A new efficient method with error analysis for solving the second kind Fredholm integral equation with Cauchy kernel

Beyrami, Hossein; Lotfi, Taher; Mahdiani, Katayoun

doi:10.1016/j.cam.2016.01.011

Cited by 11 publications

(5 citation statements)

References 27 publications

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“…Finding such solutions is key to provide a general answer on the apparently basic question posed at the beginning of this Introduction, and we provide in the below an ab initio answer, in which we put the inhomogeneous dipolar BdG equation of a quasi-1D gas into the form of an equivalent singular integral equation, and solve this equation. To the best of our knowledge, this singular integral equation is novel, in that it provides an extension of the well known Cauchy-type singular kernels [38]. Specifically, the integral kernel we obtain is a combination of Cauchy-type kernels almost everywhere, with the exception of two isolated points where the singularity is stronger and the kernel becomes hypersingular.…”

Section: Introductionmentioning

confidence: 90%

“…Our case is however different from established textbook examples of hypersingular kernels [39], where the set of singular points has nonzero measure. From a more practical perspective, no universally reliable numerical method exists for solving singular integral equations, and each case must be treated differently in order to avoid numerical instabilities [38]. We discuss a method suitable for the BdG equation in its hypersingular integral form.…”

Section: Introductionmentioning

confidence: 99%

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Nonlocal field theory of quasiparticle scattering in dipolar Bose-Einstein condensates

Ribeiro

Fischer

2023

SciPost Phys. Core

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We consider the propagation of quasiparticle excitations in a dipolar Bose-Einstein condensate, and derive a nonlocal field theory of quasiparticle scattering at a stepwise inhomogeneity of the sound speed, obtained by tuning the contact coupling part of the interaction on one side of the barrier. To solve this problem ab initio, i.e., without prior assumptions on the form of the solutions, we reformulate the dipolar Bogoliubov-de Gennes equation as a singular integral equation. The latter is of a novel hypersingular type, in having a kernel which is hypersingular at only two isolated points. Deriving its solution, we show that the integral equation reveals a continuum of evanescent channels at the sound barrier which is absent for a purely contact-interaction condensate. We furthermore demonstrate that by performing a discrete approximation for the kernel, one achieves an excellent solution accuracy for already a moderate number of discretization steps. Finally, we show that the non-monotonic nature of the system dispersion, corresponding to the emergence of a roton minimum in the excitation spectrum, results in peculiar features of the transmission and reflection at the sound barrier which are nonexistent for contact interactions.

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Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Nonlocal field theory of quasiparticle scattering in dipolar Bose-Einstein condensates

Ribeiro

Fischer

2023

SciPost Phys. Core

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“…where ξ (s) is a given function, λ, μ are constants, and x(s) is an unknown function. The CSIEs have been solved via various numerical techniques such as using orthogonal Legendre polynomial [6], Lagrangian interpolation with Gauss-Jacobi mechanical quadrature [8], spline method [12,13], Galerkin technique [14], collocation method [15][16][17], application of Jacobi polynomials [18], using Chebyshev polynomials of the second kind [19], quadrature formula [20][21][22], reproducing kernel Hilbert space method [23,24], and other schemes [25][26][27]. Recently, several types of operational matrix methods with truncated series have been proposed for solving the integral and integro-differential equations (see [16,28]).…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of certain Cauchy singular integral equations using a collocation scheme

Seifi

2020

Adv Differ Equ

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The present study is devoted to developing a computational collocation technique for solving the Cauchy singular integral equation of the second kind (CSIE-2). Although, several studies have investigated the numerical approximation solution of CSIEs, the strong singularity and accuracy of the numerical methods are still two important challenges for these integral equations. In this paper, we focus on the smooth transformation and implementation of Bessel basis polynomials (BBP). The reduction of the CSIEs-2 into a system of algebraic equations with the Gauss–Legendre collocation points simplifies this technique. The technique of performing numerical approximation of the solution is well presented and illustrated in the matrix form. Also, the convergence and error bound associated with the scheme are established. Finally, several experiments show the reliability and numerical efficiency of the proposed scheme in comparison with other methods.

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“…However, some new numerical methods have been proposed for solving one-and twodimensional FIEs, but modification of the existing methods and development of new techniques should yet be explored to obtain accurate solutions successfully. In recent years, some computational schemes have been used to compute the solutions approximately such as Gaussian radial basis functions (Alipanah and Esmaeili 2011;Islam et al 2015), Chebyshev collocation (Avazzadeh and Heydari 2012), rationalized Haar functions (Babolian et al 2011a), Block-pulse functions (Babolian et al 2011b), Gauss quadratures (Bazm and Babolian 2012), Legendre collocation and Tau (Chokri 2013;Tari and Shahmorad 2008), Galerkin (Han and Wang 2002), extrapolation (Han and Jiong 2001), Taylor Tau (Rahimi et al 2010), LS-Algorithm (Karimi and Jozi 2015), and reproducing kernel method (Beyrami et al 2016). Moreover, one can refer to other methods that were applied to solve one-and two-dimensional FIEs, such as Hanson and Phillips (1978), Liang and Lin (2010), and Xie and Lin (2009).…”

Section: Introductionmentioning

confidence: 99%

LSMR iterative method for solving one- and two-dimensional linear Fredholm integral equations

et al. 2019

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This paper is devoted to extend an iterative algorithm for sparse least-squares problems for solving one-and two-dimensional linear Fredholm integral equations. We consider the operator form of these equations and then develop the LSMR method for solving them in an appropriate manner. The proposed method is based on bidiagonalization process of Golub-Kahan and reducing the linear operator L to the lower bidiagonal matrix form. Convergence property of the numerical solution associated with the suggested scheme is also provided. The method is fast and very accurate in solving two-dimensional equations. Other advantage of the proposed method is its high accuracy for solving Fredholm integral equations with non-smooth solutions. Solving such problems with other numerical methods, for instance, spectral methods, are not accurate as our proposed LSMR method. Numerical results confirm this claim.

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A new efficient method with error analysis for solving the second kind Fredholm integral equation with Cauchy kernel

Cited by 11 publications

References 27 publications

Nonlocal field theory of quasiparticle scattering in dipolar Bose-Einstein condensates

Nonlocal field theory of quasiparticle scattering in dipolar Bose-Einstein condensates

Numerical solution of certain Cauchy singular integral equations using a collocation scheme

LSMR iterative method for solving one- and two-dimensional linear Fredholm integral equations

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