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The problem of the initial conditions and their physical meaning in linear differential equations of fractional order. (English) Zbl 1037.34004
This paper considers the practically important question of how to set up the initial conditions when solving a fractional differential equation involving fractional derivatives of Riemann-Liouville or Grunwald-Post type. Some authors have chosen to use practically simpler Caputo-type fractional derivative because here the inital conditions would be specified in terms of integer order derivatives, which have some clear physical meaning and are relatively simply measured. However, this paper considers what might be the physical meaning of initial conditions (or indeed of the evaluation of a derivative) of fractional order. The discussion leads to a new formulation of fractional derivative which emphasises the multiplicity of definitions of fractional derivative in the existing literature.

34A25Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
Full Text: DOI
[1] Oldham, K. B.; Spanier, J.: The fractional calculus. Theory and applications of differentiation and integration to arbitrary order. (1974) · Zbl 0292.26011
[2] I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering, Vol. 198, Academic Press, San Diego, CA, 1999 · Zbl 0924.34008
[3] Kransov, M. L.: Ordinary differential equations. (1983)
[4] Ghizzetti, A.; Marchetti, L.; Ossicini, A.: Lezioni di complementi di matematica. (1970)
[5] A. Ghizzetti, Lezioni di Analisi Matematica, seconda edizione, Vol. II, Veschi, Rome, 1970
[6] C. Giannantoni, Equazioni differenziali a coefficienti costanti. Nuovo Metodo Generale di Soluzione. ENEA----RT/ERG/94/23, 1994
[7] C. Giannantoni, Linear differential equations with variable coefficients. Fundamental Theorem of the Solving Kernel. ENEA----RT/ERG/95/07, 1995
[8] M.L. Chiara, G. Toraldo Di Francia, Le Teorie Fisiche, Un’ Analisi Formale, Boringhieri, 1981
[9] M.T. Brown, Workshop on Emergy Analysis, Siena, 20--25 September, 1993
[10] H.T. Odum, Ecological and general systems. An Introduction to Systems Ecology, Re. Edition, University Press Colorado, CO, 1994
[11] C. Giannantoni, Toward a mathematical formulation of the maximum Em-power principle. First Emergy Analysis Conference, Gainesville, USA, Florida, September 2--4, 1999
[12] C. Giannantoni, Advanced mathematical tools for energy analysis of complex systems. International Workshop on Advances in Energy Studies, Porto Venere, Italy, May 23--27, 2000
[13] C. Giannantoni, Multiple bifurcation as a solution of a linear differential equation of fractional order. International Congress on Qualitative Theory of Differential Equations, Siena, Italy, September 18--20, 2000
[14] C. Giannantoni, Mathematical formulation of the maximum Em-power principle. Paper to be presented at at the Second Biennial Emergy Evaluation and Research Conference, Gainesville, USA, FL, September 20--22, 2001
[15] C. Giannantoni, Mathematics for quality. Living and Non-living Systems. Paper to be presented at the Second Biennial Emergy Evaluation and Research Conference, Gainesville, USA, FL, September 20--22, 2001