1,734,404 research outputs found
Prabhakar-like fractional viscoelasticity
No full textThe aim of this paper is to present a linear viscoelastic model based on Prabhakar fractional operators. In particular, we propose a modification of the classical fractional Maxwell model, in which we replace the Caputo derivative with the Prabhakar one. Furthermore, we also discuss how to recover a formal equivalence between the new model and the known classical models of linear viscoelasticity by means of a suitable choice of the parameters in the Prabhakar derivative. Moreover, we also underline an interesting connection between the theory of Prabhakar fractional integrals and the recently introduced Caputo-Fabrizio differential operator
Applications of Hilfer-Prabhakar operator to option pricing financial model
No full textIn this paper, we focus on option pricing models based on time-fractional diffusion with generalized Hilfer-Prabhakar derivative. It is demonstrated how the option is priced for fractional cases of European vanilla option pricing models. Series representations of the pricing formulas and the risk-neutral parameter under the time-fractional diffusion are also derived.Mathematical Physic
Prabhakar Lévy processes
No full textWe introduce here a generalization of the Mittag-Leffler L ́evy process (with parameter ), obtained by extending
its L ́evy measure through the Prabhakar function (which is a Mittag-Leffler with the additional parameters and ). We prove that this so-called Prabhakar process, in the special case , can be represented as
an -stable process subordinated by an independent generalized gamma subordinator; thus it can be considered
as an extension of the geometric stable process, to which it reduces for . On the other hand,
for , it coincides with the generalized gamma process itself. Therefore, by suitably specifying the
three parameters, the Prabhakar process turns out to represent an interpolation among various well-known
and widely applied stochastic models
General Conditions for Existence of Maximal Elements via the Uncovered Set
No full textThis paper disentangles the topological assumptions of classical results (e.g., Walker (1977)) on existence of maximal elements from rationality conditions. It is known from the social choice literature that under the standard topological conditions-with no other restrictions on preferences-there is an element such that the upper section of strict preference at that element is minimal in terms of set inclusion, i.e., the uncovered set is non-empty. Adding a condition that weakens known acyclicity and convexity assumptions, each such uncovered alternative is indeed maximal. A corollary is a result that weakens the semi-convexity condition of Yannelis and Prabhakar (1983).
Prabhakar-type linear differential equations with variable coefficients
No full textLinear differential equations with variable coefficients and Prabhakar-type operators featuring Mittag-Leffler kernels are solved. In each case, the unique solution is constructed explicitly as a convergent infinite series involving compositions of Prabhakar fractional integrals. We also extend these results to Prabhakar operators with respect to functions. As an important illustrative example, we consider the case of constant coefficients, and give the solutions in a more closed form by using multivariate Mittag-Leffler functions.Linear differential equations with variable coefficients and Prabhakar-type operators featuring Mittag-Leffler kernels are solved. In each case, the unique solution is constructed explicitly as a convergent infinite series involving compositions of Prabhakar fractional integrals. We also extend these results to Prabhakar operators with respect to functions. As an important illustrative example, we consider the case of constant coefficients, and give the solutions in a more closed form by using multivariate Mittag-Leffler functions.A
Principles of Generalized Prabhakar-Hilfer Fractional Calculus and Applications
No full textHere we introduce the generalized Prabhakar fractional calculus and we also combine it with the generalized Hilfer calculus. We prove that the generalized left and right side Prabhakar fractional integrals preserve continuity and we find tight upper bounds for them. We present several left and right side generalized Prabhakar fractional inequalities of Hardy, Opial and Hilbert-Pachpatte types
Commerce Secretary Ron Brown and NIST Director Arati Prabhakar at NIST Gaithersburg in 1996
No full textCommerce Secretary Ron Brown and NIST Director Arati Prabhakar during Brown's visit to the NIST Gaithersburg, MD campus in 1996
Stability of fractional-order systems with Prabhakar derivatives
No full textFractional derivatives of Prabhakar type are capturing an increasing interest since their ability to describe anomalous relaxation phenomena (in dielectrics and other fields) showing a simultaneous nonlocal and nonlinear behaviour. In this paper we study the asymptotic stability of systems of differential equations with the Prabhakar derivative, providing an exact characterization of the corresponding stability region. Asymptotic expansions (for small and large arguments) of the solution of linear differential equations of Prabhakar type and a numerical method for nonlinear systems are derived. Numerical experiments are hence presented to validate theoretical findings
Vectorial Advanced Hilfer-Prabhakar-Hardy Fractional Inequalities
No full textWe present a variety of univariate and multivariate left and right side Hardy type fractional inequalities, many of them under convexity, and other also of Lp type, p≥ 1, in the setting of generalized Hilfer and Prabhakar fractional Calculi
Fractional Dynamics with Depreciation and Obsolescence: Equations with Prabhakar Fractional Derivatives
No full textIn economics, depreciation functions (operator kernels) are certain decreasing functions, which are assumed to be equal to unity at zero. Usually, an exponential function is used as a depreciation function. However, exponential functions in operator kernels do not allow simultaneous consideration of memory effects and depreciation effects. In this paper, it is proposed to consider depreciation of a non-exponential type, and simultaneously take into account memory effects by using the Prabhakar fractional derivatives and integrals. Integro-differential operators with the Prabhakar (generalized Mittag-Leffler) function in the kernels are considered. The important distinguishing features of the Prabhakar function in operator kernels, which allow us to take into account non-exponential depreciation and fading memory in economics, are described. In this paper, equations with the following operators are considered: (a) the Prabhakar fractional integral, which contains the Prabhakar function as the kernels; (b) the Prabhakar fractional derivative of Riemann–Liouville type proposed by Kilbas, Saigo, and Saxena in 2004, which is left inverse for the Prabhakar fractional integral; and (c) the Prabhakar operator of Caputo type proposed by D’Ovidio and Polito, which is also called the regularized Prabhakar fractional derivative. The solutions of fractional differential equations with the Prabhakar operator and its special cases are suggested. The asymptotic behavior of these solutions is discussed
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