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Nonstandard finite differences for a truncated Bratu–Picard model

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  • Zegeling, Paul Andries
  • Iqbal, Sehar

Abstract

In this paper, we consider theoretical and numerical properties of a nonlinear boundary-value problem which is strongly related to the well-known Gelfand–Bratu model with parameter λ. When approximating the nonlinear term in the model via a Taylor expansion, we are able to find new types of solutions and multiplicities, depending on the final index N in the expansion. The number of solutions may vary from 0, 1, 2 to ∞. In the latter case of infinitely many solutions, we find both periodic and semi-periodic solutions. Numerical experiments using a non-standard finite-difference (NSFD) approximation illustrate all these aspects. We also show the difference in accuracy for different denominator functions in NSFD when applied to this model. A full classification is given of all possible cases depending on the parameters N and λ.

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  • Zegeling, Paul Andries & Iqbal, Sehar, 2018. "Nonstandard finite differences for a truncated Bratu–Picard model," Applied Mathematics and Computation, Elsevier, vol. 324(C), pages 266-284.
    • Handle: RePEc:eee:apmaco:v:324:y:2018:i:c:p:266-284
    DOI: 10.1016/j.amc.2017.12.005
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    Cited by:

    1. Iqbal, Sehar & Zegeling, Paul Andries, 2020. "An efficient nonlinear multigrid scheme for 2D boundary value problems," Applied Mathematics and Computation, Elsevier, vol. 372(C).

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