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Revisited Fisher’s equation in a new outlook: A fractional derivative approach

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  • Alquran, Marwan
  • Al-Khaled, Kamel
  • Sardar, Tridip
  • Chattopadhyay, Joydev

Abstract

The well-known Fisher equation with fractional derivative is considered to provide some characteristics of memory embedded into the system. The modified model is analyzed both analytically and numerically. A comparatively new technique residual power series method is used for finding approximate solutions of the modified Fisher model. A new technique combining Sinc-collocation and finite difference method is used for numerical study. The abundance of the bird species Phalacrocorax carbois considered as a test bed to validate the model outcome using estimated parameters. We conjecture non-diffusive and diffusive fractional Fisher equation represents the same dynamics in the interval (memory index, α∈(0.8384,0.9986)). We also observe that when the value of memory index is close to zero, the solutions bifurcate and produce a wave-like pattern. We conclude that the survivability of the species increases for long range memory index. These findings are similar to Fisher observation and act in a similar fashion that advantageous genes do.

Suggested Citation

  • Alquran, Marwan & Al-Khaled, Kamel & Sardar, Tridip & Chattopadhyay, Joydev, 2015. "Revisited Fisher’s equation in a new outlook: A fractional derivative approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 438(C), pages 81-93.
    • Handle: RePEc:eee:phsmap:v:438:y:2015:i:c:p:81-93
    DOI: 10.1016/j.physa.2015.06.036
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    References listed on IDEAS

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    1. John Geweke, 1991. "Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments," Staff Report 148, Federal Reserve Bank of Minneapolis.
    2. Rana, Sourav & Bhattacharya, Sabyasachi & Pal, Joydeep & N’Guérékata, Gaston M. & Chattopadhyay, Joydev, 2013. "Paradox of enrichment: A fractional differential approach with memory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(17), pages 3610-3621.
    3. El-Ajou, Ahmad & Abu Arqub, Omar & Momani, Shaher & Baleanu, Dumitru & Alsaedi, Ahmed, 2015. "A novel expansion iterative method for solving linear partial differential equations of fractional order," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 119-133.
    4. Michael G. Neubert & Ingrid M. Parker, 2004. "Projecting Rates of Spread for Invasive Species," Risk Analysis, John Wiley & Sons, vol. 24(4), pages 817-831, August.
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    Cited by:

    1. Han, Yuxin & Huang, Xin & Gu, Wei & Zheng, Bolong, 2023. "Linearized transformed L1 finite element methods for semi-linear time-fractional parabolic problems," Applied Mathematics and Computation, Elsevier, vol. 458(C).
    2. Zhao, Dazhi & Yu, Guozhu & Tian, Yan, 2020. "Recursive formulae for the analytic solution of the nonlinear spatial conformable fractional evolution equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 537(C).
    3. Atangana, Abdon, 2016. "On the new fractional derivative and application to nonlinear Fisher’s reaction–diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 948-956.
    4. Muhammed I. Syam, 2017. "A Numerical Solution of Fractional Lienard’s Equation by Using the Residual Power Series Method," Mathematics, MDPI, vol. 6(1), pages 1-9, December.

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