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On shifted Jacobi spectral method for high-order multi-point boundary value problems
Authors:EH Doha  AH Bhrawy  RM Hafez
Institution:1. Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt;2. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia;3. Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt;4. Department of Basic Science, Institute of Information Technology, Modern Academy, Cairo, Egypt;1. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia;2. Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt;3. Department of Mathematics, Faculty of Science for Girls, King Abdulaziz University, Jeddah, Saudi Arabia;4. Applied and Industrial Mathematics Research Group, School of Physical, Environmental and Mathematical Sciences, UNSW Canberra, ACT 2600, Australia;5. Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA;1. Department of Mathematics, Physics and Geology, Cape Breton University Sydney, NS, Canada, B1P 6L2;2. Department of Biology, Cape Breton University Sydney, NS, Canada, B1P 6L2;3. College of Science, Harbin Engineering University, Harbin 150001, PR China;1. Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA;2. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia;3. Department of Engineering Sciences, Faculty of Technology and Engineering, East of Guilan, University of Guilan, P.C. 44891-63157 Rudsar-Vajargah, Iran;4. Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran;5. School of Electronics and Information Engineering, Wuhan Donghu University, Wuhan 430212, People’s Republic of China;6. Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt;7. Science Program, Texas A & M University at Qatar, P.O. Box 23874, Doha, Qatar
Abstract:This paper reports a spectral tau method for numerically solving multi-point boundary value problems (BVPs) of linear high-order ordinary differential equations. The construction of the shifted Jacobi tau approximation is based on conventional differentiation. This use of differentiation allows the imposition of the governing equation at the whole set of grid points and the straight forward implementation of multiple boundary conditions. Extension of the tau method for high-order multi-point BVPs with variable coefficients is treated using the shifted Jacobi Gauss–Lobatto quadrature. Shifted Jacobi collocation method is developed for solving nonlinear high-order multi-point BVPs. The performance of the proposed methods is investigated by considering several examples. Accurate results and high convergence rates are achieved.
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