Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 32583
Application of the Central-Difference with Half- Sweep Gauss-Seidel Method for Solving First Order Linear Fredholm Integro-Differential Equations

Authors: E. Aruchunan, J. Sulaiman


The objective of this paper is to analyse the application of the Half-Sweep Gauss-Seidel (HSGS) method by using the Half-sweep approximation equation based on central difference (CD) and repeated trapezoidal (RT) formulas to solve linear fredholm integro-differential equations of first order. The formulation and implementation of the Full-Sweep Gauss-Seidel (FSGS) and Half- Sweep Gauss-Seidel (HSGS) methods are also presented. The HSGS method has been shown to rapid compared to the FSGS methods. Some numerical tests were illustrated to show that the HSGS method is superior to the FSGS method.

Keywords: Integro-differential equations, Linear fredholm equations, Finite difference, Quadrature formulas, Half-Sweep iteration.

Digital Object Identifier (DOI):

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1739


[1] E. Yusufoglu, "Numerical solving initial value problem for Fredholm type linear integro-differential equation system", Journal of the Franklin Institute, 346, 2009, pp. 636-649.
[2] H. Jorquera, "Simple algorithm for solving linear integrodifferential equations with variable limits", Computer physics communications, 86, 1995. 91-96.
[3] M. T. Rashed, "Lagrange interpolation to compute the numerical solutions differential and integro-differential equations", Applied Mathematics and Computation, 151, 2003, pp. 869-878.
[4] S.M. Hosseini and S. Shahmorad. "Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial bases". Applied Math. Model, 27, 2003, pp. 145-154.
[5] A. I. Fedotov, "Quadrature-difference methods for solving linear and nonlinear singular integro-differential equations", Nonlinear Analysis, 71, 12, 2009, pp. e303.
[6] N.H. Sweilam. "Fourth order integro-differential equations using variational iteration method". Comput. Math. Appl., 54, 2007, pp.1086- 1091,
[7] M. Aguilar and H. Brunner. "Collocation methods for second-order Volterra integro-differential equations". Applied Numer. Math., 4, 1988, pp. 455-470.
[8] A. Yildirim, "Solution of BVPs for fourth-order integro differential equations by using homotopy perturbation method". Comput. Math. Appl., 32, 2009, pp.1711-1716.
[9] P.J. Van der Houwen, and B.P. Sommeijer, "Euler-Chebyshe methods for integro-differential equations". Applied Numer. Math., 24: 1997, pp.203-218.
[10] E. Aruchunan and J. Sulaiman, "Numerical Solution of Second-Order Linear Fredholm Integro-Differential Equation Using Generalized Minimal Residual (GMRES) Method". American Journal of Applied Sciences, Science Publication, 7 (6): 2010, pp.780-783.
[11] K. Styś and T. Styś, "A higher-order finite difference method for solving a system of integro-differential equations". Journal of Computational and Applied Mathematics, 126, 2000, pp. 33-46.
[12] K. S. Jacob, "A Zero-Stable Optimal Order Method for Direct Solution of Second Order Differential Equations", Journal of Mathematics and Statistics, 6 (3), 2010, pp.367-371.
[13] M. Sezer, "A method for the approximate solution of the second order linear differential equations in terms of Taylor polynomials", Int. J. Math. Educ. Sci. Technol. , 27(6), 1996, pp. 821-834.
[14] M. Sezer and M. Kaynak, "Chebyshev polynomial solutions of linear differential equations", Int. J.Math. Educ. Sci. Technol. 27(4), 1996, pp.607-618.
[15] R. Alexander, "Diagonally implicit Runge-Kutta methods for stiff ODES", SIAM J. Numer. Anal. 14, 1977, pp. 1006-1021.
[16] A. Rathinasamy and K. Balachandran," Mean square stability of semiimplicit Euler method for linear stochastic differential equations with multiple delays and Markovian switching". Appl Math Comput 2008,206, pp.968-979.
[17] C.T.H. Baker, "The Numerical Treatment of Integral Equations", Clarendon Press Oxford, 1977.
[18] A.D. Polyanin and A.V. Manzhirov, "Handbook of integral equations", CRC Press LCC, 1998.
[19] M.A. Abdou, "Fredholm-Volterra integral equation with singular kernel," Applied Mathematics and Computation 137, 2003, pp. 231- 243.
[20] D.P. Laurie, "Computation of Gauss-type quadrature formulas". Journal of Computational and Applied Mathematics, 127, 2001, pp. 201-217.
[21] K. Maleknejad and M.T. Kajani, "Solving second kind integral equations by Galerkin methods with hybrid Legendre and Block-Pulse functions". Appl. Math. Comput., 145, 2003, ppt 623-629.
[22] S.O.Oladejo, T.A. Mojeed, and K.A. Olurode, "The application of cubic spline collocation to the solution of integral equations". Journal of Applied Sciences Research, 4(6), 2008, pp.748-753.
[23] S.A. Ashour, "Numerical solution of integral equations with finite part integrals," Internat. J. Math. & Math. Sci. 22 (1) , 1999, pp.155-160.
[24] A. R. Abdullah, "The four point Explicit Decoupled Group (EDG) method: A fast Poisson solver". International Journal of Computer Mathematics 38, 1991, pp. 61-70.
[25] J. Sulaiman, M.K. Hasan and M. Othman, "Red-Black Half-Sweep iterative method using triangle finite element approximation for 2D Poisson equations." In Y. Shi et al. (Eds.), Computational Science, Lecture Notes in Computer Science (LNCS 4487), 2007, pp. 326-333. Springer-Verlag, Berlin.
[26] E. Aruchunan and J. Sulaiman, "Half-sweep Conjugate Gradient Method for Solving First Order Linear Fredholm Integro-differential Equations." Australian Journal of Basic and Applied Sciences, 5(3), 2011, pp.38-43.
[27] J. Sulaiman, M.K. Hasan and M. Othman, The Half-Sweep Iterative Alternating Decomposition Explicit (HSIADE) method for diffusion equation. In J. Zhang, J.-H. He and Y. Fu (Eds.), Computational and Information Science, Lecture Notes in Computer Science (LNCS 3314): 2004, pp.57-63. Springer-Verlag, Berlin.
[28] J. Sulaiman, M. Othman and M.K. Hasan. Half-Sweep Algebraic Multigrid (HSAMG) method applied to diffusion equations. In Modeling, Simulation and Optimization of Complex Processes: 2008, pp. 547-556. Springer-Verlag, Berlin.
[29] M.S. Muthuvalu, and J. Sulaiman, "Half-Sweep Arithmetic Mean method with high-order Newton-Cotes quadrature schemes to solve linear second kind Fredholm equations". Journal of Fundamental Sciences 5(1), 2009, pp. 7-16.
[30] M.S. Muthuvalu and J. Sulaiman, "Half-Sweep Arithmetic mean method with composite trapezoidal scheme for solving linear Fredholm integral equation." Applied Mathematics and Computation. 217 (2011) 5442-5448.
[31] P. Darania and A. Ebadia. "A method for numerical Solution of tintegro-differetial equations, Applied Mathematics and Computation, 188, 2007, pp. 657-668.
[32] B. Raftari, A. Ahmadi, and H. Adibi, "The Use of Finite Difference Method, Homotopy Perturbation Method and Variational Iteration Method for a Special Type of Linear Fredholm Integro-differential Equations", Australian Journal of Basic and Applied Sciences, 4(6): 2010,pp.1221-1239,.