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The Use of Cubic Splines in the Numerical Solution of Fractional Differential Equations

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  • W. K. Zahra
  • S. M. Elkholy

Abstract

Fractional calculus became a vital tool in describing many phenomena appeared in physics, chemistry as well as engineering fields. Analytical solution of many applications, where the fractional differential equations appear, cannot be established. Therefore, cubic polynomial spline-function-based method combined with shooting method is considered to find approximate solution for a class of fractional boundary value problems (FBVPs). Convergence analysis of the method is considered. Some illustrative examples are presented.

Suggested Citation

  • W. K. Zahra & S. M. Elkholy, 2012. "The Use of Cubic Splines in the Numerical Solution of Fractional Differential Equations," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2012, pages 1-16, August.
    • Handle: RePEc:hin:jijmms:638026
    DOI: 10.1155/2012/638026
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    References listed on IDEAS

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    1. Momani, Shaher & Odibat, Zaid, 2007. "Numerical comparison of methods for solving linear differential equations of fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 31(5), pages 1248-1255.
    2. Tavazoei, Mohammad Saleh & Haeri, Mohammad, 2009. "A note on the stability of fractional order systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(5), pages 1566-1576.
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    Cited by:

    1. Zahra, W.K. & Elkholy, S.M. & Fahmy, M., 2019. "Rational spline-nonstandard finite difference scheme for the solution of time-fractional Swift–Hohenberg equation," Applied Mathematics and Computation, Elsevier, vol. 343(C), pages 372-387.
    2. Chinedu Nwaigwe & Abdon Atangana, 2025. "Generalizing Averaging Techniques for Approximating Fractional Differential Equations With Caputo Derivative," Journal of Applied Mathematics, John Wiley & Sons, vol. 2025(1).
    3. 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).
    4. Hou, Jie & Ma, Zhiying & Ying, Shihui & Li, Ying, 2024. "HNS: An efficient hermite neural solver for solving time-fractional partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    5. Waseem, Waseem & Sulaiman, M. & Aljohani, Abdulah Jeza, 2020. "Investigation of fractional models of damping material by a neuroevolutionary approach," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).

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