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Solving Non-Linear Fractional Variational Problems Using Jacobi Polynomials

Author

Listed:
  • Harendra Singh

    (Department of Mathematics, Post Graduate College, Ghazipur 233001, India)

  • Rajesh K. Pandey

    (Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Varanasi 221005, India
    Centre for Advanced Biomaterials and Tissue Engineering, Indian Institute of Technology (BHU) Varanasi, Varanasi 221005, India)

  • Hari Mohan Srivastava

    (Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
    Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan)

Abstract

The aim of this paper is to solve a class of non-linear fractional variational problems (NLFVPs) using the Ritz method and to perform a comparative study on the choice of different polynomials in the method. The Ritz method has allowed many researchers to solve different forms of fractional variational problems in recent years. The NLFVP is solved by applying the Ritz method using different orthogonal polynomials. Further, the approximate solution is obtained by solving a system of nonlinear algebraic equations. Error and convergence analysis of the discussed method is also provided. Numerical simulations are performed on illustrative examples to test the accuracy and applicability of the method. For comparison purposes, different polynomials such as 1) Shifted Legendre polynomials, 2) Shifted Chebyshev polynomials of the first kind, 3) Shifted Chebyshev polynomials of the third kind, 4) Shifted Chebyshev polynomials of the fourth kind, and 5) Gegenbauer polynomials are considered to perform the numerical investigations in the test examples. Further, the obtained results are presented in the form of tables and figures. The numerical results are also compared with some known methods from the literature.

Suggested Citation

  • Harendra Singh & Rajesh K. Pandey & Hari Mohan Srivastava, 2019. "Solving Non-Linear Fractional Variational Problems Using Jacobi Polynomials," Mathematics, MDPI, vol. 7(3), pages 1-24, February.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:3:p:224-:d:209627
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    References listed on IDEAS

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    1. Samer S. Ezz-Eldien & Ramy M. Hafez & Ali H. Bhrawy & Dumitru Baleanu & Ahmed A. El-Kalaawy, 2017. "New Numerical Approach for Fractional Variational Problems Using Shifted Legendre Orthonormal Polynomials," Journal of Optimization Theory and Applications, Springer, vol. 174(1), pages 295-320, July.
    2. Mehdi Dehghan & Mehdi Tatari, 2006. "The use of Adomian decomposition method for solving problems in calculus of variations," Mathematical Problems in Engineering, Hindawi, vol. 2006, pages 1-12, June.
    3. Ricardo Almeida, 2017. "Variational Problems Involving a Caputo-Type Fractional Derivative," Journal of Optimization Theory and Applications, Springer, vol. 174(1), pages 276-294, July.
    4. Behroozifar, M. & Sazmand, A., 2017. "An approximate solution based on Jacobi polynomials for time-fractional convection–diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 296(C), pages 1-17.
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    Cited by:

    1. Pandey, Divyansh & Pandey, Rajesh K. & Agarwal, R.P., 2023. "Numerical approximation of fractional variational problems with several dependent variables using Jacobi poly-fractonomials," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 28-43.
    2. Waleed Mohamed Abd-Elhameed, 2022. "Novel Formulae of Certain Generalized Jacobi Polynomials," Mathematics, MDPI, vol. 10(22), pages 1-25, November.
    3. Hari Mohan Srivastava & Khaled M. Saad, 2020. "A Comparative Study of the Fractional-Order Clock Chemical Model," Mathematics, MDPI, vol. 8(9), pages 1-14, August.

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