Fractional differential equations solved by using Mellin transform

In this paper, the solution of the multi-order differential equations, by using Mellin Transform, is proposed. It is shown that the problem related to the shift of the real part of the argument of the transformed function, arising when the Mellin integral operates on the fractional derivatives, may be overcame. Then, the solution may be found for any fractional differential equation involving multi-order fractional derivatives (or integrals). The solution is found in the Mellin domain, by solving a linear set of algebraic equations, whose inverse transform gives the solution of the fractional differential equation at hands.Comment: 19 pages, 2 figure

Fractional Spectral Moments for Digital Simulation of Multivariate Wind Velocity Fields

In this paper, a method for the digital simulation of wind velocity fields by Fractional Spectral Moment function is proposed. It is shown that by constructing a digital filter whose coefficients are the fractional spectral moments, it is possible to simulate samples of the target process as superposition of Riesz fractional derivatives of a Gaussian white noise processes. The key of this simulation technique is the generalized Taylor expansion proposed by the authors. The method is extended to multivariate processes and practical issues on the implementation of the method are reported.Comment: 12 pages, 2 figure

The fractal model of non-local elasticity with long-range interactions

The mechanically-based model of non-local elasticity with long-range interactions is framed, in this study, in a fractal mechanics context. Non-local interactions are modelled introducing long-range central body forces between non-adjacent volume elements of the elastic continuum. Such long-range interactions are modelled as proportional to the product of interacting volumes, to the relative displacements of the centroids and to a distance-decaying function that is monotonically-decreasing with the distance. The choice of the decaying function is a key aspect of the model and it has been proved that any continuous function, strictly positive, is thermodynamically consistent and it leads to a material that satisfy the Drucker stability criterion [2]. Such a mathematical model of non-local elasticity has an interesting mechanical counterpart that is described by a point-spring network with multiple springs with distance-decaying stiffness. As the functional class of the distance-decaying function is modelled as a power-law function of the distance of interacting particles, then, in the 1D case, the governing operators are Marchaud-type fractional derivatives as proved by the authors in previous studies [1]. In this study we aim to show that, as we assume that the stiffness associated to long-range interactions is modelled as a self-similar transformation of the Euclidean distance with anomalous and real scaling exponent, the mechanical model of the non-local elasticity is a self-similar fractal object. In more detail, assuming a non-integer power-law decay of the long-range forces between adjacent volumes of an ideal next nearest (NN) model, the scaling law of the stiffness of the long-range bonds is readily obtained. The Hausdorff-Besitckovich (HB) fractal dimension provides the appropriate bounds of the decay coefficient necessary to maintain the self-similarity of the obtained fractal set. The NN model, however leads to mathematically inconsistent governing operator for general class of continuous displacement function and it is proved that in this case only one integer value of the long-range force decay is admissible leading to classical second-order differential operators. A different scenario is involved as we introduce, on mechanical grounds, the long-range interaction concept so that as we refine observation scale, the interactions between particles is still involving the presence of all the new, non-adjacent particles so that the original NN lattice is turned into a more refined and realistic next to the nearest next (NNN) lattice model. Such a model is equivalent to the mechanical model of the long-range interactions introduced by the authors to describe non-local elasticity. The model is constituted of self-similar copies of elastic chains and henceforth it may be considered as a mechanical fractal as we assume an unbounded domain. This fractal set is not coalescing with usual fractals since it retains all the informations of previous observation scales and henceforth it has been dubbed as multiscale fractal. In this context the HB dimension of the mechanical fractal may be obtained as a function of the decaying exponent of the long-range interactions and it may be proved that the governing equation of 1 the problem are Marchaud fractional-type differential operator as already postulated by the authors in a previous study [1]. Some conclusions about the use of fractional operators in the context of multiscale approach to non-local mechanics may be also withdrawn from previous considerations

The multiscale stochastic model of fractional hereditary materials (FHM)

In a recent paper the authors proposed a mechanical model corresponding, exactly, to fractional hereditary materials (FHM). Fractional derivation index β; ∈ [0, 1/2] corresponds to a mechanical model composed by a column of massless newtonian fluid resting on a bed of independent linear springs. Fractional derivation index β ∈ [1/2, 1], corresponds, instead, to a mechanical model constituted by massless, shear-type elastic column resting on a bed of linear independent dashpots. The real-order of derivation is related to the exponent of the power-law decay of mechanical characteristics. In this paper the authors aim to introduce a multiscale fractance description of FHM in presence of stochastic fluctuations of model parameters. In this setting the random multiscale fractance may be used to capture the fluctuations of material parameters observed in experimental tests by means of proper analytical evaluation of the model statistic

Exact Mechanical Models of Fractional Hereditary Materials (FHM)

Fractional Viscoelasticity is referred to materials, whose constitutive law involves fractional derivatives of order β R such that 0 β 1. In this paper, two mechanical models with stress-strain relation exactly restituting fractional operators, respectively, in ranges 0 β 1 / 2 and 1 / 2 β 1 are presented. It is shown that, in the former case, the mechanical model is described by an ideal indefinite massless viscous fluid resting on a bed of independent springs (Winkler model), while, in the latter case it is a shear-type indefinite cantilever resting on a bed of independent viscous dashpots. The law of variation of all mechanical characteristics is of power-law type, strictly related to the order of the fractional derivative. Because the critical value 1/2 separates two different behaviors with different mechanical models, we propose to distinguish such different behavior as elasto-viscous case with 0< β <1 / 2 and visco-elastic case for 1 / 2 <β <1. The motivations for this different definitions as well as the comparison with other existing mechanical models available in the literature are presented in the pape

Probabilistic characterization of nonlinear systems under α-stable white noise via complex fractional moments

The probability density function of the response of a nonlinear system under external α-stable Lévy white noise is ruled by the so called Fractional Fokker-Planck equation. In such equation the diffusive term is the Riesz fractional derivative of the probability density function of the response. The paper deals with the solution of such equation by using the complex fractional moments. The analysis is performed in terms of probability density for a linear and a non-linear half oscillator forced by Lévy white noise with different stability indexes α. Numerical results are reported for a wide range of non-linearity of the mechanical system and stability index of the Lévy white noise

Einstein-Smoluchowsky equation handled by complex fractional moments

In this paper the response of a non linear half oscillator driven by a-stable white noise in terms of probability density function (PDF) is investigated. The evolution of the PDF of such a system is ruled by the so called Einstein-Smoluchowsky equation involving, in the diffusive term, the Riesz fractional derivative. The solution is obtained by the use of complex fractional moments of the PDF, calculated with the aid of Mellin transform operator. It is shown that solution can be found for various values of stability index a and for any nonlinear function f (X; t)