Abstract
From the point of view of the general theory of the hyper-Bessel operators, we consider a particular operator that is suitable to generalize the standard process of relaxation by taking into account both memory effects of power law type and time variability of the characteristic coefficient. According to our analysis, the solutions are still expressed in terms of functions of the Mittag-Leffler type as in case of fractional relaxation with constant coefficient but exhibit a further stretching in the time argument due to the presence of Erdélyi-Kober fractional integrals in our operator. We present solutions, both singular and regular in the time origin, that are locally integrable and completely monotone functions in order to be consistent with the physical phenomena described by non-negative relaxation spectral distributions.
Similar content being viewed by others
References
I. Ali, V. Kiryakova, S. Kalla, Solutions of fractional multi-order integral and differential equations using a Poisson-type transform. J. Math. Anal. and Appl. 269, No 1 (2002), 172–199; DOI:10.1016/S0022-247X(02)00012-4.
L. Beghin, Fractional relaxation equations and Brownian crossing probabilities of a random boundary. Advances in Applied Probability 44 (2012), 479–505.
E. Capelas de Oliveira, F. Mainardi, J. Vaz Jr., Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics. J. Eur. Phys.-Special Topics 193, No 1 (2011), 161–171. Revised version in E-print arXiv:1106.1761.
I. Dimovski, On an operational calculus for a differential operator. Compt. Rendues de l’Acad. Bulg. des Sci. 21, No 6 (1966), 513–516.
I. Dimovski, V. Kiryakova, Transmutations, convolutions and fractional powers of Bessel-type operators via Meijer’s G-function. In: Proc. Intern. Conf. ”Complex Analysis and Applications, Varna’ 1983”, Bulg. Acad. Sci., Sofia, 45–66.
V.A. Ditkin, A.P. Prudnikov, On the theory of operational calculus, generated by the Bessel equation. Zhournal Vych. Mat. i Mat. Fiziki 3, No 2 (1963), 223–238 (In Russian).
G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation. Springer Verlag, Berlin (1974).
R. Gorenflo and F. Mainardi, Fractional calculus: integral and differential equations of fractional order. In: A. Carpinteri and F. Mainardi (Editors), Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, Wien (1997), 223–276. [E-print: arXiv:0805.3823].
F. Jarad, R. Abdeljawad and D. Baleanu, Caputo-type modification of the Hadamard fractional derivatives. Advances in Difference Equations 2012 (2012), 1–8.
V. Kiryakova, An application of Meijer’s G-function to Bessel type operators. In: Proc. Internat. Conf. “Constr. Function Theory, Varna 1981”, Bulg. Acad. Sci., Sofia, 457–462.
V. Kiryakova, Generalized Fractional Calculus and Applications. Longman — J. Wiley, Harlow — N. York (1994).
V. Kiryakova, Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus. J. Comput. Appl. Mathematics 118 (2000), 241–259; DOI:10.1016/S0377-0427(00)00292-2.
V. Kiryakova, Yu. Luchko, Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators. Central European J. of Physics 11, No 10 (2013), 1314–1336; DOI:10.2478/s11534-013-0217-1.
A.A. Kilbas and A.A. Titioura, Nonlinear differential equations with Marchaud-Hadamard-type fractional derivative in the weighted space of summable functions. Mathematical Modelling and Analysis 12, No 3 (2007), 343–356.
W. Lamb, Fractional Powers of Operators on Fréchet Spaces with Applications. Ph.D. Thesis, University of Strathclyde, Glasgow, 1980.
F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press — World Sci., London — Singapore (2010).
A.C. McBride, Fractional Calculus and Integral Transforms of Generalised Functions. Pitman, London (1979).
A.C. McBride, A theory of fractional integration for generalized functions. SIAM J. on Mathematical Analysis 6, No 3 (1975), 583–599.
A.C. McBride, Fractional powers of a class of ordinary differential operators. Proc. London Mathematical Society (3) 45 (1982), 519–546.
K.S. Miller and S.G. Samko, A note on the complete monotonicity of the generalized Mittag-Leffler function. Real Anal. Exchange 23 (1997), 753–755.
K.S. Miller and S.G. Samko, Completely monotonic functions. Integral Transforms and Special Functions 12 (2001), 389–402.
H. Pollard, The completely monotonic character of the Mittag-Leffler function E α(−x). Bull. Amer. Math. Soc. 54 (1948), 1115–1116.
S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Gordon and Breach Science, Yverdon (1993).
W.R. Schneider, Completely monotone generalized Mittag-Leffler functions. Expositiones Mathematicae 14 (1996), 3–16.
E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals. Oxford University Press, Oxford (1937).
D.V. Widder, The Laplace Transform. Princeton University Press, Princeton (1946).
S. Yakubovich, Yu. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions. Ser.: Mathematics and Its Applications 287, Kluwer Acad. Publ., Dordrecht-Boston-London (1994).
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Garra, R., Giusti, A., Mainardi, F. et al. Fractional relaxation with time-varying coefficient. fcaa 17, 424–439 (2014). https://doi.org/10.2478/s13540-014-0178-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.2478/s13540-014-0178-0