### Abstract

We consider a class of nonlinear singular Hammerstein Volterra integral equations. In general, these equations will have kernels containing both an end point and an Abel-type singularity, with exact solutions being typically nonsmooth. Under certain conditions, a uniformly convergent iterative solution is obtained on a small interval near the origin. In this work, two product integration methods are proposed and analyzed where the integral over a small initial interval is calculated analytically, allowing the optimal convergence rates to be achieved. This is illustrated by some numerical examples.

Original language | English |
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Pages (from-to) | 2-17 |

Number of pages | 16 |

Journal | Applied Numerical Mathematics |

Volume | 114 |

Issue number | SI |

DOIs | |

Publication status | Published - Apr 2017 |

### Keywords

- Hammerstein equations
- Iterative solution
- Nonlinear Volterra integral equation
- Product integration method
- Weakly singular kernel

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## Cite this

Allaei, S. S., Diogo, T., & Rebelo, M. (2017). Analytical and computational methods for a class of nonlinear singular integral equations.

*Applied Numerical Mathematics*,*114*(SI), 2-17. https://doi.org/10.1016/j.apnum.2016.06.001