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Using generating functions to convert an implicit (3,3) finite difference method to an explicit form on diffusion equation with different boundary conditions

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Abstract

In this article, our main goal is to develop an idea to convert an implicit (3,3) 𝜃-scheme finite difference method to an explicit form for both linear and nonlinear diffusion equations and also for nonlinear advection-diffusion equation with different boundary conditions. Accordingly, we assist power series generating functions which are a routine method in discrete mathematics. Also, the stability analysis of 𝜃–scheme to implement in nonlinear advection–diffusion equation has been investigated. Finally, the new approach has been implemented for Fisher, reaction–diffusion, Burgers and coupled Burgers equations as test problems to verify the ability and efficiency of the method proposed in this paper.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., 15914, Tehran, Iran

    Saeed Kazem, Ali Hatam & Mehdi Dehghan

  2. School of Computing, Science and Engineering, University of Salford, Salford, M5 4WT, UK

    Edmund Chadwick

Authors
  1. Saeed Kazem
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  2. Edmund Chadwick
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  3. Ali Hatam
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  4. Mehdi Dehghan
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Correspondence to Ali Hatam.

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Kazem, S., Chadwick, E., Hatam, A. et al. Using generating functions to convert an implicit (3,3) finite difference method to an explicit form on diffusion equation with different boundary conditions. Numer Algor 71, 827–854 (2016). https://doi.org/10.1007/s11075-015-0026-2

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  • DOI: https://doi.org/10.1007/s11075-015-0026-2

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