Sains Malaysiana 44(1)(2015): 139–146

Quarter-Sweep Iteration Concept on Conjugate Gradient Normal Residual Method

via Second Order Quadrature - Finite Difference Schemes for Solving Fredholm Integro-Differential Equations

(Konsep Lelaran Sapuan Suku ke atas Kaedah Kecerunan Konjugat Sisa Biasa menerusi Kesukuan Peringkat Kedua - Beza Terhingga bagi Menyelesaikan Persamaan Integro-pembezaan Fredholm)

ELAYARAJA ARUCHUNAN¹*, MOHANA SUNDARAM MUTHUVALU² & JUMAT SULAIMAN³

¹Department of Mathematics and Statistics, Faculty of Science and Engineering

Curtin University, Perth WA6845, Australia

²Department of Fundamental and Applied Sciences, Universiti Teknologi PETRONAS

31750 Tronoh, Perak, Malaysia

³Faculty of Science and Natural Resources, Universiti Malaysia Sabah, 88400 Kota Kinabalu,

Sabah, Malaysia

Diserahkan: 14 Ogos 2012/Diterima: 30 September 2014

ABSTRACT

In this paper, we have examined the effectiveness of the quarter-sweep iteration concept on conjugate gradient normal residual (CGNR) iterative method by using composite Simpson's (CS) and finite difference (FD) discretization schemes in solving Fredholm integro-differential equations. For comparison purposes, Gauss- Seidel (GS) and the standard or full- and half-sweep CGNR methods namely FSCGNR and HSCGNR are also presented. To validate the efficacy of the proposed method, several analyses were carried out such as computational complexity and percentage reduction on the proposed and existing methods.

Keywords: Conjugate gradients normal residual method; linear Fredholm integro-differential equations; quarter-sweep iteration

ABSTRAK

Dalam kertas ini, kami telah menganalisis keberkesanan konsep lelaran sapuan suku ke atas kaedah lelaran kecerunan konjugat sisa biasa (CGNR) dengan menggunakan komposit Simpson's (CS) dan beza terhingga (FD) dalam menyelesaikan persamaan integro-pembezaan Fredholm. Bagi tujuan perbandingan, Gauss-Seidel (GS) dan kaedah CGNR biasa atau penuh dan separuh sapuan iaitu FSCGNR dan HSCGNR juga turut dibincangkan. Bagi mengesahkan keberkesanan kaedah yang dicadangkan, beberapa analisis seperti kekompleksan pengiraan dan pengurangan peratusan untuk kedua-dua kaedah yang dicadangkan dan sedia ada telah dijalankan.

Kata kunci: Kaedah lelaran kecerunan konjugat sisa biasa; lelaran sapuan suku; persamaan integro-pembezaan linear Fredholm

RUJUKAN

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

Abdullah, A.R. & Ali, N.H.M. 1996. A comparative study of parallel strategies for the solution of elliptic pdes. Parallel Algorithms and Applications 10: 93-103.

Agarwal, R.P. 1983. Boundary value problems for higher order integro-differential equations, Nonlinear Analysis, Theory, Methods and Applications 7: 259-270.

Aruchunan, E. & Sulaiman, J. 2013. Half-sweep quadrature-difference schemes with iterative method in solving linear Fredholm integro-differential equations. Progress in Applied Mathematics 5(1): 11-21.

Aruchunan, E. & Sulaiman, J. 2012a. Application of the central-difference scheme with half-sweep Gauss-Seidel method for solving first order linear Fredholm integro-differential equations. International Journal of Engineering and Applied Sciences 6: 296-300.

Aruchunan, E. & Sulaiman, J. 2012b. Comparison of closed repeated Newton-Cotes quadrature schemes with half-sweep iteration concept in solving linear Fredholm integro-differential equations. International Journal of Science and Engineering Investigations 1(8): 296-300.

Aruchunan, E. & Sulaiman, J. 2011a. Half-sweep conjugate gradient method for solving first order linear Fredholm integro-differential equations. Australian Journal of Basic and Applied Sciences 5: 38-43.

Aruchunan, E. & Sulaiman, J. 2011b. Quarter sweep Gauss-Seidel method for solving first order linear Fredholm integro-differential equations. Matematika 27: 199-208.

Aruchunan, E. & Sulaiman, J. 2010. Numerical solution of second-order linear Fredholm integro-differential equation using generalized minimal residual (GMRES) method. American Journal of Applied Sciences 7(6): 780-783.

