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A Biorthogonal Hermite Cubic Spline Galerkin Method for Solving Fractional Riccati Equation

Author

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  • Haifa Bin Jebreen

    (Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
    These authors contributed equally to this work.)

  • Ioannis Dassios

    (FRESLIPS, University College Dublin, D04 V1W8 Dublin, Ireland
    These authors contributed equally to this work.)

Abstract

This paper is devoted to the wavelet Galerkin method to solve the Fractional Riccati equation. To this end, biorthogonal Hermite cubic Spline scaling bases and their properties are introduced, and the fractional integral is represented based on these bases as an operational matrix. Firstly, we obtain the Volterra integral equation with a weakly singular kernel corresponding to the desired equation. Then, using the operational matrix of fractional integration and the Galerkin method, the corresponding integral equation is reduced to a system of algebraic equations. Solving this system via Newton’s iterative method gives the unknown solution. The convergence analysis is investigated and shows that the convergence rate is O ( 2 − s ) . To demonstrate the efficiency and accuracy of the method, some numerical simulations are provided.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:9:p:1461-:d:803259
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. X. Y. Li & B. Y. Wu & R. T. Wang, 2014. "Reproducing Kernel Method for Fractional Riccati Differential Equations," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-6, April.
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    Cited by:

    1. Archna Kumari & Vijay K. Kukreja, 2023. "Survey of Hermite Interpolating Polynomials for the Solution of Differential Equations," Mathematics, MDPI, vol. 11(14), pages 1-28, July.
    2. Fathy, Mohamed & Abdelgaber, K.M., 2022. "Approximate solutions for the fractional order quadratic Riccati and Bagley-Torvik differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).

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