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Sinc Nyström Method for Singularly Perturbed Love's Integral Equation

Published online by Cambridge University Press:  28 May 2015

Fu-Rong Lin ,
Xin Lu  and
Xiao-Qing Jin
Fu-Rong Lin*
Affiliation:
Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, China
Xin Lu
Affiliation:
Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, China
Xiao-Qing Jin*
Affiliation:
Department of Mathematics, University of Macau, Macao, China
*
Corresponding author. Email: [email protected]
Corresponding author. Email: [email protected]
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Abstract

An efficient numerical method is proposed for the solution of Love's integral equation

where c > 0 is a small parameter, by using a sinc Nyström method based on a double exponential transformation. The method is derived using the property that the solution ƒ(x) of Love's integral equation satisfies ƒ (x) → 0.5 for x ∈ (−1, 1) when the parameter c → 0. Numerical results show that the proposed method is very efficient.

Keywords

45L1065R20Love's integral equationsinc functionNyström methodDE-sinc quadrature
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 3 , Issue 1 , February 2013 , pp. 48 - 58
Copyright
Copyright © Global-Science Press 2013

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References

[1]Agida, M. and Kumar, A. S., A Boubaker polynomials expansion scheme solution to random Love's equation in the case of a rational kernel, Electr. J. Theor. Phys., 7 (24) (2010), pp. 319–326.Google Scholar
[2]El-Gendi, S. E., Chebyshev solution of differential, integral and integro-differential equations, Comput. J., 12 (1969-1970), pp. 282–287.CrossRefGoogle Scholar
[3]Elliott, D., A Chebyshev series method for the numerical solution of Fredholm integral equations, Comput. J., 6 (1963-1964), pp. 102–111.Google Scholar
[4]Fox, L. and Goodwin, E. T., The numerical solution of non-singular linear integral equations, Phil. Trans. R. Soc. Lond. A, 245 (1953), pp. 501–534.Google Scholar
[5]Love, E. R., The electrostatic field of two equal circular co-axial conducting disks, Quart. J. Mech. Appl. Math., 2 (1949), pp. 428–451.CrossRefGoogle Scholar
[6]Monegato, G. and Orsi, A. P., Product formulas for Fredholm integral equations with rational kernel functions, in: Numerical Integration, III Oberwolfach, 1987, in: Internat. Schriftenreihe Numer. Math., vol. 85, Birkhäuser, Basel, 1988, pp. 140–156.Google Scholar
[7]Mori, M., Nurmuhammad, A., and Murai, T., Numerical solution of Volterra integral equations with weakly singular kernel based on the DE-sinc method, Japan J. Indust. Appl. Math., 25 (2008), pp. 165–183.CrossRefGoogle Scholar
[8]Mori, M. and Sugihara, M., The double exponential transformation in numerical analysis, J. Comput. Appl. Math., 127 (2001), pp. 287–296.Google Scholar
[9]Tanaka, K., Sugihara, M., Murota, K., and Mori, M., Function classes for double exponential integration formulas, Numer. Math., 111 (2009), pp. 631–655.CrossRefGoogle Scholar
[10]Pastore, P., The numerical treatment of Love's integral equation having very small parameter, J. Comput. Appl. Math., 236 (2011), pp. 1267–1281.CrossRefGoogle Scholar
[11]Phillips, J. L., The use of collocation as aprojection method for solving linear operator equations, SIAM J. Numer. Anal., 9 (1972), pp. 14–28.CrossRefGoogle Scholar
[12]Sastry, S. S., Numerical solution of non-singular Fredholm integral equations of the second kind, Indian J. Pure Appl. Math., 6 (1975), pp. 773–783.Google Scholar
[13]Stenger, F., Summary of sinc numerical methods, J. Comput. Appl. Math., 121 (2000), pp. 379–420.Google Scholar
[14]Sugihara, M., Optimality of the double exponential formula-functional analysis approach, Numer. Math., 75 (1997), pp. 379–395.CrossRefGoogle Scholar
[15]Sugihara, M. and Matsuo, T., Recent developments of the sinc numerical methods, J. Comput. Appl. Math., 164-165 (2004), pp. 673–689.Google Scholar
[16]Takahasi, H. and Mori, M., Double exponential formulas for numerical integration, Publ. Res. Inst. Math. Sci., 9 (1974), pp. 721–741.Google Scholar
[17]Wolfe, M. A., The numerical solution of non-singular integral and integrodifferential equations by iteration with Chebyshev series, Comput. J., 12 (1969-1970), pp. 193–196.Google Scholar
[18]Young, A., The application of approximate product integration to the numerical solution of integral equations, Proc. R. Soc. Lond. A, 224 (1954), pp. 561–573.Google Scholar