On Redlich-Kister Finite Difference Solution Of Two-Point Boundary Value Problems Using Half-Sweep Kaudd Successive Over Relaxation Iteration

**Citation**

**MLA Style: **Mohd Norfadli Suardi, Jumat Sulaiman "On Redlich-Kister Finite Difference Solution Of Two-Point Boundary Value Problems Using Half-Sweep Kaudd Successive Over Relaxation Iteration" International Journal of Engineering Trends and Technology 69.2(2021):77-82.

**APA Style:**Mohd Norfadli Suardi, Jumat Sulaiman. On Redlich-Kister Finite Difference Solution Of Two-Point Boundary Value Problems Using Half-Sweep Kaudd Successive Over Relaxation Iteration. International Journal of Engineering Trends and Technology, 69(2), 77-82.

**Abstract**

This paper deals with the application of two newly established Redlich-Kister Finite Difference (RKFD) discretization schemes for approximating and solving two-point boundary value problems (TPBVPs). To get the Redlich-Kister Finite Difference Solution of the proposed problem, firstly, two newly second-order half-sweep RKFD discretization schemes are established and used to discretize overall derivative terms of the TPBVPs regarding getting the second-order half-sweep RKFD approximation equation. Then this RKFD approximation equation leads to the construct of the linear system. Due to the increase in the convergence rate iteratively in solving this linear system, the combination of the Kaudd Successive Over Relaxation (KSOR) method with a half-sweep approach is formulated and then known as Half-sweep Kaudd Successive Over Relaxation (HSKSOR) method. With the purpose of evaluating the efficiency of the HSKSOR method, other methods such as Full-sweep Kaudd Successive Over Relaxation (FSKSOR) and Full-sweep Gauss-Seidel (FSGS) are also presented as a control method. The results of the examples of TPBVPs are tested to prove that the HSKSOR iteration is more efficient compared with FSGS and FSKSOR iterations in terms of iterations, execution time, and maximum norm.

**Reference**

[1] Q. Fang, T. Tsuchiya, and T. Yamamomoto., Finite difference, finite element and finite volume methods applied to two-point boundary value problems, Journal of Computational and Applied Mathematics, 139(2002) 9-19.

[2] M. El-Gamel. Comparison of the solution obtained by domain decomposition and wavelet-galerkin methods of boundary-value problems, Applied Mathematics and Computation, 186(1)(2007) 652-664.

[3] F. Geng, and M. Cui. A novel method for nonlinear two-point boundary value problems: Combination of ADM and RKM., Applied Mathematics and Computation, 217, 4676-4681, (2011).

[4] J. Aarao, B. H. Bradshaw-Hajek, S. J. Miklavcic, and D. A. Ward. “The extended domain eigenfunction method for solving elliptic boundary value problems with annular domains.” Journal of Physics A: Mathematical and Theoretical, 43(2010) 185-202.

[5] T. N. Robertson. The linear two-point boundary value problem on an infinite interval. Mathematics Of Computation, 25(115)(1971) 475-481.

[6] Y. M. Wang, and B. Y. Guo., Fourth-order compact finite difference method for fourth-order nonlinear elliptic boundary value problems. Journal of Computational and Applied Mathematics, 221(1)(2008) 76-97.

[7] B. Jang., Two-point boundary value problems by extended a domain decomposition method. Computational and Applied Mathematics, 219(1)(2007) 253-262.

[8] A. Mohsen, and M. E. Gamel., On the galerkin and collocation methods for two-point boundary value problems using sinc bases., Computer and Mathematics with Applications, 56(2008) 930-941.

[9] Y. Lin, J. A. Enszer, and M. A. Stadtherr., Enclosing all solutions of two-point boundary value problems for ODEs. Computer and Chemical Engineering, 32(8)(2008) 1714-1725.

[10] C. Nazan, and C. Hikmet., B-spline methods for solving a linear system of second-order boundary value problems. Computers and Mathematics with Application, 57(5)(2008) 757-762.

[11] M. N. Suardi, N. Z. F. M. Radzuan, and J. Sulaiman., Cubic b-spline solution for two-point boundary value problem with AOR iterative method. Journal of Physics: Conference Series, 890(2017) 12015.

