tailieunhanh - báo cáo hóa học:" Research Article Uniform Second-Order Difference Method for a Singularly Perturbed Three-Point Boundary Value Problem"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Uniform Second-Order Difference Method for a Singularly Perturbed Three-Point Boundary Value Problem | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 102484 13 pages doi 2010 102484 Research Article Uniform Second-Order Difference Method for a Singularly Perturbed Three-Point Boundary Value Problem Musa Cakir Department of Mathematics Faculty of Sciences YUzUnctl Yil University 65080 Van Turkey Correspondence should be addressed to Musa Cakir cakirmusa@ Received 21 June 2010 Accepted 15 October 2010 Academic Editor Paul Eloe Copyright 2010 Musa Cakir. 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. We consider a singularly perturbed one-dimensional convection-diffusion three-point boundary value problem with zeroth-order reduced equation. The monotone operator is combined with the piecewise uniform Shishkin-type meshes. We show that the scheme is second-order convergent in the discrete maximum norm independently of the perturbation parameter except for a logarithmic factor. Numerical examples support the theoretical results. 1. Introduction We consider the following singularly perturbed three-point boundary value problem Lu 2u x a x u x - b x u x f x 0 x f u 0 A L0u u - yu i B 0 f1 f where e 0 1 is the perturbation parameter and A B and Y are given constants. The functions a x 0 b x p 0 and f x are sufficiently smooth. For 0 1 the function u x has in general boundary layers at x 0 and x f. Equations of this type arise in mathematical problems in many areas of mechanics and physics. Among these are the Navier-Stokes equations of fluid flow at high Reynolds number mathematical models of liquid crystal materials and chemical reactions shear in second-order fluids control theory electrical networks and other physical models 1 2 . 2 Advances in Difference Equations Differential equations with a small parameter 0 e 1 multiplying the highest order