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Báo cáo hóa học: " Research Article Robust Monotone Iterates for Nonlinear Singularly Perturbed Boundary Value Problems"

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 Robust Monotone Iterates for Nonlinear Singularly Perturbed Boundary Value Problems | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 320606 17 pages doi 10.1155 2009 320606 Research Article Robust Monotone Iterates for Nonlinear Singularly Perturbed Boundary Value Problems Igor Boglaev Institute of Fundamental Sciences Massey University Private Bag 11-222 4442 Palmerston North New Zealand Correspondence should be addressed to Igor Boglaev i.boglaev@massey.ac.nz Received 8 April 2009 Accepted 11 May 2009 Recommended by Donal O Regan This paper is concerned with solving nonlinear singularly perturbed boundary value problems. Robust monotone iterates for solving nonlinear difference scheme are constructed. Uniform convergence of the monotone methods is investigated and convergence rates are estimated. Numerical experiments complement the theoretical results. Copyright 2009 Igor Boglaev. 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. 1. Introduction We are interested in numerical solving of two nonlinear singularly perturbed problems of elliptic and parabolic types. The first one is the elliptic problem -f7u f x u 0 x e w 0 1 u 0 0 u 1 0 1.1 fu c const 0 x u e w X -TO to fu df du where f is a positive parameter and f is sufficiently smooth function. For f 1 this problem is singularly perturbed and the solution has boundary layers near x 0 and x 1 see 1 for details . The second problem is the one-dimensional parabolic problem -f7uxx ut f x t u 0 x t e Q w X 0 T w 0 1 u 0 t 0 u 1 t 0 u x 0 u0 x x e w fu 0 x t u e Q X -TO to 1.2 2 Boundary Value Problems where ạ is a positive parameter. Under suitable continuity and compatibility conditions on the data a unique solution of this problem exists. For ạ 1 problem 1.2 is singularly perturbed and has boundary layers near the lateral boundary of Q see 2 for details . In the study of numerical methods for nonlinear .