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Numerical Experiments on Eigenvalues of Weakly Singular Integral Equations Using Product Simpson’s Rule

Mohamad Rashidi Razali, and Mohamed M. S. Nasser, (2002) Numerical Experiments on Eigenvalues of Weakly Singular Integral Equations Using Product Simpson’s Rule. Matematika, 18 (1). pp. 9-20. ISSN 01278274

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Official URL: http://www.fs.utm.my/matematika/images/stories/matematika/200218101.pdf

Affiliations

Universiti Teknologi Malaysia, Faculty of Science, Dept. of Mathematics
Universiti Teknologi Malaysia, Faculty of Science, Dept. of Mathematics

Abstract

This paper discusses the use of Product Simpson’s rule to solve the integral equation eigenvalue problem λf(x) =R1−1 k(|x− y|)f(y)dy wherek(t) = ln|t| or k(t) = t-α, 0 < α < 1, λ, f and are unknowns which we wish to obtain. The function f(y) in the integral above is replaced by an interpolating function Lfn(y) = Pni=0 f(xi)ϕi(y), where ϕi(y) are Simpson interpolating elements and x0, x1, . . . ,xn are the interpolating points and they are chosen to be the appropriate non uniform mesh points in [−1, 1]. The product integration formula R1−1 k(y)f(y)dy ≈ Pni=0 wif(xi) is used, where the weights wi are chosen such that the formula is exact when f(y) is replaced by Lfn(y) and k(y) as given above. The five eigenvalues with largest moduli of the two kernels K(x, y) = ln|x − y| and K(x, y) = |x − y|−α, 0 < α < 1 are given.

Item Type:Journal
Keywords:eigenvalue, product integration, singular kernel, integral equation
Subjects:Q Science, Computer Science
ID Code:5684

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