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: GENERALIZED QUASILINEARIZATION METHOD AND HIGHER ORDER OF CONVERGENCE FOR SECOND-ORDER BOUNDARY VALUE PROBLEMS | GENERALIZED QUASILINEARIZATION METHOD AND HIGHER oRdER of convergence for SECOND-ORDER BOUNDARY VALUE PROBLEMS TANYA G. MELTON AND A. S. VATSALA Received 24 March 2005 Revised 13 September 2005 Accepted 19 September 2005 The method of generalized quasilinearization for second-order boundary value problems has been extended when the forcing function is the sum of 2-hyperconvex and 2-hyperconcave functions. We develop two sequences under suitable conditions which converge to the unique solution of the boundary value problem. Furthermore the convergence is of order 3. Finally we provide numerical examples to show the application of the generalized quasilinearization method developed here for second-order boundary value problems. Copyright 2006 T. G. Melton and A. S. Vatsala. 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 The method of quasilinearization 1 2 combined with the technique of upper and lower solutions is an effective and fruitful technique for solving a wide variety of nonlinear problems. It has been referred to as a generalized quasilinearization method. See 9 for details. The method is extremely useful in scientific computations due to its accelerated rate of convergence as in 10 11 . In 4 13 the authors have obtained a higher order of convergence an order more than 2 for initial value problems. They have considered situations when the forcing function is either hyperconvex or hyperconcave. In 12 we have obtained the results of higher order of convergence for first order initial value problems when the forcing function is the sum of hyperconvex and hyperconcave functions with natural and coupled lower and upper solutions. In this paper we extend the result to the second-order boundary value problems when the forcing function is a sum of 2-hyperconvex and 2-hyperconcave .
