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High-Order Approximation to Generalized Caputo Derivatives and Generalized Fractional Advection–Diffusion Equations

Author

Listed:
  • Sarita Kumari

    (Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Varanasi 221005, Uttar Pradesh, India)

  • Rajesh K. Pandey

    (Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Varanasi 221005, Uttar Pradesh, India)

  • Ravi P. Agarwal

    (Department of Mathematics, Texas A & M University-Kingsville, Kingsville, TX 77553, USA)

Abstract

In this article, a high-order time-stepping scheme based on the cubic interpolation formula is considered to approximate the generalized Caputo fractional derivative (GCFD). Convergence order for this scheme is ( 4 − α ) , where α ( 0 < α < 1 ) is the order of the GCFD. The local truncation error is also provided. Then, we adopt the developed scheme to establish a difference scheme for the solution of the generalized fractional advection–diffusion equation with Dirichlet boundary conditions. Furthermore, we discuss the stability and convergence of the difference scheme. Numerical examples are presented to examine the theoretical claims. The convergence order of the difference scheme is analyzed numerically, which is ( 4 − α ) in time and second-order in space.

Suggested Citation

  • Sarita Kumari & Rajesh K. Pandey & Ravi P. Agarwal, 2023. "High-Order Approximation to Generalized Caputo Derivatives and Generalized Fractional Advection–Diffusion Equations," Mathematics, MDPI, vol. 11(5), pages 1-24, February.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:5:p:1200-:d:1084008
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    References listed on IDEAS

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    1. Owolabi, Kolade M., 2020. "High-dimensional spatial patterns in fractional reaction-diffusion system arising in biology," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    2. Baleanu, Dumitru & Wu, Guo–Cheng & Zeng, Sheng–Da, 2017. "Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 99-105.
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