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Quartic non-polynomial spline solution for solving two-point boundary value problems by using Conjugate Gradient iterative method

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Solving two-point boundary value problems has become a scope of interest among many researchers due to its significant contributions in the field of science, engineering, and economics which is evidently apparent in many previous literary publications. This present paper aims to discretize the two-point boundary value problems by using a quartic non-polynomial spline before finally solving them iteratively with Conjugate Gradient (CG) method. Then, the performances of the proposed approach in terms of iteration number, execution time and maximum absolute error are compared with Gauss-Seidel (GS) and Successive Over-Relaxation (SOR) iterative methods. Based on the performances analysis, the two-point boundary value problems are found to have the most favorable results when solved using CG compared to GS and SOR methods.
Rocznik
Strony
41--50
Opis fizyczny
Biliogr. 23 poz., rys., tab.
Twórcy
autor
  • Mathematics with Economics Programme, University Malaysia Sabah Sabah, Malaysia
  • Mathematics with Economics Programme, University Malaysia Sabah Sabah, Malaysia
autor
  • Mathematics with Economics Programme, University Malaysia Sabah Sabah, Malaysia
Bibliografia
  • [1] Ozışık M.N., Boundary value problems of heat conduction, Courier Corporation, 1989.
  • [2] Goffe W.L., A user’s guide to the numerical solution of two-point boundary value problems arising in continuous time dynamic economic models, Computational Economics 1993, 6(3-4), 249-255.
  • [3] Chew J.V., Sulaiman J., Implicit solution of 1D nonlinear porous medium equation using the four-point Newton-EGMSOR iterative method, Journal of Applied Mathematics and Computational Mechanics 2016, 15(2), 11-21.
  • [4] Jang B., Two-point boundary value problems by the extended Adomian decomposition method, Journal of Computational and Applied Mathematics 2008, 219(1), 253-262.
  • [5] Kong D.X., Sun Q.Y., Two-point boundary value problems and exact controllability for several kinds of linear and nonlinear wave equations, In Journal of Physics: Conference Series 2011, 290(1), 012008. IOP Publishing.
  • [6] Chen B., Tong L., Gu Y., Precise time integration for linear two-point boundary value problems, Applied Mathematics and Computation 2006, 175(1), 182-211.
  • [7] Ha S.N., A nonlinear shooting method for two-point boundary value problems, Computers & Mathematics with Applications 2001, 42(10), 1411-1420.
  • [8] Mohsen A., El-Gamel M., On the Galerkin and collocation methods for two-point boundary value problems using sinc bases, Computers & Mathematics with Applications 2008, 56(4), 930-941.
  • [9] Fang Q., Tsuchiya T., Yamamoto T., Finite difference, finite element and finite volume methods applied to two-point boundary value problems, Journal of Computational and Applied Mathematics 2002, 139(1), 9-19.
  • [10] Albasiny E.L., Hoskins W.D., Cubic spline solutions to two-point boundary value problems, The Computer Journal 1969, 12(2), 151-153.
  • [11] Ramadan M.A., Lashien I.F., Zahra W.K., Quintic nonpolynomial spline solutions for fourth order two-point boundary value problem, Communications in Nonlinear Science and Numerical Simulation 2009, 14(4), 1105-1114.
  • [12] Kelley C.T., Iterative Methods for Linear and Nonlinear Equations, SIAM, Philadelphia 1995.
  • [13] Beauwens R., Iterative solution methods, Applied Numerical Mathematics 2004, 51(4), 437-450.
  • [14] Hestenes M.R., Stiefel E., Methods of conjugate gradients for solving linear systems, NBS 1952, 49, 1.
  • [15] Saad Y., Iterative methods for sparse linear systems, SIAM, 2003.
  • [16] Hackbusch W., Iterative Solution of Large Sparse Systems of Equations, Springer Science & Business Media, 2012.
  • [17] Young D.M., Iterative solution of large linear systems, Elsevier, 2014.
  • [18] Young D.M., Second-degree iterative methods for the solution of large linear systems, Journal of Approximation Theory 1972, 5(2),137-148.
  • [19] Abdullah A.R., Ibrahim A., Solving the two-dimensional diffusion-convection equation by the four point explicit decoupled group (edg) iterative method, International Journal of Computer Mathematics 1995, 58(1-2), 61-71.
  • [20] Evans D.J., Yousif W.S., The implementation of the explicit block iterative methods on the balance 8000 parallel computer, Parallel Computing 1990, 16(1), 81-97.
  • [21] Islam S.U., Tirmizi I.A., A smooth approximation for the solution of special non-linear thirdorder boundary-value problems based on non-polynomial splines, International Journal of Computer Mathematics 2006, 83(4), 397-407.
  • [22] Ramadan M.A., Lashien I.F., Zahra W.K., Polynomial and nonpolynomial spline approaches to the numerical solution of second order boundary value problems, Applied Mathematics and computation 2007, 184(2), 476-484.
  • [23] Siddiqi S.S., Akram G., Solution of fifth order boundary value problems using nonpolynomial spline technique, Applied Mathematics and Computation 2006, 175(2), 1574-1581.
Uwagi
Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-aff3670d-b8b8-4123-9d07-fb79480cc289
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