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Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives

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Abstract

On a series of examples from the field of viscoelasticity we demonstrate that it is possible to attribute physical meaning to initial conditions expressed in terms of Riemann–Liouville fractional derivatives, and that it is possible to obtain initial values for such initial conditions by appropriate measurements or observations.

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References

  • Beris AN, Edwards BJ (1993) On the admissibility criteria for linear viscoelastic kernels. Rheol Acta 32:505–510

    Article  CAS  Google Scholar 

  • Caputo M, Mainardi F (1971) Linear models of dissipation in anelastic solids. Riv Nuovo Cimento (Ser II) 1:161–198

    Article  Google Scholar 

  • Diethelm K, Ford NJ, Freed AD, Luchko Yu (2005) Algorithms for the fractional calculus: a selection of numerical methods. Comput Methods Appl Mech Engrg 194:743–773

    Article  Google Scholar 

  • Glöckle WG, Nonnenmacher TF (1991) Fractional integral operators and Fox functions in the theory of viscoelasticity. Macromolecules 24:6426–6434

    Article  Google Scholar 

  • Heymans N (1996) Hierarchical models for viscoelasticity: dynamic behaviour in the linear range. Rheol Acta 35:508–519

    Article  CAS  Google Scholar 

  • Heymans N, Bauwens J-C (1994) Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheol Acta 33:210–219

    Article  CAS  Google Scholar 

  • Heymans N, Kitagawa M (2004) Modelling “unusual” behaviour after strain reversal with hierarchical fractional models. Rheol Acta 43:383–389

    Article  CAS  Google Scholar 

  • Koeller RC (1984) Applications of fractional calculus to the theory of viscoelasticity. J Appl Mech 51:299–307

    Article  Google Scholar 

  • Podlubny I (1999) Fractional differential equations. Academic Press, San Diego

    Google Scholar 

  • Podlubny I (2002) Geometric and physical interpretation of fractional integration and fractional differentiation. Fract Calc Appl Anal 5:367–386

    Google Scholar 

  • Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives: theory and applications. Gordon and Breach, Amsterdam

    Google Scholar 

  • Schiessel H, Blumen A (1993) Hierarchical analogues to fractional relaxation equations. J Phys A: Math Gen 26:5057–5069

    Article  Google Scholar 

  • Schiessel H, Blumen A (1995) Mesoscopic pictures of the sol–gel transition: ladder models and fractal networks. Macromolecules 28:4013–4019

    Article  CAS  Google Scholar 

  • Westerlund S (2002) Dead matter has memory! Causal consulting. Kalmar, Sweden

    Google Scholar 

Download references

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Correspondence to Igor Podlubny.

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Dedicated to professor Hari M. Srivastava on the occasion of his 65th birthday

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Heymans, N., Podlubny, I. Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol Acta 45, 765–771 (2006). https://doi.org/10.1007/s00397-005-0043-5

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  • DOI: https://doi.org/10.1007/s00397-005-0043-5

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