A note on a modified fractional Maxwell model

Meccanica, 2014

We revisit the Kilbas and Saigo functions of the Mittag-Leffler type of a real variable t, with two independent real order-parameters. These functions, subjected to the requirement to be completely monotone for t [ 0, can provide suitable models for the responses and for the corresponding spectral distributions in anomalous (non-Debye) relaxation processes, found e.g. in dielectrics. Our analysis includes as particular cases the classical models referred to as Cole-Cole (the one-parameter Mittag-Leffler function) and to as Kohlrausch (the stretched exponential function). After some remarks on the Kilbas and Saigo functions, we discuss a class of fractional differential equations of order a 2 ð0; 1 with a characteristic coefficient varying in time according to a power law of exponent b, whose solutions will be presented in terms of these functions. We show 2D plots of the solutions and, for a few of them, the corresponding spectral distributions, keeping fixed one of the two orderparameters. The numerical results confirm the complete monotonicity of the solutions via the nonnegativity of the spectral distributions, provided that the parameters satisfy the additional condition 0\a þ b 1, assumed by us.

Fractional Calculus and Applied Analysis, 2014

From the point of view of the general theory of the hyper-Bessel operators, we consider a particular operator that is suitable to generalize the standard process of relaxation by taking into account both memory effects of power law type and time variability of the characteristic coefficient. According to our analysis, the solutions are still expressed in terms of functions of the Mittag-Leffler type as in case of fractional relaxation with constant coefficient but exhibit a further stretching in the time argument due to the presence of Erdélyi-Kober fractional integrals in our operator. We present solutions, both singular and regular in the time origin, that are locally integrable and completely monotone functions in order to be consistent with the physical phenomena described by non-negative relaxation spectral distributions.

Rheologica Acta, 2020

In view of the increasing attention to the time responses of complex fluids described by power-laws in association with the need to capture inertia effects that manifest in high-frequency microrheology, we compute the five basic time-response functions of in-series or in-parallel connections of two elementary fractional derivative elements known as the Scott-Blair (springpot) element. The order of fractional differentiation in each Scott-Blair element is allowed to exceed unity reaching values up to 2 and at this limit-case the Scott-Blair element becomes an inerter-a mechanical analogue of the electric capacitor that its output force is proportional only to the relative acceleration of its end-nodes. With this generalization, inertia effects may be captured beyond the traditional viscoelastic behavior. In addition to the relaxation moduli and the creep compliances, we compute closed-form expressions of the memory functions, impulse fluidities (impulse response functions) and impulse strain-rate response functions of the generalized fractional derivative Maxwell fluid, the generalized fractional derivative Kelvin-Voigt element and their special cases that have been implemented in the literature. Central to these calculations is the fractional derivative of the Dirac delta function which makes possible the extraction of singularities embedded in the fractional derivatives of the two-parameter Mittag-Leffler function that emerges invariably in the time-response functions of fractional derivative rheological models.

Fractal and Fractional

Fractional calculus has performed an important role in the fields of mathematics, physics, electronics, mechanics, and engineering in recent years. The modeling methods involving fractional operators have been continuously generalized and enhanced, especially during the last few decades. Many operations in physics and engineering can be defined accurately by using systems of differential equations containing different types of fractional derivatives.

Chaos, Solitons & Fractals, 2017

We present a new approach based on linear integro-differential operators with logarithmic kernel related to the Hadamard fractional calculus in order to generalize, by a parameter ν ∈ (0, 1], the logarithmic creep law known in rheology as Lomnitz law (obtained for ν = 1). We derive the constitutive stress-strain relation of this generalized model in a form that couples memory effects and time-varying viscosity. Then, based on the hereditary theory of linear viscoelasticity, we also derive the corresponding relaxation function by solving numerically a Volterra integral equation of the second kind. So doing we provide a full characterization of the new model both in creep and in relaxation representation, where the slow varying functions of logarithmic type play a fundamental role as required in processes of ultra slow kinetics. Chaos, Solitons and Fractals (2017): Special Issue on Future Directions in Fractional Calculus.

