Sinc Nyström Method for Singularly Perturbed Love's Integral Equation

AbstractAn efficient numerical method is proposed for the solution of Love's integral equationwhere 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.

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Numerical solution of the fourth-order partial integro-differential equation with multi-term kernels by the Sinc-collocation method based on the double exponential transformation

Applied Mathematics and Computation ◽

10.1016/j.amc.2020.125693 ◽

2021 ◽

Vol 392 ◽

pp. 125693

Author(s):

Wenlin Qiu ◽

Da Xu ◽

Jing Guo

Keyword(s):

Differential Equation ◽

Numerical Solution ◽

Collocation Method ◽

Fourth Order ◽

Integro Differential Equation ◽

Exponential Transformation ◽

Double Exponential Transformation ◽

Double Exponential

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On the Equivalence of RWG Method of Moments and the Locally Corrected Nyström Method for Solving the Electric Field Integral Equation

IEEE Transactions on Antennas and Propagation ◽

10.1109/tap.2013.2291905 ◽

2014 ◽

Vol 62 (2) ◽

pp. 772-782 ◽

Cited By ~ 8

Author(s):

Mohammad Shafieipour ◽

Ian Jeffrey ◽

Jonatan Aronsson ◽

Vladimir I. Okhmatovski

Keyword(s):

Integral Equation ◽

Electric Field ◽

Method Of Moments ◽

Nyström Method ◽

Electric Field Integral Equation ◽

Nystrom Method ◽

Field Integral

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Numerical solution of nonlinear Hammerstein integral equations via Sinc collocation method based on double exponential transformation

Mathematical Sciences ◽

10.1186/2251-7456-7-30 ◽

2013 ◽

Vol 7 (1) ◽

pp. 30 ◽

Cited By ~ 1

Author(s):

Mohammad Ali Fariborzi Araghi ◽

Ghasem Kazemi Gelian

Keyword(s):

Numerical Solution ◽

Integral Equations ◽

Collocation Method ◽

Exponential Transformation ◽

Hammerstein Integral Equations ◽

Double Exponential Transformation ◽

Double Exponential

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Sinc-Nyström method for numerical solution of one-dimensional Cauchy singular integral equation given on a smooth arc in the complex plane

Mathematics of Computation ◽

10.1090/s0025-5718-1988-0942147-x ◽

1988 ◽

Vol 51 (183) ◽

pp. 133-133

Author(s):

Bernard Bialecki ◽

Frank Stenger

Keyword(s):

Integral Equation ◽

Singular Integral Equation ◽

Numerical Solution ◽

Complex Plane ◽

Singular Integral ◽

Nyström Method ◽

One Dimensional ◽

Nystrom Method ◽

Cauchy Singular Integral ◽

Cauchy Singular Integral Equation

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A formally second‐order backward differentiation formula Sinc‐collocation method for the Volterra integro‐differential equation with a weakly singular kernel based on the double exponential transformation

Numerical Methods for Partial Differential Equations ◽

10.1002/num.22703 ◽

2020 ◽

Author(s):

Wenlin Qiu ◽

Da Xu ◽

Jing Guo

Keyword(s):

Differential Equation ◽

Collocation Method ◽

Singular Kernel ◽

Weakly Singular Kernel ◽

Differentiation Formula ◽

Exponential Transformation ◽

Backward Differentiation Formula ◽

Weakly Singular ◽

Double Exponential Transformation ◽

Double Exponential

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Discovery of the double exponential transformation and its developments

Publications of the Research Institute for Mathematical Sciences ◽

10.2977/prims/1145474600 ◽

2005 ◽

Vol 41 (4) ◽

pp. 897-935 ◽

Cited By ~ 47

Author(s):

Masatake Mori

Keyword(s):

Exponential Transformation ◽

Double Exponential Transformation ◽

Double Exponential

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Error behavior of the Nyström method with singularity subtraction applied to a singular Fredholm integral equation of the second kind

10.1063/1.5044088 ◽

2018 ◽

Author(s):

Josef Rak

Keyword(s):

Integral Equation ◽

Fredholm Integral Equation ◽

Nyström Method ◽

Nystrom Method ◽

Error Behavior

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Solving fuzzy Fredholm linear integral equations using Sinc method and double exponential transformation

Soft Computing ◽

10.1007/s00500-014-1316-7 ◽

2014 ◽

Vol 19 (4) ◽

pp. 1063-1070 ◽

Cited By ~ 5

Author(s):

M. A. Fariborzi Araghi ◽

Gh. Kazemi Gelian

Keyword(s):

Integral Equations ◽

Sinc Method ◽

Exponential Transformation ◽

Double Exponential Transformation ◽

Double Exponential ◽

Linear Integral ◽

Linear Integral Equations

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Density-Weighted Nyström Method for Computing Large Kernel Eigensystems

Neural Computation ◽

10.1162/neco.2009.11-07-651 ◽

2009 ◽

Vol 21 (1) ◽

pp. 121-146 ◽

Cited By ~ 49

Author(s):

Kai Zhang ◽

James T. Kwok

Keyword(s):

Principal Component Analysis ◽

Integral Equation ◽

Spectral Clustering ◽

Principal Component ◽

Kernel Principal Component Analysis ◽

Nyström Method ◽

Weighted Version ◽

Nystrom Method ◽

Equal Importance ◽

Large Kernel

The Nyström method is a well-known sampling-based technique for approximating the eigensystem of large kernel matrices. However, the chosen samples in the Nyström method are all assumed to be of equal importance, which deviates from the integral equation that defines the kernel eigenfunctions. Motivated by this observation, we extend the Nyström method to a more general, density-weighted version. We show that by introducing the probability density function as a natural weighting scheme, the approximation of the eigensystem can be greatly improved. An efficient algorithm is proposed to enforce such weighting in practice, which has the same complexity as the original Nyström method and hence is notably cheaper than several other alternatives. Experiments on kernel principal component analysis, spectral clustering, and image segmentation demonstrate the encouraging performance of our algorithm.

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