Limit Values of Derivatives of the Cauchy Integrals and Computation of the Logarithmic Potentials

Xu, Yuesheng; Chen, H.-L.; Zou, Qinmeng

doi:10.1007/s00607-003-0072-4

Cited by 7 publications

(12 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When S is closed, the limit values of higher order derivatives of the Cauchy integrals were derived in [23]. We next give similar results for the case when S is an open arc.…”

Section: Theorem 62mentioning

confidence: 72%

“…The key idea used in computing the potential (6.11) is to develop the Taylor expansion of u in terms of its limit values of the normal derivatives. To this end, as in [23], we begin with giving the limit values of higher order derivatives of the Cauchy integrals defined on open arcs. Let S be an arc in C and have a parametrization as S = {ζ(τ ), a τ b}.…”

Section: Theorem 62mentioning

confidence: 99%

“…[1,2,24]). A fast numerical algorithm for computing the logarithmic potentials defined on a closed curve was proposed in [23]. It has an optimal convergence order and linear computational complexity.…”

Section: Theorem 62mentioning

confidence: 99%

“…In [13], a fully discrete algorithm for the fast Fourier-Galerkin method is developed which needs only a quasi-linear number of multiplications for generating the compressed matrix and still has the optimal convergence order of the original method. A fast numerical algorithm for computing the logarithmic potential defined on a closed curve was proposed in [23]. Ideas in these papers are used in this paper in developing the fast algorithm.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Fast Fourier-Galerkin methods for first-kind logarithmic-kernel integral equations on open arcs

Wang

2010

Sci. China Ser. A-Math.

View full text Add to dashboard Cite

We propose a fully discrete fast Fourier-Galerkin method for solving an integral equation of the first kind with a logarithmic kernel on a smooth open arc, which is a reformulation of the Dirichlet problem of the Laplace equation in the plane. The optimal convergence order and quasi-linear complexity order of the proposed method are established. A precondition is introduced. Combining this method with an efficient numerical integration algorithm for computing the single-layer potential defined on an open arc, we obtain the solution of the Dirichlet problem on a smooth open arc in the plane. Numerical examples are presented to confirm the theoretical estimates and to demonstrate the efficiency and accuracy of the proposed method. Keywords Dirichlet problem, open arc, singular boundary integral equations, Fourier-Galerkin methods, logarithmic potentials MSC(2000): 65R20, 45E05, 41A55, 65F35 Citation: Wang B, Wang R, Xu Y S. Fast Fourier-Galerkin methods for first-kind logarithmic-kernel integral equations on open arcs.

show abstract

“…When S is closed, the limit values of higher order derivatives of the Cauchy integrals were derived in [23]. We next give similar results for the case when S is an open arc.…”

Section: Theorem 62mentioning

confidence: 72%

Section: Theorem 62mentioning

confidence: 99%

Section: Theorem 62mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Fast Fourier-Galerkin methods for first-kind logarithmic-kernel integral equations on open arcs

Wang

2010

Sci. China Ser. A-Math.

View full text Add to dashboard Cite

show abstract

“…Ideally, such a quadrature strategy should preserve the convergence order for the compression method and use only O(N L ) number of functional evaluations in computing all nonzero entries of the matrix V L . The quadrature strategy proposed in this section is influenced by the work in [21,25,40,47].…”

Section: A Numerical Quadrature Strategymentioning

confidence: 99%

Multiparameter regularization for Volterra kernel identification via multiscale collocation methods

Brenner

Jiang

2008

Adv Comput Math

View full text Add to dashboard Cite

Identification of the Volterra system is an ill-posed problem. We propose a regularization method for solving this ill-posed problem via a multiscale collocation method with multiple regularization parameters corresponding to the multiple scales. Many highly nonlinear problems such as flight data analysis demand identifying the system of a high order. This task requires huge computational costs due to processing a dense matrix of a large order. To overcome this difficulty a compression strategy is introduced to approximate the full matrix resulted in collocation of the Volterra kernel by an appropriate sparse matrix. A numerical quadrature strategy is designed to efficiently compute the entries of the compressed matrix. Finally, numerical results of three simulation experiments are presented to demonstrate the accuracy and efficiency of the proposed method. Communicated by Juan Manuel Peña. 422 M. Brenner et al.

show abstract

Fast Fourier–Galerkin methods for solving singular boundary integral equations: Numerical integration and precondition

Jiang

2010

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

Limit Values of Derivatives of the Cauchy Integrals and Computation of the Logarithmic Potentials

Cited by 7 publications

References 16 publications

Fast Fourier-Galerkin methods for first-kind logarithmic-kernel integral equations on open arcs

Fast Fourier-Galerkin methods for first-kind logarithmic-kernel integral equations on open arcs

Multiparameter regularization for Volterra kernel identification via multiscale collocation methods

Fast Fourier–Galerkin methods for solving singular boundary integral equations: Numerical integration and precondition

Contact Info

Product

Resources

About