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Eigenvalue problems for fractional ordinary differential equations

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  • Duan, Jun-Sheng
  • Wang, Zhong
  • Liu, Yu-Lu
  • Qiu, Xiang

Abstract

The eigenvalue problems are considered for the fractional ordinary differential equations with different classes of boundary conditions including the Dirichlet, Neumann, Robin boundary conditions and the periodic boundary condition. The eigenvalues and eigenfunctions are characterized in terms of the Mittag–Leffler functions. The eigenvalues of several specified boundary value problems are calculated by using MATLAB subroutine for the Mittag–Leffler functions. When the order is taken as the value 2, our results degenerate to the classical ones of the second-ordered differential equations. When the order α satisfies 1<α<2 the eigenvalues can be finitely many.

Suggested Citation

  • Duan, Jun-Sheng & Wang, Zhong & Liu, Yu-Lu & Qiu, Xiang, 2013. "Eigenvalue problems for fractional ordinary differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 46(C), pages 46-53.
    • Handle: RePEc:eee:chsofr:v:46:y:2013:i:c:p:46-53
    DOI: 10.1016/j.chaos.2012.11.004
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    References listed on IDEAS

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    1. Giona, Massimiliano & Eduardo Roman, H., 1992. "Fractional diffusion equation for transport phenomena in random media," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 185(1), pages 87-97.
    2. Metzler, Ralf & Glöckle, Walter G. & Nonnenmacher, Theo F., 1994. "Fractional model equation for anomalous diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 211(1), pages 13-24.
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    Cited by:

    1. Ahmad, Bashir & Ntouyas, Sotiris K. & Alsaedi, Ahmed, 2016. "On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions," Chaos, Solitons & Fractals, Elsevier, vol. 83(C), pages 234-241.
    2. John W. Hanneken & B. N. Narahari Achar & David M. Vaught, 2013. "An Alpha‐Beta Phase Diagram Representation of the Zeros and Properties of the Mittag‐Leffler Function," Advances in Mathematical Physics, John Wiley & Sons, vol. 2013(1).
    3. He, Ying & Zuo, Qian, 2021. "Jacobi-Davidson method for the second order fractional eigenvalue problems," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    4. Gupta, Sandipan & Ranta, Shivani, 2022. "Legendre wavelet based numerical approach for solving a fractional eigenvalue problem," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    5. Jun-Sheng Duan, 2013. "The Periodic Solution of Fractional Oscillation Equation with Periodic Input," Advances in Mathematical Physics, John Wiley & Sons, vol. 2013(1).
    6. Aljoudi, Shorog & Ahmad, Bashir & Nieto, Juan J. & Alsaedi, Ahmed, 2016. "A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 39-46.
    7. Li, Jing & Qi, Jiangang, 2015. "Eigenvalue problems for fractional differential equations with right and left fractional derivatives," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 1-10.
    8. Al-Mdallal, Qasem M., 2018. "On fractional-Legendre spectral Galerkin method for fractional Sturm–Liouville problems," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 261-267.

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