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Mittag–Leffler stability for a fractional Euler–Bernoulli problem

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  • Tatar, Nasser-eddine

Abstract

We investigate the stability of an Euler–Bernoulli type problem of fractional order. By adding a fractional term of lower-order, namely of order half of the order of the leading fractional derivative, the problem will generalize the well-known telegraph equation. It is shown that this term is capable of stabilizing the system to rest in a Mittag–Leffler manner. Moreover, we consider a much weaker dissipative term consisting of a memory term in the form of a convolution known as viscoelastic term. It is proved that we can still obtain Mittag–Leffler stability under a smallness condition on the involved kernels. The results rely heavily on some established properties of fractional derivatives and some newly introduced functionals.

Suggested Citation

  • Tatar, Nasser-eddine, 2021. "Mittag–Leffler stability for a fractional Euler–Bernoulli problem," Chaos, Solitons & Fractals, Elsevier, vol. 149(C).
    • Handle: RePEc:eee:chsofr:v:149:y:2021:i:c:s0960077921004318
    DOI: 10.1016/j.chaos.2021.111077
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    References listed on IDEAS

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    1. Tatar, Nasser-eddine, 2008. "Nonexistence results for a fractional problem arising in thermal diffusion in fractal media," Chaos, Solitons & Fractals, Elsevier, vol. 36(5), pages 1205-1214.
    2. Zhong Bo Fang & Liru Qiu, 2013. "Global Existence and Uniform Energy Decay Rates for the Semilinear Parabolic Equation with a Memory Term and Mixed Boundary Condition," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-12, November.
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