An iterative finite difference method for approximating the two-branched solution of Bratu's problem

Mohamed Ben-Romdhane, Helmi Temimi, Mahboub Baccouch

Research output: Contribution to journalArticle

Abstract

In this paper, we propose an iterative finite difference (IFD) scheme to simultaneously approximate both branches of a two-branched solution to the one-dimensional Bratu's problem. We first introduce a transformation to convert Bratu's problem into a simpler one. The transformed nonlinear ordinary differential equation is discretized using the Newton–Raphson–Kantorovich approximation in function space. The convergence of the sequence of approximations is proved to be quadratic. Then, we apply the classical finite difference method to approximate the sequence of approximations. The proposed new scheme has two main advantages. First, it produces accurate numerical solutions with low computational cost. Second, it is able to compute the two branches of the solution of Bratu's problem, even for small values of the transition parameter λ where the numerical computation of the upper branch of the solution becomes challenging. Numerical examples are provided to show the efficiency and accuracy of the proposed scheme.

Original languageEnglish (US)
Pages (from-to)62-76
Number of pages15
JournalApplied Numerical Mathematics
Volume139
DOIs
StatePublished - May 1 2019

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Finite difference method
Ordinary differential equations
Difference Method
Finite Difference
Branch
Approximation
Costs
Nonlinear Ordinary Differential Equations
Finite Difference Scheme
Numerical Computation
Function Space
Convert
Computational Cost
Numerical Solution
Numerical Examples

Keywords

  • Bratu's problem
  • Iterative finite difference method
  • Newton–Raphson–Kantorovich approximation
  • Two-branched solution

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this

An iterative finite difference method for approximating the two-branched solution of Bratu's problem. / Ben-Romdhane, Mohamed; Temimi, Helmi; Baccouch, Mahboub.

In: Applied Numerical Mathematics, Vol. 139, 01.05.2019, p. 62-76.

Research output: Contribution to journalArticle

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