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Neuronal dynamics and electrophysiology fractional model: A modified wavelet approach

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  • Usman, Muhammad
  • Hamid, Muhammad
  • Khan, Zafar Hayat
  • Haq, Rizwan Ul

Abstract

The modeling and numerical simulations of fractional-order cable type problems can provide a proper structure of anomalous-diffusion in the measure of ions in complex neuronal dynamics and electrophysiology. In the present work, a novel approach of Gegenbauer wavelets (GWM) based on the operational matrices with their derivatives is proposed. New operational matrices are established for fractional-order derivative and variable-order derivative with the help of piecewise functions. The given problem is converted into a system of nonlinear equations via Gegenbauer wavelets in the suggested method. The fractional cable equation of variable-order is taken to account and successfully solved using a new algorithm. The present study also contains the convergence and error bound analysis of the proposed approach. Solutions obtained via operational matrix-based algorithm are validating the accuracy, efficiency and reliability of the suggested method. The comparative study in tabular form, as well as graphical plots (2D and 3D) for solutions and absolute error, has been reported. Hence, outcomes are validating the suggested method as an accurate and efficient tool and could be adopted for other types of fractional-order nonlinear complex dynamical problems.

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  • Usman, Muhammad & Hamid, Muhammad & Khan, Zafar Hayat & Haq, Rizwan Ul, 2021. "Neuronal dynamics and electrophysiology fractional model: A modified wavelet approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 570(C).
    • Handle: RePEc:eee:phsmap:v:570:y:2021:i:c:s0378437121000777
    DOI: 10.1016/j.physa.2021.125805
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    References listed on IDEAS

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    1. Atangana, Abdon & Koca, Ilknur, 2016. "Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 89(C), pages 447-454.
    2. Saeed, Umer & Rehman, Mujeeb ur & Iqbal, Muhammad Asad, 2015. "Modified Chebyshev wavelet methods for fractional delay-type equations," Applied Mathematics and Computation, Elsevier, vol. 264(C), pages 431-442.
    3. Hamid, Muhammad & Usman, Muhammad & Zubair, Tamour & Haq, Rizwan Ul & Shafee, Ahmad, 2019. "An efficient analysis for N-soliton, Lump and lump–kink solutions of time-fractional (2+1)-Kadomtsev–Petviashvili equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 528(C).
    4. Usman, M. & Hamid, M. & Zubair, T. & Haq, R.U. & Wang, W. & Liu, M.B., 2020. "Novel operational matrices-based method for solving fractional-order delay differential equations via shifted Gegenbauer polynomials," Applied Mathematics and Computation, Elsevier, vol. 372(C).
    5. Hamid, M. & Usman, M. & Haq, R.U. & Wang, W., 2020. "A Chelyshkov polynomial based algorithm to analyze the transport dynamics and anomalous diffusion in fractional model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 551(C).
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    1. Shahni, Julee & Singh, Randhir, 2022. "Numerical simulation of Emden–Fowler integral equation with Green’s function type kernel by Gegenbauer-wavelet, Taylor-wavelet and Laguerre-wavelet collocation methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 430-444.

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