Zeitschrift für Angewandte Mathematik und Physik, 2019

Inspired by the article "Anomalous relaxation model based on the fractional derivative with a Prabhakar-like kernel" (Z. Angew. Math. Phys. (2019) 70:42) whose authors Zhao and Sun studied the integro-differential equation with the kernel given by the Prabhakar function e −γ α,β (t, λ), we provide the solution to this equation which is complementary to that obtained up to now. Our solution is valid for effective relaxation times whose admissible range extends the limits given in Zhao and Sun (Z Angew Math Phys 70:42, 2019, Theorem 3.1) to all positive values. For special choices of parameters entering the equation itself and/or characterizing the kernel, the solution comprises to known phenomenological relaxation patterns, e.g., to the Cole-Cole model (if γ = 1, β = 1 − α) or to the standard Debye relaxation.

Progress in Fractional Differentiation and Applications, 2017

Starting from the Cattaneo constitutive relation with exponential kernel applied to mass diffusion the derivation of a new form the diffusion equation with a relaxation term expressed through the Caputo-Fabrizio time-fractional operator (derivative) has been developed. The developed equation reduces to the fractional Dodson equation for large relaxation times corresponding to low fractional order of the Caputo-Fabrizio derivative. The approach separates large time effects resulting in the classical Dodson equation with exponentially decaying in time diffusivity and the short time relaxation process modeled by Caputo-Fabrizio time fractional derivative. The solution developed allows seeing a new physical background of the Caputo-Fabrizio time-fractional operator (derivative) and to demonstrate a new interpretation of the Dodson equation incorporating fading memory effects. Moreover a new model with two memories corresponding to large and short time relation effects has been conceived. Defining the diffusion process parameters then the fractional order of the Caputo-Fabrizio time fractional derivative can be determined in a straightforward manner as a function of the Deborah number calculated as a ratio of the relaxation time to the characteristic diffusion time of the process.

Fractal and Fractional

Modelling, simulation, and applications of Fractional Calculus have recently become increasingly popular subjects, with impressive growth concerning applications. The founding and limited ideas on fractional derivatives have achieved an incredibly valuable status. The variety of applications in mathematics, physics, engineering, economics, biology, and medicine, have opened new, challenging fields of research. For instance, in soil mechanics, a suitable definition of the fractional operator has shed some light on viscoelasticity, explaining memory effects on materials. Needless to say, these applications require the development of practical mathematical tools in order to extract quantitative information from models, newly reformulated in terms of fractional differential equations. Even confining ourselves to the field of ordinary differential equations, the well-known Bagley-Torvik model showed that fractional derivatives may actually arise naturally within certain physical models, and are not merely fanciful mathematical generalizations. This Special Issue focuses on the most recent advances in fractional calculus, applied to dynamic problems, linear and nonlinear fractional ordinary and partial differential equations, integral fractional differential equations, and stochastic integral problems arising in all fields of science, engineering, and other applied fields. In this issue, we have collected several significant papers devoted to applications of fractional methods with a focus on dynamical aspects. The applications range from theoretical mathematical-numerical aspects [1,2] to bio-medical subjects [3-7]. Applications to complex materials are investigated in [8], aiming at proposing a generalized definition of fractional operators. Special diffusion models are studied in [9-11].

Current Topics in Biophysics, 2015

It is the goal of this paper to present general strategy for using fractional operators to model the magnetic relaxation in complex environments revealing time and spacial disorder. Such systems have anomalous temporal and spacial response (non-local interactions and long memory) compared to systems without disorder. The systems having no memory can be modeled by linear differential equations with constant coefficients (exponential relaxation); the differential equations governing the systems with memory are known as Fractional Order Differential Equations (FODE). The relaxation of the spin system is best described phenomenologically by so-called Bloch's equations, which detail the rate of change of the magnetization M of the spin system. The Ordinary Order Bloch's Equations (OOBE) are a set of macroscopic differential equations of the first order describing the magnetization behavior under influence of static, varying magnetic fields and relaxation. It is assumed that spins...