[12] S. Babu, R. Trabelsi, T. Srinivasa Krishna, N. Ouerfelli, and A. Toumi. Reduced redlich–Kister functions and interaction studies of dehpa+ petrofin binary mixtures at 298.15 K. Physics and Chemistry of Liquids, 57(4)(2019) 536-546.

[13] A. Gayathri, T. Venugopal, and K. Venkatramanan. Redlich-Kister coefficients on the analysis of Physico-chemical characteristics of functional polymers. Materials Today: Proceedings, 17(2019) 2083-2087.

[14] N. P. Komninos, and E. D. Rogdakis., Geometric investigation of the three-coefficient Redlich-Kister expansion global phase diagram for binary mixtures. Fluid Phase Equilibria, 112728, (2020).

[15] M. K. Hasan, J. Sulaiman, S. Ahmad, M. Othman, and S. A. ABDUL KARIM. Approximation of iteration number for Gauss-Seidel using Redlich-Kister polynomial. American Journal of Applied Sciences, 7(2010) 956-962.

[16] M. M. Chawla, and C. P. Katti., Finite difference methods for two-point boundary value problems involving high order differential equations., BIT Numerical Mathematics, 19(1)(1979) 27-33.

[17] E. M. Elbarbary, and M. El-Kady., Chebyshev finite difference approximation for the boundary value problems. Applied Mathematics and Computation, 139,2-3(2003) 513-523.

[18] P. K. Pandey., Rational finite difference approximation of high order accuracy for nonlinear two-point boundary value problems. Sains Malaysiana, 43(7)(2014) 1105-1108.

[19] P. K. Pandey., Solving two-point boundary value problems for ordinary differential equations using exponential finite difference method., Boletim da Sociedade Paranaense de Matemática, 34(1)(2016) 45-52.

[20] D. M. Young. Iterative solution of large linear systems. London: Academic Press, (1971).

[21] W. Hackbusch. Iterative solution of large sparse systems of equations. Springer-Verlag, (1995).

[22] Y. Saad. Iterative methods for sparse linear systems. International Thomas Publishing, (1996).

[23] J. Sulaiman, M. K. Hasan, M. Othman, and S. A. A. Karim. MEGSOR iterative method for the triangle element solution of 2D Poisson equations. Procedia Computer Science, 1(1)(2010) 377-385.

[24] A. Saudi, and J. Sulaiman., Robot path planning using four point-explicit groups via nine-point laplacian (4EG9L) iterative method”. Procedia Engineering, 41(2012) 182-188.

[25] A. Sunarto, J. Sulaiman, and A. Saudi., Implicit finite difference solution for time-fractional diffusion equations using AOR method, In Journal of Physics: Conference Series, 495(1)(2014) 012032.

[26] A. Saudi, and J. Sulaiman., Path planning simulation using potential harmonic fields through our-point-edgsor method via 9-point laplacian. Jurnal Teknologi, 78(2016) 8-2.

[27] N. Z. F. M. Radzuan, M. N. Suardi, and J. Sulaiman. , KSOR iterative method with quadrature scheme for solving the system of Fredholm integral equations of the second kind. Journal of Fundamental and Applied Sciences, 9(5S), (2017) 609-623.

[28] R. Rahman, N. A. M. Ali, J. Sulaiman, and F. A. Muhiddin., Caputo’s finite-difference solution of fractional two-point boundary value problems using SOR iteration. , In AIP Conference Proceedings, 1(2013) 020034, 2018.

[29] M. N. Suardi, N. Z. F. M. Radzuan, and J. Sulaiman., KAOR iterative method with a cubic b-spline approximation for solving two-point boundary value problems. Proceedings of 25th National Symposium on Mathematical Sciences (SKSM25): Mathematical Science as The Core of Intellectual Excellence, 1974(1)(2018) 020094.

[30] K. Ghazali, J. Sulaiman, Y. Dasril, and D. Gabda., Application of Newton- 4EGSOR Iteration for solving large-scale unconstrained optimization problems with a tridiagonal hessian matrix. In Computational Science and Technology, (2019) 401-411.

[31] F. A. Muhiddin, J. Sulaiman, and A. Sunarto. Implementation of the 4EGKSOR for solving one-dimensional time-fractional parabolic equations with grünwald implicit difference scheme., In Computational Science and Technology, (2020) 511-520.

