### 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 |
---|---|

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 |

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### Keywords

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

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*Computing (Vienna/New York)*,

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

**Numerical verification of solutions of Nekrasov's integral equation.** / Murashige, S.; Oishi, Shinichi.

Research output: Contribution to journal › Article

*Computing (Vienna/New York)*, vol. 75, no. 1 SPEC. ISS., pp. 15-25. https://doi.org/10.1007/s00607-004-0112-0

}

TY - JOUR

T1 - Numerical verification of solutions of Nekrasov's integral equation

AU - Murashige, S.

AU - Oishi, Shinichi

PY - 2005/7

Y1 - 2005/7

N2 - 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.

AB - 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.

KW - Nekrasov's integral equation

KW - Numerical verification

KW - Singular integral equation

UR - http://www.scopus.com/inward/record.url?scp=23744517384&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23744517384&partnerID=8YFLogxK

U2 - 10.1007/s00607-004-0112-0

DO - 10.1007/s00607-004-0112-0

M3 - Article

AN - SCOPUS:23744517384

VL - 75

SP - 15

EP - 25

JO - Computing

JF - Computing

SN - 0010-485X

IS - 1 SPEC. ISS.

ER -