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Numerical Investigation of the Fractional Oscillation Equations under the Context of Variable Order Caputo Fractional Derivative via Fractional Order Bernstein Wavelets

Author

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  • Ashish Rayal

    (Department of Mathematics, School of Applied and Life Sciences, Uttaranchal University, Dehradun 248007, India)

  • Bhagawati Prasad Joshi

    (Department of Mathematics, Graphic Era Hill University, Bhimtal 263136, India)

  • Mukesh Pandey

    (School of Computer Science, University of Petroleum & Energy Studies, Dehradun 248007, India)

  • Delfim F. M. Torres

    (Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal)

Abstract

This article describes an approximation technique based on fractional order Bernstein wavelets for the numerical simulations of fractional oscillation equations under variable order, and the fractional order Bernstein wavelets are derived by means of fractional Bernstein polynomials. The oscillation equation describes electrical circuits and exhibits a wide range of nonlinear dynamical behaviors. The proposed variable order model is of current interest in a lot of application areas in engineering and applied sciences. The purpose of this study is to analyze the behavior of the fractional force-free and forced oscillation equations under the variable-order fractional operator. The basic idea behind using the approximation technique is that it converts the proposed model into non-linear algebraic equations with the help of collocation nodes for easy computation. Different cases of the proposed model are examined under the selected variable order parameters for the first time in order to show the precision and performance of the mentioned scheme. The dynamic behavior and results are presented via tables and graphs to ensure the validity of the mentioned scheme. Further, the behavior of the obtained solutions for the variable order is also depicted. From the calculated results, it is observed that the mentioned scheme is extremely simple and efficient for examining the behavior of nonlinear random (constant or variable) order fractional models occurring in engineering and science.

Suggested Citation

  • Ashish Rayal & Bhagawati Prasad Joshi & Mukesh Pandey & Delfim F. M. Torres, 2023. "Numerical Investigation of the Fractional Oscillation Equations under the Context of Variable Order Caputo Fractional Derivative via Fractional Order Bernstein Wavelets," Mathematics, MDPI, vol. 11(11), pages 1-22, May.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:11:p:2503-:d:1158815
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    References listed on IDEAS

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    1. Heydari, M.H., 2020. "Chebyshev cardinal functions for a new class of nonlinear optimal control problems generated by Atangana–Baleanu–Caputo variable-order fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    2. Rayal, Ashish & Ram Verma, Sag, 2020. "Numerical analysis of pantograph differential equation of the stretched type associated with fractal-fractional derivatives via fractional order Legendre wavelets," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    3. Gómez-Aguilar, J.F., 2018. "Analytical and Numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 494(C), pages 52-75.
    4. Mohammed K. A. Kaabar & Ahmed Refice & Mohammed Said Souid & Francisco Martínez & Sina Etemad & Zailan Siri & Shahram Rezapour, 2021. "Existence and U-H-R Stability of Solutions to the Implicit Nonlinear FBVP in the Variable Order Settings," Mathematics, MDPI, vol. 9(14), pages 1-17, July.
    5. Min Cai & Changpin Li, 2020. "Numerical Approaches to Fractional Integrals and Derivatives: A Review," Mathematics, MDPI, vol. 8(1), pages 1-53, January.
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    Cited by:

    1. Abdellatif Benchaib & Abdelkrim Salim & Saïd Abbas & Mouffak Benchohra, 2023. "New Stability Results for Abstract Fractional Differential Equations with Delay and Non-Instantaneous Impulses," Mathematics, MDPI, vol. 11(16), pages 1-19, August.

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