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

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. MSC 2010: 34B05

[1]  Takuya Tsuchiya,et al.  Finite difference, finite element and finite volume methods applied to two-point boundary value problems , 2002 .

[2]  Louis A. Hageman,et al.  Iterative Solution of Large Linear Systems. , 1971 .

[3]  Abdul Rahman Abdullah,et al.  Solving the two-dimensional diffusion-convection equation by the four point explicit decoupled group (edg) iterative method , 1995, Int. J. Comput. Math..

[4]  Liyong Tong,et al.  Precise time integration for linear two-point boundary value problems , 2006, Appl. Math. Comput..

[5]  S. Ha.  A nonlinear shooting method for two-point boundary value problems , 2001 .

[6]  D. Tarzia,et al.  BOUNDARY VALUE PROBLEMS IN HEAT CONDUCTION , 2014 .

[7]  David J. Evans,et al.  The implementation of the explicit block iterative methods on the Balance 8000 parallel computer , 1990, Parallel Comput..

[8]  Mohamed El-Gamel,et al.  On the Galerkin and collocation methods for two-point boundary value problems using sinc bases , 2008, Comput. Math. Appl..

[9]  W. Hackbusch.  Iterative Solution of Large Sparse Systems of Equations , 1993 .

[10]  David M. Young,et al.  Second-degree iterative methods for the solution of large linear systems , 1972 .

[11]  Bongsoo Jang.  Two-point boundary value problems by the extended Adomian decomposition method , 2008 .

[12]  Shahid S. Siddiqi,et al.  Solution of fifth order boundary value problems using nonpolynomial spline technique , 2006, Appl. Math. Comput..

[13]  Waheed K. Zahra,et al.  Polynomial and nonpolynomial spline approaches to the numerical solution of second order boundary value problems , 2007, Appl. Math. Comput..

[14]  C. Kelley.  Iterative Methods for Linear and Nonlinear Equations , 1987 .

[15]  Implicit solution of 1d nonlinear porous medium equation using the four-point Newton- EGMSOR iterative method , 2016 .

[16]  I. F. Lashien,et al.  Quintic nonpolynomial spline solutions for fourth order two-point boundary value problem , 2009 .

[17]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[18]  Siraj-ul-Islam,et al.  A smooth approximation for the solution of special non-linear third-order boundary-value problems based on non-polynomial splines , 2006, Int. J. Comput. Math..

[19]  Barbara Kaltenbacher,et al.  Iterative Solution Methods , 2015, Handbook of Mathematical Methods in Imaging.

[20]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[21]  W. D. Hoskins,et al.  Cubic spline solutions to two-point boundary value problems , 1969, Comput. J..

[22]  William L. Goffe.  A user's guide to the numerical solution of two-point boundary value problems arising in continuous time dynamic economic models , 1993 .

[23]  D. Kong,et al.  Two-point boundary value problems and exact controllability for several kinds of linear and nonlinear wave equations , 2009, 0912.5018.