Computing

, Volume 75, Issue 1, pp 15–25 | Cite as

Numerical Verification of Solutions of Nekrasov’s Integral Equation

Article

Abstract.

This paper describes numerical verification of solutions of Nekrasov’s integral equation which is a mathematical model of two-dimensional water waves. This nonlinear and periodic integral equation includes a logarithmic singular kernel which is typically found in some two-dimensional potential problems. We propose the verification method using some properties of the singular integral for trigonometric polynomials and Schauder’s fixed point theorem in the periodic Sobolev space. A numerical example shows effectiveness of the present method.

AMS Subject Classifications:

45G05 65G20 65R20 

Keywords

Nekrasov’s integral equation singular integral equation numerical verification 

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Copyright information

© Springer-Verlag Wien 2005

Authors and Affiliations

  1. 1.Department of Complexity Science and EngineeringThe University of TokyoTokyoJapan
  2. 2.School of Science and EngineeringWaseda UniversityTokyoJapan

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