Complex Variables and Elliptic Equations Vol. 53, No. 4, April 2008, 355–364 A solution of the Neumann–Dirichlet boundary value problem for generalized bi-axially symmetric Helmholtz equation M. S. SALAKHITDINOV and A. HASANOV* Institute of Mathematics, Uzbek Academy of Sciences, 29 F. Hodjaev street, Tashkent 700125, Uzbekistan Communicated by R. P. Gilbert (Received in final form 19 September 2007) In the paper ‘‘Fundamental solutions of generalized bi-axially symmetric Helmholtz equation’’, Complex Variables and Elliptic Equations, 52(8), 2007, 673–683, fundamental solutions of generalized bi-axially symmetric Helmholtz equation (GBSHE), H ! ,  ðuÞ u xx þ u yy þ ð2=xÞu x þð2=yÞu y  ! 2 u ¼ 0, 0 < 2,2< 1, , , ! ¼ const, were constructed in R þ 2  fðx, yÞ : x > 0, y > 0g, which express confluent hypergeometric functions Kummer from three variables. In this article, using one of the constructed fundamental solutions, in the domain   R þ 2 the boundary value problem N is solved. A solution of a boundary value problem is defined in an explicit form. Keywords: GBSHE; Singular partial differential equation; Fundamental solutions; A Green function; Neumann–Dirichlet problem; Confluent hypergeometric function Kummer from three variables AMS Subject Classifications: 35A08; 35J70 1. Introduction In the monograph of Gilbert [1], by applying a method of complex analysis, integral representation of solutions of the generalized bi-axially Helmholtz equation (GBSHE) H ! ,  ðuÞ u xx þ u yy þ 2 x u x þ 2 y u y  ! 2 u ¼ 0, ðH ! ,  Þ *Corresponding author. Email: anvarhasanov@yahoo.com Complex Variables and Elliptic Equations ISSN 1747-6933 print/ISSN 1747-6941 online ß 2008 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/17476930701769041