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Fractional Relaxation and Fractional Oscillation Models Involving Erdélyi-Kober Integrals

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Abstract

We consider fractional relaxation and fractional oscillation equations involving Erdélyi-Kober integrals. In terms of the Riemann-Liouville integrals, the equations we analyze can be understood as equations with time-varying coefficients. Replacing the Riemann-Liouville integrals with Erdélyi-Kober-type integrals in certain fractional oscillation models, we obtain some more general integro-differential equations. The corresponding Cauchy-type problems can be solved numerically, and, in some cases analytically, in terms of the Saigo-Kilbas Mittag-Leffler functions. The numerical results are obtained by a treatment similar to that developed by K. Diethelm and N.J. Ford to solve the Bagley-Torvik equation. Novel results about the numerical approach to the fractional damped oscillator equation with time-varying coefficients are also presented.

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Authors and Affiliations

  1. Department of Mathematics and Physics, Roma Tre University, 1, Largo S. Leonardo Murialdo, 00146, Rome, Italy

    Moreno Concezzi & Renato Spigler

  2. Department of Statistical Sciences, “Sapienza” University of Rome, 5, P. le Aldo Moro, 00185, Rome, Italy

    Roberto Garra

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  1. Moreno Concezzi
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  2. Roberto Garra
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  3. Renato Spigler
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Correspondence to Moreno Concezzi.

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Concezzi, M., Garra, R. & Spigler, R. Fractional Relaxation and Fractional Oscillation Models Involving Erdélyi-Kober Integrals. FCAA 18, 1212–1231 (2015). https://doi.org/10.1515/fca-2015-0070

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