Fractional differential equations as alternative models to nonlinear differential equations. (English) Zbl 1120.34323

Bonilla, B.; Rivero, M.; Rodríguez-Germá, L.; Trujillo, J. J.

Appl. Math. Comput. 187, No. 1, 79-88 (2007).

Summary: The main objective of this paper is to demonstrate the possibility of using fractional differential equations to simulate the dynamics of anomalous processes whose analytical representations are continuous but strongly not differentiable, like Weierstrass-type functions. This allows for the possibility of modeling phenomena which traditional differential modeling cannot accomplish. To this end we shall see how some functions of this kind have a fractional derivative at every point in a real interval, and are therefore solutions to fractional differential equations.

MSC:

34C60

Qualitative investigation and simulation of ordinary differential equation models

Keywords:

fractional differential equations; fractional models; Weierstrass type functions

References:

[1]	Bisquert, J., Interpretation of a fractional diffusion equation with nonconserved probability density in terms of experimental systems with trapping or recombination, Phys. Rev. E, 72, 011109 (2005)
[2]	Caputo, M.; Mainardi, F., Linear models of dissipation in anelastic solids, Riv. Nuovo Cimento (Ser. II), 1, 161-198 (1971)
[3]	Gorenflo, R.; Luchko, Yu.; Loutchko, J., Computation of the Mittag-Leffler function \(E_{a,b}(z)\) and its derivative, Fract. Calc. Appl. Anal., 5, 491-518 (2002) · Zbl 1027.33016
[4]	Gorenflo, R.; Mainardi, F., Fractional calculus: integral and differential equations of fractional order, (Carpinteri, A.; Mainardi, F., Fractal and Fractional Calculus in Continuum Mechanics. Fractal and Fractional Calculus in Continuum Mechanics, CISM Courses and Lectures, vol. 378 (1997), Springer: Springer New York), 277-290 · Zbl 1438.26010
[5]	Hu, J.; Zähle, M., Jump processes and nonlinear fractional heat equations, Math. Nachr., 279, 150-163 (2006) · Zbl 1111.28005
[6]	Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Fractional differential equations: an emergent field in applied and mathematical sciences, (Samko, S.; Lebre, A.; dos Santos, A. F., Factorization. Factorization, Singular Operators and Related Problems (2003), Kluwer Acad. Pub.: Kluwer Acad. Pub. Dordrecht), 151-173 · Zbl 1053.35038
[7]	Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204 (2006), Elsevier: Elsevier Holland · Zbl 1092.45003
[8]	Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamic approach, Phys. Rep., 339, 1-77 (2000) · Zbl 0984.82032
[9]	Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), John Wiley and Sons: John Wiley and Sons New York · Zbl 0789.26002
[10]	Podlubny, I., Fractional Differential Equations. Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198 (1999), Academic Press: Academic Press London · Zbl 0918.34010
[11]	Rocco, A.; West, B., Fractional Calculus and the Evolution of Fractal Phenomena, Physica A, 265, 535-546 (1999)
[12]	Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers Switzerland · Zbl 0818.26003
[13]	Zähle, M.; Ziezold, H., Fractional derivatives of Weierstrass-type functions, J. Comput. Appl. Math., 76, 265-275 (1996) · Zbl 0870.65011