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Cauchy and source problems for an advection-diffusion equation with Atangana–Baleanu derivative on the real line

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  • Avcı, Derya
  • Yetim, Aylin

Abstract

In this paper, a linear advection-diffusion equation involving Atangana–Baleanu derivative described with the Mittag–Leffler kernel is considered on the real line. Different kinds of diffusive transports in the nature obey the exponential/generalized exponential and Mittag–Leffler functions rather than the power law. By this reality, the current study is devoted to investigate the fundamental solutions of the Cauchy and source problems. For this purpose, Laplace and exponential Fourier transforms are applied. The results are achieved in terms of one and two-parameter Mittag–Leffler functions. The results show that the Atangana–Baleanu derivative is an effective alternative to Caputo derivative to model the diffusion with advection processes because the continuous structure of Mittag–Leffler kernel removes the computational complexities. Thus, it is rather practical to achieve analytical solutions.

Suggested Citation

  • Avcı, Derya & Yetim, Aylin, 2019. "Cauchy and source problems for an advection-diffusion equation with Atangana–Baleanu derivative on the real line," Chaos, Solitons & Fractals, Elsevier, vol. 118(C), pages 361-365.
    • Handle: RePEc:eee:chsofr:v:118:y:2019:i:c:p:361-365
    DOI: 10.1016/j.chaos.2018.11.035
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    References listed on IDEAS

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    1. Dumitru Baleanu & Amin Jajarmi & Mojtaba Hajipour, 2017. "A New Formulation of the Fractional Optimal Control Problems Involving Mittag–Leffler Nonsingular Kernel," Journal of Optimization Theory and Applications, Springer, vol. 175(3), pages 718-737, December.
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    5. Solís-Pérez, J.E. & Gómez-Aguilar, J.F. & Atangana, A., 2018. "Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 175-185.
    6. Alkahtani, B.S.T. & Atangana, A., 2016. "Controlling the wave movement on the surface of shallow water with the Caputo–Fabrizio derivative with fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 89(C), pages 539-546.
    7. Atangana, Abdon & Gómez-Aguilar, J.F., 2018. "Fractional derivatives with no-index law property: Application to chaos and statistics," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 516-535.
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