### 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.

Original language | English |
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Pages (from-to) | 15-25 |

Number of pages | 11 |

Journal | Computing (Vienna/New York) |

Volume | 75 |

Issue number | 1 SPEC. ISS. |

DOIs | |

Publication status | Published - 2005 Jul 1 |

### Keywords

- Nekrasov's integral equation
- Numerical verification
- Singular integral equation

### ASJC Scopus subject areas

- Software
- Theoretical Computer Science
- Numerical Analysis
- Computer Science Applications
- Computational Theory and Mathematics
- Computational Mathematics

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## Cite this

Murashige, S., & Oishi, S. (2005). Numerical verification of solutions of Nekrasov's integral equation.

*Computing (Vienna/New York)*,*75*(1 SPEC. ISS.), 15-25. https://doi.org/10.1007/s00607-004-0112-0