×

General uniqueness and monotone iterative technique for fractional differential equations. (English) Zbl 1161.34031

The authors present a very interesting work on how can be explored the solution of certain class of hyperbolic evolution equation, with limited smoothness, using multi-scale approaching techniques. The construction considered by the authors in this paper implies a full-wave description. The study suggests a novel computational algorithm. Also the authors present some applications of such numerical approach.

MSC:

34G20 Nonlinear differential equations in abstract spaces
26A33 Fractional derivatives and integrals
34A35 Ordinary differential equations of infinite order
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Caputo, M., Linear models of dissipation whose Q is almost independent, II, Geophys. J. Roy. Astron., 13, 529-539 (1967)
[2] Glöckle, W. G.; Nonnenmacher, T. F., A fractional calculus approach to self similar protein dynamics, Biophys. J., 68, 46-53 (1995)
[3] Diethelm, K.; Ford, N. J., Analysis of fractional differential equations, J. Math. Anal. Appl., 265, 229-248 (2002) · Zbl 1014.34003
[4] Diethelm, K.; Ford, N. J., Multi-order fractional differential equations and their numerical solution, Appl. Math. Comput., 154, 621-640 (2004) · Zbl 1060.65070
[5] Diethelm, K.; Freed, A. D., On the solution of nonlinear fractional differential equations used in the modeling of viscoplasticity, (Keil, F.; Mackens, W.; Vob, H.; Werther, J., Scientific Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties (1999), Springer: Springer Heidelberg), 217-224
[6] Kiryakova, V., Generalized fractional calculus and applications, (Pitman Res. Notes Math. Ser., vol. 301 (1994), Longman-Wiley: Longman-Wiley New York) · Zbl 1189.33034
[7] Ladde, G. S.; Lakshmiakntham, V.; Vatsala, A. S., Monotone Iterative Techniques for Nonlinear Differential Equations (1985), Pitman Advanced Publishing Program: Pitman Advanced Publishing Program Boston · Zbl 0658.35003
[8] Lakshmikantham, V.; Leela, S., Differential and Integral Inequalities, vol. I (1969), Academic Press: Academic Press New York · Zbl 0177.12403
[9] V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. TMA (2007) (in press); V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. TMA (2007) (in press) · Zbl 1159.34006
[10] V. Lakshmikantham, A.S. Vatsala, Theory of fractional differential inequalities and applications, Commun. Appl. Anal. (2007) (in press); V. Lakshmikantham, A.S. Vatsala, Theory of fractional differential inequalities and applications, Commun. Appl. Anal. (2007) (in press) · Zbl 1159.34006
[11] Metzler, R.; Schick, W.; Kilian, H. G.; Nonnenmacher, T. F., Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys., 103, 7180-7186 (1995)
[12] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[13] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives, Theory and Applications (1993), Gordon and Breach: Gordon and Breach Yverdon · Zbl 0818.26003
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.