Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article The Jump Problem for Mixed-Type Equations with Defects on the Type Change Line Ahmed Maher1, 2 | Hindawi Publishing Corporation Boundary Value Problems Volume 2010 Article ID 681709 15 pages doi 2010 681709 Research Article The Jump Problem for Mixed-Type Equations with Defects on the Type Change Line Ahmed Maher1 2 1 Department of Mathematics Faculty of Science Assiut University Assiut 71516 Egypt 2 Department of Mathematics The Teacher s College in Makkah Umm Al-Qura University . Box 2064 Makkah Saudi Arabia Correspondence should be addressed to Ahmed Maher a_maher69@ Received 2 January 2010 Accepted 4 June 2010 Academic Editor Zhitao Zhang Copyright 2010 Ahmed Maher. 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. The jump problem and problems with defects on the type change line for model mixed-type equations in the mixed domains are investigated. The explicit solutions of the jump problem are obtained by the method of integral equations and by the Fourier transformation method. The problems with defects are reduced to singular integral equations. Some results for the solution of the equation under consideration are discussed concerning the existence and uniqueness for the solution of the suggested problem. 1. Introduction Consider the jump problem and problems with defects on the type change line for the mixed-type equation of the first kind sgny yl s2 Ệ 0 m 0. This equation is a model equation among mixed-type equations of the first kind. For m 0 and m 1 coincides with Lavrent ev-Bitsadze equation and the Tricomi equation respectively. For even m coincides with the Gellerstedt equation see 1-9 . Equation is elliptic for y 0 and hyperbolic for y 0. In the formulation of the boundary value problems in the mixed domain it is usually required that the unknown solution u x y and its normal derivative should be continuous on the type change line y 0 that is the 2 .
