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On spectral polar fractional Laplacian

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  • Ansari, Alireza
  • Derakhshan, Mohammad Hossein

Abstract

In this paper, we introduce the fractional Sturm–Liouville operators and present the polar fractional Laplacian as a particular case of these operators. We also consider the distributed order space–time fractional diffusion equation involving the polar fractional Laplacian and propose three approaches for studying this problem on the circle and ring domains. First, we apply the integral transforms as the analytical methods to get the solution in terms of the Laplace-type integrals using the Titchmarsh theorem. Second, we use the finite difference method for discretization of the space variable and employ the matrix transfer technique to obtain a system of the distributed order fractional equations. For this system, we modify the Putzer’s algorithm and get the solution in terms of the eigenvalues of discretization matrix. Third, we employ the backward Euler numerical method for discretization of the time variable and find the corresponding error bound. Moreover, we compare and verify the approximate solutions with the exact solutions in analytical form.

Suggested Citation

  • Ansari, Alireza & Derakhshan, Mohammad Hossein, 2023. "On spectral polar fractional Laplacian," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 636-663.
    • Handle: RePEc:eee:matcom:v:206:y:2023:i:c:p:636-663
    DOI: 10.1016/j.matcom.2022.12.008
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    References listed on IDEAS

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    1. Boyadjiev, Lyubomir & Luchko, Yuri, 2017. "Mellin integral transform approach to analyze the multidimensional diffusion-wave equations," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 127-134.
    2. Yuri Luchko, 2017. "On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equation," Mathematics, MDPI, vol. 5(4), pages 1-16, December.
    3. Azamat Dzarakhohov & Yuri Luchko & Elina Shishkina, 2021. "Special Functions as Solutions to the Euler–Poisson–Darboux Equation with a Fractional Power of the Bessel Operator," Mathematics, MDPI, vol. 9(13), pages 1-18, June.
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    Cited by:

    1. Derakhshan, Mohammad Hossein & Rezaei, Hamid & Marasi, Hamid Reza, 2023. "An efficient numerical method for the distributed order time-fractional diffusion equation with error analysis and stability," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 214(C), pages 315-333.

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