Half-Sweep Geometric Mean Iterative Method for the Repeated Simpson Solution of Second Kind Linear Fredholm Integral Equations
DOI:
https://doi.org/10.4067/S0716-09172012000100007Keywords:
Linear Fredholm equations, half-sweep iteration, repeated Simpson, geometric mean, ecuaciones lineares de Fredholm, iteración de medio barrido, Simpson repetida, media geométrica.Abstract
In previous studies, the effectiveness of the Half-Sweep Geometric Mean (HSGM) iterative method has been shown in solving first and second kind linear Fredholm integral equations using repeated trapezoidal (RT) discretization scheme. In this work, we investigate the efficiency of the HSGM method to solve dense linear system generated from the discretization of the second kind linear Fredholm integral equations by using repeated Simpson's 1/3(RS1) scheme. The formulation and implementation ofthe proposed method are also presented. In addition, several numerical simulations and computational complexity analysis were also included to verify the efficiency of the proposed method.References
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[16] Muthuvalu, M. S., and J. Sulaiman. Numerical solution of second kind linear Fredholm integral equations using QSGS iterative method with high-order Newton-Cotes quadrature schemes. Malaysian Journal of Mathematical Sciences 5: pp. 85-100, (2011).
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[24] Sulaiman, J., M. Othman, N. Yaacob, and M. K. Hasan. Half-Sweep Geometric Mean (HSGM) method using fourth-order finite difference scheme for two-point boundary problems. Proceedings of the First International Conference on Mathematics and Statistics: pp. 25-33, (2006).
[25] Wang, W. A new mechanical algorithm for solving the second kind of Fredholm integral equation. Applied Mathematics and Computation 172: pp. 946-962, (2006).
[2] Abdullah, M. H., J. Sulaiman, A. Saudi, M. K. Hasan, and M. Othman. A numerical simulation on water quality model using Half-Sweep Geometric Mean method. Proceedings of the Second Southeast Asian Natural Resources and Environment Management Conference: pp. 25-
29, (2006).
[3] Allahviranloo, T., E. Ahmady, N. Ahmady, and K. S. Alketaby. Block Jacobi two-stage method with Gauss-Sidel inner iterations for fuzzy system of linear equations. Applied Mathematics and Computation 175: pp. 1217-1228, (2006).
[4] Atkinson, K. E. The Numerical Solution of Integral Equations of the Second Kind. Cambridge: Cambridge University Press. (1997).
[5] Cattani, C., and A. Kudreyko. Harmonic wavelet method towards solution of the Fredholm type integral equations of the second kind. Applied Mathematics and Computation 215: pp. 4164-4171, (2010).
[6] Chen, Z., B. Wu, and Y. Xu. Fast numerical collocation solutions of integral equations. Communications on Pure and Applied Analysis 6: pp. 643-666, (2007).
[7] Evans, D. J. The Alternating Group Explicit (AGE) matrix iterative method. Applied Mathematical Modelling 11: pp. 256-263, (1987).
[8] Kang, S. -Y., I. Koltracht, and G. Rawitscher. Nyström-ClenshawCurtis Quadrature for Integral Equations with Discontinuous Kernels. Mathematics of Computation 72: pp. 729-756, (2003).
[9] Liu, Y. Application of the Chebyshev polynomial in solving Fredholm integral equations. shape Mathematical and Computer Modelling 50: pp. 465-469, (2009).
[10] Long, G., M. M. Sahani, and G. Nelakanti. Polynomially based multiprojection methods for Fredholm integral equations of the second kind. Applied Mathematics and Computation 215: pp. 147-155, (2009).
[11] Mastroianni, G., and G. Monegato. Truncated quadrature rules over and Nyström-type methods. SIAM Journal on Numerical Analysis 41: pp. 1870-1892, (2004).
[12] Mirzaee, F., and S. Piroozfar. (2010). Numerical solution of linear Fredholm integral equations via modified Simpson’s quadrature rule. Journal of King Saud University - Science 23: (2011).
[13] Muthuvalu, M. S., and J. Sulaiman. Half-Sweep Geometric Mean method for solution of linear Fredholm equations. Matematika 24: pp.
75-84, (2008).
[14] Muthuvalu, M. S., and J. Sulaiman. Numerical solutions of second kind Fredholm integral equations using Half-Sweep Geometric Mean method. Proceedings of the IEEE International Symposium on Information Technology: pp. 1927-1934, (2008).
[15] Muthuvalu, M. S., 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: pp. 7-16, (2009).
[16] Muthuvalu, M. S., and J. Sulaiman. Numerical solution of second kind linear Fredholm integral equations using QSGS iterative method with high-order Newton-Cotes quadrature schemes. Malaysian Journal of Mathematical Sciences 5: pp. 85-100, (2011).
[17] Nichols, N. K. On the convergence of two-stage iterative process for solving linear equations. SIAM Journal on Numerical Analysis 10: pp.
460-469, (1973).
[18] Polyanin, A. D., and A. V. Manzhirov. Handbook of Integral Equations. CRC Press LCC, (1998).
[19] Ruggiero, V., and E. Galligani. An iterative method for large sparse systems on a vector computer. Computers & Mathematics with Applications 20: pp. 25-28, (1990).
[20] Saberi-Nadjafi, J., and M. Heidari. Solving integral equations of the second kind with repeated modified trapezoid quadrature method. Applied Mathematics and Computation 189: pp. 980-985, (2007).
[21] Sahimi, M. S., A. Ahmad, and A. A. Bakar. The Iterative Alternating Decomposition Explicit (IADE) method to solve the heat conduction equation. International Journal of Computer Mathematics 47: pp. 219-
229, (1993).
[22] Sahimi, M. S., and M. Khatim. The Reduced Iterative Alternating Decomposition Explicit (RIADE) method for diffusion equation. Pertanika Journal of Science & Technology 9: pp. 13-20, (2001).
[23] Sulaiman, J., M. Othman, and M. K. Hasan. 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 Application in the Development of Science and Technology: pp. 169-173, (2004).
[24] Sulaiman, J., M. Othman, N. Yaacob, and M. K. Hasan. Half-Sweep Geometric Mean (HSGM) method using fourth-order finite difference scheme for two-point boundary problems. Proceedings of the First International Conference on Mathematics and Statistics: pp. 25-33, (2006).
[25] Wang, W. A new mechanical algorithm for solving the second kind of Fredholm integral equation. Applied Mathematics and Computation 172: pp. 946-962, (2006).
Published
2012-01-29
How to Cite
[1]
M. S. Muthuvalu and J. Sullaiman, “Half-Sweep Geometric Mean Iterative Method for the Repeated Simpson Solution of Second Kind Linear Fredholm Integral Equations”, Proyecciones (Antofagasta, On line), vol. 31, no. 1, pp. 65-79, Jan. 2012.
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