[32] A. R. Abdullah., The four-point explicit decoupled group (EDG) method: a fast Poisson solver., International Journal Computer Mathematics, 38,(1-2)(1991) 61-70.

[33] M. K. M. Akhir, M. Othman, J. Sulaiman, Z. A. Majid, and M. Suleiman., Half-sweep modified successive over-relaxation for solving two-dimensional helmoltz equations. Australian Journal of Basic and Applied Science, 15(12), (2011) 3033-3039.

[34] M. U. Alibubin, A. Sunarto, M. K. M. Akhir, and J. Sulaiman., Performance analysis of half-sweep sor iteration with rotated nonlocal arithmetic mean scheme for 2d nonlinear elliptic problem., Global Journal of Pure and Applied Mathematics, 12(4)(2016) 3415-3424.

[35] J. V. L. Chew, and J. Sulaiman., Half-sweep newton-gauss-seidel for implicit finite difference solution of 1d nonlinear porous medium equations., Global Journal of Pure and Applied Mathematics, 12(3)(2016) 2745-2752.

[36] M. N. Suardi, N. Z. F. M. Radzuan, and J. Sulaiman., Cubic B-spline solution of two-point boundary value problem using HSKSOR iteration., Global Journal of Pure and Applied Mathematics, 13(11)(2017) 7921-7934.

[37] L. Ali, J. Sulaiman, and S. Hashim., Numerical solution of fuzzy fredholm integral equations of a second kind using half-sweep gauss-seidel iteration., Journal of Engineering Science and Technology, 15(5)(2020) 3303-3313.

[38] I. K. Youssef, and A. A. Taha., On modified successive overrelaxation method., Applied Mathematics and Computation, 219(2013) 4601-4613.

[39] N. Z. F. M. Radzuan, M. N. Suardi, and J. Sulaiman., Numerical solution for the system of second kind fredholm integral equation by using quadrature and HSKSOR iteration., Global Journal of Pure and Applied Mathematics,13(11)(2017) 7935-7946.

[40] I. Youssef., On the successive overrelaxation method. Journal of Mathematics and Statistics, 8(2)(2012) 176-184.

[41] H. N. Caglar, S. H. Caglar, and K. Elfaituri., B-spline interpolation compared with a finite difference, finite element and finite volume methods which applied to two-point boundary value problems., Applied Mathematics and Computation, 175(1)(2006) 72-79.

[42] M. A. Ramadan, I. F. Lashien, and W. K. Zahra., Polynomial and nonpolynomial spline approaches to the numerical solution of second-order boundary value problems., Applied Mathematics and Computation, vol. 184(2007) 476-484.

[43] A. A. Dahalan, M. S. Muthuvalu, and J. Sulaiman., Numerical solutions of two-point fuzzy boundary value problem using half-sweep alternating group explicit method., American Institute of Physics, 1557(1)(2013) 103-107.

[44] A. A. Dahalan, J. Sulaiman, and M. S. Muthuvalu., Performance of HSAGE method with Seikkala derivative for 2-D fuzzy Poisson equation., Applied Mathematical Sciences,8(17-20)(2014) 885-899.

[45] M. K. Hasan, J. L. Sulaiman, S. A. Abdul Karim, and M. Othman., Development of some numerical methods applying complexity reduction approach for solving a scientific problem., (2010).

[46] N. I. M. Fauzi, and J. Sulaiman., Quarter-Sweep Modified SOR iterative algorithm and cubic spline basis for the solution of second-order two-point boundary value problems., Journal of Applied Sciences (Faisalabad),12, (17)(2012) 1817-1824.

[47] Muhiddin, F. A., Sulaiman, J., & Sunarto, A., Numerical evaluation of quarter-sweep KSOR method to solve time-fractional parabolic equations., International Journal of Engineering Trends and Technology, (2020) 63-69.

[48] J. V. Lung, and J. Sulaiman., On quarter-sweep finite difference scheme for one-dimensional porous medium equations., International Journal of Applied Mathematics, 33(3)(2020) 439.

**Keywords**

Boundary value problems, Redlich-Kister Finite Difference scheme, KSOR iteration, Half-sweep concept.