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Jacobi collocation method for the approximate solution of some fractional-order Riccati differential equations with variable coefficients

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  • Singh, Harendra
  • Srivastava, H.M.

Abstract

This paper presents a computational method for the approximate solution of arbitrary-order non-linear fractional Riccati differential equations with variable coefficients. Proposed computational method is a combination of the operational matrix of integration method and the collocation method associated with the Jacobi polynomials. Convergence analysis of the proposed method is provided. Numerical results for different fractional orders of the Riccati differential equations are discussed. Figures and tables are used to show the numerical results derived from the proposed computational method for particular cases of Jacobi polynomials such as the Legendre polynomials, the Chebyshev polynomials of the second kind, the Chebyshev polynomials of the third kind, the Chebyshev polynomial of the fourth kind, and the Gegenbauer (or ultraspherical) polynomials. Numerical results from the proposed methods are compared from those derived by using the existing analytical and numerical methods. It is observed that the results from the proposed method are more accurate. Maximum absolute error and the root-mean square error tables are given for all five kinds of polynomials for comparison purposes.

Suggested Citation

  • Singh, Harendra & Srivastava, H.M., 2019. "Jacobi collocation method for the approximate solution of some fractional-order Riccati differential equations with variable coefficients," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 1130-1149.
    • Handle: RePEc:eee:phsmap:v:523:y:2019:i:c:p:1130-1149
    DOI: 10.1016/j.physa.2019.04.120
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    References listed on IDEAS

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    1. Singh, Jagdev & Kumar, Devendra & Baleanu, Dumitru & Rathore, Sushila, 2018. "An efficient numerical algorithm for the fractional Drinfeld–Sokolov–Wilson equation," Applied Mathematics and Computation, Elsevier, vol. 335(C), pages 12-24.
    2. Kumar, Devendra & Singh, Jagdev & Baleanu, Dumitru & Sushila,, 2018. "Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 492(C), pages 155-167.
    3. Odibat, Zaid & Momani, Shaher, 2008. "Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 36(1), pages 167-174.
    4. Kashkari, Bothayna S.H. & Syam, Muhammed I., 2016. "Fractional-order Legendre operational matrix of fractional integration for solving the Riccati equation with fractional order," Applied Mathematics and Computation, Elsevier, vol. 290(C), pages 281-291.
    5. Singh, Jagdev & Kumar, Devendra & Hammouch, Zakia & Atangana, Abdon, 2018. "A fractional epidemiological model for computer viruses pertaining to a new fractional derivative," Applied Mathematics and Computation, Elsevier, vol. 316(C), pages 504-515.
    6. 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. Singh, Harendra, 2020. "Analysis for fractional dynamics of Ebola virus model," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    2. Izadi, Mohammad & Srivastava, H.M., 2021. "Numerical approximations to the nonlinear fractional-order Logistic population model with fractional-order Bessel and Legendre bases," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    3. Srivastava, H.M. & Saad, Khaled M. & Khader, M.M., 2020. "An efficient spectral collocation method for the dynamic simulation of the fractional epidemiological model of the Ebola virus," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    4. Haifa Bin Jebreen & Ioannis Dassios, 2022. "A Biorthogonal Hermite Cubic Spline Galerkin Method for Solving Fractional Riccati Equation," Mathematics, MDPI, vol. 10(9), pages 1-14, April.
    5. Singh, Harendra, 2021. "Analysis of drug treatment of the fractional HIV infection model of CD4+ T-cells," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    6. Saad, Khaled M. & Srivastava, H.M. & Gómez-Aguilar, J.F., 2020. "A Fractional Quadratic autocatalysis associated with chemical clock reactions involving linear inhibition," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    7. Singh, Harendra & Baleanu, Dumitru & Singh, Jagdev & Dutta, Hemen, 2021. "Computational study of fractional order smoking model," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).

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