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Numerical algorithm for a general fractional diffusion equation

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  • Maurya, Rahul Kumar
  • Li, Dongxia
  • Singh, Anant Pratap
  • Singh, Vineet Kumar

Abstract

In this manuscript, the analytical solution to the general time fractional diffusion equation followed by the regularity analysis of its solution is presented. Further, a hybrid efficient numerical algorithm for a general fractional derivative of order α∈(0,1) is designed. The physical application of the developed approximation formula is employed to design a hybrid numerical algorithm to determine the numerical solution of the corresponding general time fractional diffusion equation. The proposed numerical techniques are subjected to solvability, numerical stability, and convergence analysis. Numerical example is utilized to verify that (3−α)-th order convergence is attained in time which is higher than the existing scheme (Xu et al., 2013). In the spatial direction, the fourth order convergence is obtained utilizing the compact finite difference method. Besides, such a hybrid numerical algorithm is also used in the stochastic model.

Suggested Citation

  • Maurya, Rahul Kumar & Li, Dongxia & Singh, Anant Pratap & Singh, Vineet Kumar, 2024. "Numerical algorithm for a general fractional diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 223(C), pages 405-432.
    • Handle: RePEc:eee:matcom:v:223:y:2024:i:c:p:405-432
    DOI: 10.1016/j.matcom.2024.04.018
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    1. Robert J. Elliott & John Van Der Hoek, 2003. "A General Fractional White Noise Theory And Applications To Finance," Mathematical Finance, Wiley Blackwell, vol. 13(2), pages 301-330, April.
    2. Fan, Enyu & Li, Changpin & Stynes, Martin, 2023. "Discretised general fractional derivative," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 501-534.
    3. Tarasov, Vasily E. & Zaslavsky, George M., 2007. "Fractional dynamics of systems with long-range space interaction and temporal memory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 383(2), pages 291-308.
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    1. Savadkoohi, Fateme Rezaei & Rabbani, Mohsen & Allahviranloo, Tofigh & Malkhalifeh, Mohsen Rostamy, 2024. "A fractional multi-wavelet basis in Banach space and solving fractional delay differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 186(C).

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