Mathematics, 2018

In this note, we show how an initial value problem for a relaxation process governed by a differential equation of a non-integer order with a constant coefficient may be equivalent to that of a differential equation of the first order with a varying coefficient. This equivalence is shown for the simple fractional relaxation equation that points out the relevance of the Mittag-Leffler function in fractional calculus. This simple argument may lead to the equivalence of more general processes governed by evolution equations of fractional order with constant coefficients to processes governed by differential equations of integer order but with varying coefficients. Our main motivation is to solicit the researchers to extend this approach to other areas of applied science in order to have a deeper knowledge of certain phenomena, both deterministic and stochastic ones, investigated nowadays with the techniques of the fractional calculus.

International Journal of Plasticity, 2003

Following the modelling of Zener, we establish a connection between the fractional Fokker-Planck equation and the anomalous relaxation dynamics of a class of viscoelastic materials which exhibit scale-free memory. On the basis of fractional relaxation, generalisations of the classical rheological model analogues are introduced, and applications to stress-strain relaxation in filled and unfilled polymeric materials are discussed. A possible generalisation of Reiner's Deborah number is proposed for systems which exhibit a diverging characteristic relaxation time. #

Journal of Engineering Mathematics

This paper deals with the computational aspect of the investigation of the relaxation properties of viscoelastic materials. The constitutive fractional Zener model is considered under continuous deformation with a jump at the origin. The analytical solution of this equation is obtained by the Laplace transform method. It is derived in a closed form in the terms of the Mittag-Leffler function. The method of numerical evaluation of the Mittag-Leffler function for arbitrary negative arguments which corresponds to physically meaningful interpretation is demonstrated. A numerical example is given to illustrate the effectiveness of this result.

Journal of Rheology

This paper examines the oscillatory behavior of complex viscoelastic systems with power law like relaxation behavior. Specifically, we use the fractional Maxwell model, consisting of a spring and fractional dashpot in series, which produces a power-law creep behavior and a relaxation law following the Mittag-Leffler function. The fractional dashpot is characterized by a parameter b, continuously moving from the pure viscous behavior when b ¼ 1 to the purely elastic response when b ¼ 0. In this work, we study the general response function and focus on the oscillatory behavior of a fractional Maxwell system in four regimes: Stress impulse, strain impulse, step stress, and driven oscillations. The solutions are presented in a format analogous to the classical oscillator, showing how the fractional nature of relaxation changes the long-time equilibrium behavior and the short-time transient solutions. We specifically test the critical damping conditions in the fractional regime, since these have a particular relevance in biomechanics. V

arXiv preprint nlin/0602029, 2006

We consider the fractional generalizations of equation that defines the medium mass. We prove that the fractional integrals can be used to describe the media with noninteger mass dimensions. Using fractional integrals, we derive the fractional generalization of the Chapman-Kolmogorov equation ͑Smolukhovski equation͒. In this paper fractional Fokker-Planck equation for fractal media is derived from the fractional Chapman-Kolmogorov equation. Using the Fourier transform, we get the Fokker-Planck-Zaslavsky equations that have fractional coordinate derivatives. The Fokker-Planck equation for the fractal media is an equation with fractional derivatives in the dual space.

Mathematical Problems in Engineering, 2015

In many fields, theoretical and applied researches have strengthened the belief that fractional calculus is not only a branch of the mathematical analysis, but also a useful and powerful tool for engineers. Namely, fractional calculus allows both a better modeling of a wide class of systems with anomalous dynamic behavior and a better understanding of the facets of both physical phenomena and artificial processes. Hence the mathematical models derived from differential equations with noninteger/fractional order derivatives or integrals are becoming a fundamental research issue for scientists and engineers. In particular, fractional models are successful when describing power-law long-term memory or hereditary properties. They are also successful when anomalous diffusion, transport phenomena, and waves propagation in complex media require nonlocal operators, or when fractal-like properties are evident in some processes. These potentialities and the related benefits have attracted many experts and practitioners of different fields to reconsider mathematical models and engineering methods both for linear and for nonlinear systems.