### Abstract

In 2005 Corrêa and Filho established existence and uniqueness results for the nonlinear PDE: -δu = g(x,u)α; ∫_{Ω} f(x,u) ^{β};, which arises in physical models of thermodynamical equilibrium via Coulomb potential, among others [3]. In this work we discuss a numerical method for a special case of this equation: -α(∫^{1} _{0} u(t)dt u" = f(x), 0 < x < 1, u(0) = a, u(1) = b. We first consider the existence and uniqueness of the analytic problem using a fixed point argument and the contraction mapping theorem. Next, we evaluate the solution of the numerical problem via a finite difference scheme. From there, the existence and convergence of the approximate solution will be addressed as well as a uniqueness argument, which requires some additional restrictions. Finally, we conclude the work with some numerical examples where an interval-halving technique was implemented.

Original language | English (US) |
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Pages (from-to) | 243-261 |

Number of pages | 19 |

Journal | Journal of Integral Equations and Applications |

Volume | 20 |

Issue number | 2 |

DOIs | |

State | Published - Dec 1 2008 |

### All Science Journal Classification (ASJC) codes

- Numerical Analysis
- Applied Mathematics