Numerical treatment of the generalized Love integral equation
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Numerical treatment of the generalized Love integral equation Luisa Fermo1
· Maria Grazia Russo2 · Giada Serafini2
Received: 31 January 2020 / Accepted: 20 May 2020 / © The Author(s) 2020
Abstract In this paper, the generalized Love integral equation has been considered. In order to approximate the solution, a Nystr¨om method based on a mixed quadrature rule has been proposed. Such a rule is a combination of a product and a “dilation” quadrature formula. The stability and convergence of the described numerical procedure have been discussed in suitable weighted spaces and the efficiency of the method is shown by some numerical tests. Keywords Love integral equation · Nearly singular kernels · Nystr¨om method · Quadrature formula Mathematics Subject Classification 2010 65R20 · 65D32 · 45B05
1 Introduction In 1949, Love investigated for the first time on a mathematical model describing the capacity of a circular plane condenser consisting of two identical coaxial discs placed
The authors are members of the INdAM Research Group GNCS.This research has been accomplished within the RITA “Research ITalian network on Approximation”. Luisa Fermo
[email protected] Maria Grazia Russo [email protected] Giada Serafini [email protected] 1
Department of Mathematics and Computer Science, University of Cagliari, Via Ospedale 72, 09124, Cagliari, Italy
2
Department of Mathematics, Computer Science and Economics, University of Basilicata, Via dell’Ateneo Lucano 10, 85100, Potenza, Italy
Numerical Algorithms
at a distance q and having a common radius r. In his paper [9], he proved that the capacity of each disk is given by: r 1 C= f (x)dx, π −1 where f is the solution of the following integral equations of the second kind: ω−1 1 1 f (x)dx = 1 (1.1) f (y) ± π −1 (x − y)2 + ω−2 with ω = q/r a real positive parameter. Then, he proved that (1.1) has a unique, continuous, real, and even solution which analytically has the following form: 1 ∞ j f (y) = 1 + (∓1) Kj (x, y) dx, −1
j =1
where the iterated kernels are given by: 1 ω−1 , K1(x, y) = π (x −y)2 +ω−2
Kj (x,y) =
1
−1
Kj −1(x, s)K1(s, y)ds,
j = 2, . . . .
From a numerical point of view, the developed methods [7, 8, 12, 15, 17] for the undisputed most interesting case (i.e., when ω−1 → 0) have followed the very first methods [4, 5, 16, 19, 20], and the most recent ones [11], proposed for the case when ω = 1. If ω−1 → 0 the kernel function is “close” to be singular on the bisector x = y. This kind of kernel belongs to the so-called nearly singular kernels class. Moreover, Phillips noted in [16] that: 1 1 ω−1 f (x)dx → f (y) if ω−1 → 0. (1.2) π −1 (x − y)2 + ω−2 Hence, for ω sufficiently large, the left-hand side of (1.1), in the case “−” is considered, becomes approximately zero which does not coincide with the right-hand side of (1.1). In [12], the authors presented a numerical approach based on a suitable transformation to move away from the poles x = y ± ω−1 i from the real axis. The numerical method produced very accurate results in the
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