Foundations of Fractional Dynamics: A Short Account

Hilfer, R.

doi:10.1142/9789814340595_0009

Cited by 7 publications

(9 citation statements)

References 78 publications

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“…For the example of broadband dielectric spectroscopy in glasses, generalized relaxation functions and susceptibilities based on Equation (70) have already been successfully compared to experiments [23,[29][30][31][32]. Theoretical, mathematical and experimental studies are encouraged to further explore the consequences of the generalized concept.…”

Section: Discussionmentioning

confidence: 99%

“…The limit gives rise to a family of one-parameter semigroups T h α (with family index α and parameter h) of ultra-long-time evolution operators [22,23].…”

Section: Equation (46) Impliesmentioning

confidence: 99%

“…In general, the integration of infinitesimal system changes leads to convolutions instead of just translations along the orbit [22,23]. Of course, translations are a special case of convolutions, to which they reduce in the case when the parameter α approaches unity.…”

Section: Equation (46) Impliesmentioning

confidence: 99%

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Time Automorphisms on C*-Algebras

Hilfer

2015

Mathematics

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Applications of fractional time derivatives in physics and engineering require the existence of nontranslational time automorphisms on the appropriate algebra of observables. The existence of time automorphisms on commutative and noncommutative C * -algebras for interacting many-body systems is investigated in this article. A mathematical framework is given to discuss local stationarity in time and the global existence of fractional and nonfractional time automorphisms. The results challenge the concept of time flow as a translation along the orbits and support a more general concept of time flow as a convolution along orbits. Implications for the distinction of reversible and irreversible dynamics are discussed. The generalized concept of time as a convolution reduces to the traditional concept of time translation in a special limit.

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Section: Discussionmentioning

confidence: 99%

“…The limit gives rise to a family of one-parameter semigroups T h α (with family index α and parameter h) of ultra-long-time evolution operators [22,23].…”

Section: Equation (46) Impliesmentioning

confidence: 99%

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Time Automorphisms on C*-Algebras

Hilfer

2015

Mathematics

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“…Hilfer showed that this transition from ordinary time derivative to fractional time derivative indeed arises in physical problems [34,37,38,40].…”

Section: Preliminariesmentioning

confidence: 96%

Hilfer fractional advection–diffusion equations with power-law initial condition; a numerical study using variational iteration method

Ali

Malik

2014

Computers & Mathematics with Applications

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We propose a Hilfer advection-diffusion equation of order 0 < α < 1 and type 0 ≤ β ≤ 1, and find the power series solution by using variational iteration method. Power series solutions are expressed in a form that is easy to implement numerically and in some particular cases, solutions are expressed in terms of Mittag-Leffler function. Absolute convergence of power series solutions is proved and the sensitivity of the solutions is discussed with respect to changes in the values of different parameters. For power law initial conditions it is shown that the Hilfer advectiondiffusion PDE gives the same solutions as the Caputo and Riemann-Liouville advection-diffusion PDE. To leading order, the fractional solution compared to the non-fractional solution increases rapidly with α for α > 0.7 at a given time t; but for α < 0.7 this factor is weakly sensitive to α. We also show that the truncation errors, arising when using the partial sum as approximate solutions, decay exponentially fast with the number of terms n used. We find that for α < 0.7 the number of terms needed is weakly sensitive to the accuracy level and to the fractional order, n ≈ 20; but for α > 0.7 the required number of terms increases rapidly with the accuracy level and also with the fractional order α.

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“…It should be noted that in the general case fractional differential equations are difficult to solve exactly even with special functions. A fractional embedding of an ordinary differential equation is not unique and different embeddings correspond to different microscopic physical models [8]. It also introduces sensitivity of the boundary conditions, which requires regularization procedures leading to Caputotype of derivatives.…”

Section: Introductionmentioning

confidence: 99%

Characterization of strongly non-linear and singular functions by scale space analysis

Prodanov¹

2016

Chaos, Solitons & Fractals

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Abstract. A central notion of physics is the rate of change. While mathematically the concept of derivative represents an idealization of the linear growth, power law types of non-linearities even in noiseless physical signals cause derivative divergence. As a way to characterize change of strongly nonlinear signals, this work introduces the concepts of scale space embedding and scale-space velocity operators. Parallels with the scale relativity theory and fractional calculus are discussed. The approach is exemplified by an application to De Rham's function. It is demonstrated how scale space embedding presents a simple way of characterizing the growth of functions defined by means of iterative function systems.

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Foundations of Fractional Dynamics: A Short Account

Abstract: Applications of fractional dynamics have received a steadily increasing amount of attention during the past decade. Its foundations have found less interest. This chapter briefly reviews the physical foundations of fractional dynamics.

Cited by 7 publications

References 78 publications

Time Automorphisms on C*-Algebras

Time Automorphisms on C*-Algebras

Hilfer fractional advection–diffusion equations with power-law initial condition; a numerical study using variational iteration method

Characterization of strongly non-linear and singular functions by scale space analysis

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