Aruchunan, E., Muthuvalu, M.S. & Sulaiman, J. 2013. Application of quarter-sweep iteration for first order linear Fredholm integro-differential equations. AIP Conference Proceedings 1522: 168-175.

Aruchunan, E., Muthuvalu, M.S., Sulaiman, J., Koh, W.S. & Akhir, M.K.M. 2014. An iterative solution for second order linear Fredholm integro-differential equations. Malaysian Journal of Mathematical Science 8(2): 158-170.

Avudainayagam, A. & Vani, C. 2000. Wavelet-Galerkin method for integro-differential equations. Applied Numerical Mathematics 32: 247-254.

Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C. & van der Vorst, H. 1993. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. Philadelphia: Society for Industrial and Applied Mathematics.

Fedotov, A.I. 2009. Quadrature-difference methods for solving linear and nonlinear singular integro-differential equations. Nonlinear Analysis 71: e303-e308.

Hosseini, S.M. & Shahmorad, S. 2003. Numerical solution of a class of integro-differential equations by the Tau method with error estimation. Applied Mathematics and Computation 136: 559-570.

Kajani, M.T. & Vencheh, A. 2007. Solving linear integro-differential equation with Legendre wavelets. International Journal of Computer Mathematics 81(6): 719-726.

Karamete, A. & Sezer, M. 2002. A Taylor collocation method for the solution of linear integro-differential equations. International Journal of Computer Mathematics 79(9): 987-1000.

Kurt, N. & Sezer, M. 2008. Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients. Journal of the Franklin Institute 345: 839-850.

Maleknejad, K., Mirzaee, F. & Abbasbandy, S. 2004. Solving linear integro-differential equations system by using rationalized Haar functions method. Applied Mathematics and Computation 155: 317-328.

Morchalo, J. 1975. On two point boundary value problem for integro-differential equation of second order. Fasciculi Mathematici 9: 51-56.

Muthuvalu, M.S. & Sulaiman, J. 2011. Half-sweep arithmetic mean method with composite trapezoidal scheme for solving linear Fredholm integral equations. Applied Mathematics and Computation 217: 5442-5448.

Muthuvalu, M.S., Aruchunan, E. & Sulaiman, J. 2013. Solving first kind linear Fredholm integral equations with semi-smooth kernal using 2-point half-sweep block aritmetic mean method. AIP Conference Proceedings 1557: 350-354.

Othman, M. & Abdullah, A.R. 2000. An efficient four points modified explicit group Poisson solver. International Journal of Computer Mathematics 76: 203-217.

Rashed, M.T. 2003. Lagrange interpolation to compute the numerical solutions differential and integro-differential equations. Applied Mathematics and Computation 151: 869-878.

Ren, Y., Zhang, B. & Qiao, H. 1999. A simple Taylor-series expansion method for a class of second kind integral equations. Journal of Computational and Applied Mathematics 110: 15-24.

Shahsavaran, A. 2012. On the convergence of Lagrange interpolation to solve special type of second kind Fredholm integro differential equations. Applied Mathematical Sciences 6: 343-348.

Sulaiman, J., Othman, M. & Hasan, M.K. 2004a. A new half-sweep arithmetic mean (HSAM) algorithm for two-point boundary value problems. Proceedings of the International Conference on Statistics and Mathematics and Its Applications in Development of Science and Technology. pp. 169-173.

Sulaiman, J., Othman, M. & Hasan, M.K. 2004b. Quarter-sweep iterative alternating decomposition explicit algorithm applied to diffusion equations. International Journal of Computer Mathematics 81: 1559-1565.

Sulaiman, J., Othman, M. & Hasan, M.K. 2009. A new quarter-sweep arithmetic mean (QSAM) method to solve diffusion equations. Chamchuri Journal of Mathematics 1: 93-103.

Wang, W. & Lin, C. 2005. A new algorithm for integral of trigonometric functions with mechanization. Applied Mathematics and Computation 164: 71-82.

Yalcinbas, S. & Sezer, M. 2000. The approximate solution of high-order linear Volterra-Fredholm integro- differential equations in terms of Taylor polynomials. Applied Mathematics and Computation 112: 291-308.

Yalcinbas, S. 2002. Taylor polynomial solution of nonlinear Volterra-Fredholm integral equations. Applied Mathematics and Computation 127: 195-206.

*Pengarang untuk surat-menyurat; email: earuchunan@yahoo.com