .