Fractional calculus and some intermediate physical processes. (English) Zbl 1049.35002
Fractional calculus is used to formulate the models which represents the two intermediate physical processes, namely convection-diffusion and transport-diffusion processes. Two mixed problems involving the fractional, derivatives \(D^v\) and Weyl fractional derivative \(W^v\), \(v\in (1,2]\) are considered. The existence and uniqueness of the solutions of these two problems is established.
Reviewer: Chaman Lal Koul (Jaipur)
MSC:
| 35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |
| 26A33 | Fractional derivatives and integrals |
References:
| [1] | Caputo, M., Linear model of dissipation whose \(Q\) is almost frequency independent-II, Geophys. J. R. Astr. Soc., 13, 529-539 (1967) |
| [2] | El-Sayed, A. M.A., Fractional order evolution equations, J. Frac. Calculus, 7, 89-100 (1995) · Zbl 0839.34069 |
| [3] | El-Sayed, A. M.A., Finite Weyl fractional calculus and abstract fractional differential equations, J. Frac. Calculus, 9, May, 59-68 (1996) · Zbl 0862.34010 |
| [4] | El-Sayed, A. M.A., Fractional-order diffusion-wave equation, Int. J. Theo. Phys., 35, 2, 311-322 (1996) · Zbl 0846.35001 |
| [5] | El-Sayed, A. M.A., On some basic properties and applications of the fractional calculus, Pure Math. Appl., 8, 2-4, 259-274 (1997) · Zbl 0910.26003 |
| [6] | El-Sayed, A. M.A., Nonlinear functional differential equations of arbitrary orders, Nonlinear Anal.: Theory Method Appl., 33, 2, 181-186 (1998) · Zbl 0934.34055 |
| [7] | El-Sayed, A. M.A., Fractional-order evolutionary integral equations, Appl. Math. Comput., 98, 2-3, 139-146 (1999) · Zbl 0933.34063 |
| [8] | Gelfand, I. M.; Shilov, G. E., Generalized functions, Moscow, 1 (1958) · Zbl 0091.11103 |
| [9] | R. Gorenflo, S. Vessella, Abel integral equations, Analysis and applications, Lecture Notes in Mathematics # 1461, Springer-Verlag, Berlin, 1991; R. Gorenflo, S. Vessella, Abel integral equations, Analysis and applications, Lecture Notes in Mathematics # 1461, Springer-Verlag, Berlin, 1991 · Zbl 0717.45002 |
| [10] | R. Gorenflo, R. Rutman, On ultraslow and on intermediate Processes, Fachbereich Mathematik Reprint No. A-28/94, 1994; R. Gorenflo, R. Rutman, On ultraslow and on intermediate Processes, Fachbereich Mathematik Reprint No. A-28/94, 1994 |
| [11] | F. Mainardi, Fractional relaxation and fractional diffusion equations, Mathematical aspects, in: Proceedings of the 14th IMACS World Congress, vol. 1, 1994, pp. 329-332; F. Mainardi, Fractional relaxation and fractional diffusion equations, Mathematical aspects, in: Proceedings of the 14th IMACS World Congress, vol. 1, 1994, pp. 329-332 |
| [12] | Mainardi, F., Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Soliton Fract., 7, 6, 1-17 (1996) |
| [13] | Miller, K. S.; Ross, B., An introduction to the fractional calculus and fractional differential equations (1993), John Wiley & Sons Inc: John Wiley & Sons Inc New York · Zbl 0789.26002 |
| [14] | Oden, J. T., Applied functional analysis (1979), Prentice-Hall Inc: Prentice-Hall Inc Englewwood Cliffs, NJ · Zbl 0468.46007 |
| [15] | I. Podlubny, A.M.A. El-Sayed, On two definitions of fractional calculus, Slovak Academy Of Sciences Institute Of Experimental Physics UEF-03-96 ISBN 80-7099-252-2, 1996; I. Podlubny, A.M.A. El-Sayed, On two definitions of fractional calculus, Slovak Academy Of Sciences Institute Of Experimental Physics UEF-03-96 ISBN 80-7099-252-2, 1996 |
| [16] | Podlubny, I., Fractional Differential Equation (1999), Acadamic Press: Acadamic Press San Diego, New York, London · Zbl 0893.65051 |
| [17] | S.G. Samko, A.A. Kilbas, O.I. Marichev, Integrals and derivatives of the fractional order and some of their applications, Nauka i Tehnika, Minsh (1987) (in Russian); English translation 1993; S.G. Samko, A.A. Kilbas, O.I. Marichev, Integrals and derivatives of the fractional order and some of their applications, Nauka i Tehnika, Minsh (1987) (in Russian); English translation 1993 · Zbl 0617.26004 |
| [18] | Schneider, W. R.; Wyss, W., Fractional diffusion and wave equations, J. Math. Phys., 30, 134 (1989) · Zbl 0692.45004 |
| [19] | Tanabe, H., Equations of Evolution (1979), Pitman: Pitman London · Zbl 0417.35003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
