### Abstract

We intend to numerically solve the famous two-dimensional Bratu's problem through developing a new iterative finite difference algorithm. The proposed scheme is capable of accurately approximating both branches of the solution of the considered problem. We first introduce a new transformation of Bratu's problem conserving the solution bifurcated behavior. Then, using Newton-Raphson-Kantorovich approximation in function space, we develop an iterative finite difference method yielding a simple algorithm for approximating the sequence of numerical solutions. We also perform a convergence analysis proving that our algorithm converges quadratically to the exact solution of the problem. Finally, we present numerical simulation results showing the capability of our algorithm to accurately compute the two-branches of the solution for the two-dimensional Bratu's problem.

Original language | English (US) |
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Pages (from-to) | 202-216 |

Number of pages | 15 |

Journal | Applied Numerical Mathematics |

Volume | 153 |

DOIs | |

State | Published - Jul 2020 |

### Keywords

- Finite difference
- Iterative method
- Newton-Raphson-Kantorovich approximation method
- Two-dimensional Bratu's problem

### ASJC Scopus subject areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

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## Cite this

*Applied Numerical Mathematics*,

*153*, 202-216. https://doi.org/10.1016/j.apnum.2020.02.010