Reproducing kernel pseudospectral method for the numerical investigation of nonlinear multi-point boundary value problem
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Reproducing kernel pseudospectral method for the numerical investigation of nonlinear multi-point boundary value problems Mohammad Nabati1 · Mahdi Emamjome2 · Mehdi Jalalvand3 Received: 19 April 2018 / Revised: 20 July 2018 / Accepted: 23 August 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018
Abstract This paper presents a computational method to solve nonlinear boundary value problems with multi-point boundary conditions. These problems have important applications in the theoretical physics and engineering problems. The method is based on reproducing kernel Hilbert spaces operational matrices and an iterative technique is used to overcome the nonlinearity of the problem. Furthermore, a rigorous convergence analysis is provided and some numerical tests reveal the high efficiency and versatility of the proposed method. The results of numerical experiments are compared with analytical solutions and the best results reported in the literature to confirm the good accuracy of the presented method. Keywords Multi-point boundary condition · Reproducing kernel Hilbert spaces · Nonlinear Bitsadze–Samarskii boundary value problem · Iterative method · Convergence Mathematics Subject Classification 74S25 · 34B15 · 65L20 · 65L10
1 Introduction The developments of the numerical methods for the solution of multi-point boundary value problems are important since such problems arise in many branches of science as mathematical models of various real-world processes. Multi-point boundary value problems arise in several branches of engineering, applied mathematical sciences and physics, for instance
Communicated by Antonio José Silva Neto.
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Mohammad Nabati [email protected]
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Department of Basic Sciences, Abadan Faculty of Petroleum Engineering, Petroleum University of Technology, Abadan, Iran
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Department of Mathematics, Golpayegan University of Technology, P.O. Box 8771765651, Golpayegan, Iran
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Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran
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M. Nabati et al.
modeling large-size bridges (Geng and Cui 2010), problems in the theory of elastic stability (Timoshenko 1961) and the flow of fluid such as water, oil and gas through ground layers and fluid flow through multi-layer porous medium (Hajji 2009). Bitsadze and Samarskii (1969) have studied a new problem in which the multi-point boundary conditions depend on the values of the solution in the interior and boundary of the domain. The Bitsadze–Samarskii multi-point boundary value problems (Bitsadze and Samarskii 1969) arise in mathematical modeling of plasma physics processes. The well-posedness, existence, uniqueness and multiplicity of solutions of Bitsadze–Samarskii-type multi-point boundary value problems have been investigated by many authors, see Hajji (2009), Kapanadze (1987), Ma (2004), Ashyralyev and Ozturk (2014) and the references given there. However, research for numerical solutions of the Bitsadze–Samarskii-type boundary value problems, has proceed
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