General Fractional Derivative Research Articles

General fractional-order anomalous diffusion with non-singular power-law kernel

In this paper, we investigate general fractional derivatives with a non-singular power-law kernel. The anomalous diffusion models with non-singular power-law kernel are discussed in detail. The results are efficient for modelling the anomalous behaviors within the frameworks of the Riemann-Liouville and Liouville-Caputo general fractional derivatives.

General fractional calculus in non-singular power-law kernel applied to model anomalous diffusion phenomena in heat transfer problems

In this paper we address the general fractional calculus of Liouville-Weyl and Liouville-Caputo general fractional derivative types with non-singular power-law kernel for the first time. The Fourier transforms and the anomalous diffusions are discussed in detail. The formulations are adopted to describe complex phenomena of the heat transfer problems.

Right General Fractional Monotone Approximation

Here are introduced left general fractional derivatives Caputo style with respect to a base absolutely continuous strictly increasing function g. We give various examples of such fractional derivatives for different g.

Generalized fractional derivatives and their applications to mechanical systems

New fractional derivatives, termed henceforth generalized fractional derivatives (GFDs), are introduced. Their definition is based on the concept that fractional derivatives (FDs) interpolate the integer- order derivatives. This idea generates infinite classes of FDs. The new FDs provide, beside the fractional order, any number of free parameters to better calibrate the response of a physical system or procedure. Their usefulness and consequences are subject of further investigation. Like the Caputo FD, the GFDs allow the application of initial conditions having direct physical significance. A numerical method is also developed for the solution of differential equations involving GFDs. Mechanical systems including fractional oscillators, viscoelastic plane bodies and plates described by such equations are analyzed.

Numerical Algorithms for the Forward and Backward Fractional Feynman–Kac Equations

The Feynman-Kac equations are a type of partial differential equations describing the distribution of functionals of diffusive motion. The probability density function (PDF) of Brownian functionals satisfies the Feynman-Kac formula, being a Schr\"{o}dinger equation in imaginary time. The functionals of no-Brownian motion, or anomalous diffusion, follow the fractional Feynman-Kac equation [J. Stat. Phys. 141, 1071-1092, 2010], where the fractional substantial derivative is involved. Based on recently developed discretized schemes for fractional substantial derivatives [arXiv:1310.3086], this paper focuses on providing algorithms for numerically solving the forward and backward fractional Feynman-Kac equations; since the fractional substantial derivative is non-local time-space coupled operator, new challenges are introduced comparing with the general fractional derivative. Two ways (finite difference and finite element) of discretizing the space derivative are considered. For the backward fractional Feynman-Kac equation, the numerical stability and convergence of the algorithms with first order accuracy are theoretically discussed; and the optimal estimates are obtained. For all the provided schemes, including the first order and high order ones, of both forward and backward Feynman-Kac equations, extensive numerical experiments are performed to show their effectiveness.

A class of multivalent functions defined by generalized Ruscheweyh derivatives involving a general fractional derivative operator

The main aim of the present paper is to obtain a new class of multivalent functions which is defined by making use of the generalized Ruscheweyh derivatives involving a general fractional derivative operator.We study the region of starlikeness and convexity of the class . Also we apply the Fractional calculus techniques to obtain the applications of the class . Finally, the familiar concept of δ-neighborhoods of p-valent functions for above mentioned class are employed.

Numerical solutions and analysis of diffusion for new generalized fractional advection-diffusion equations

Abstract In this paper we study a class of new Generalized Fractional Advection-Diffusion Equations (GFADEs) with a new Generalized Fractional Derivative (GFD) proposed last year. The new GFD is defined in the Caputo sense using a weight function and a scale function. The GFADE is discussed in a bounded domain, and numerical solutions for two examples consisting of a linear and a nonlinear GFADE are obtained using an implicit finite difference approach. The stability of the numerical scheme is investigated, and the order of convergence is estimated numerically. Numerical results illustrate that the finite difference scheme is simple and effective for solving the GFADEs. We investigate the influence of weight and scale functions on the diffusion of GFADEs. Linear and nonlinear stretching and contracting functions are considered. It is found that an increasing weight function increases the rate of diffusion, and a scale function can stretch or contract the diffusion on the time domain.

Numerical solutions of fractional advection–diffusion equations with a kind of new generalized fractional derivative

In the current paper, the numerical solutions for a class of fractional advection–diffusion equations with a kind of new generalized time-fractional derivative proposed last year are discussed in a bounded domain. The fractional derivative is defined in the Caputo type. The numerical solutions are obtained by using the finite difference method. The stability of numerical scheme is also investigated. Numerical examples are solved with different fractional orders and step sizes, which illustrate that the numerical scheme is stable, simple and effective for solving the generalized advection–diffusion equations. The order of convergence of the numerical scheme is evaluated numerically, and the first-order convergence rate has been observed.

Reflection symmetric formulation of generalized fractional variational calculus

Abstract We define generalized fractional derivatives (GFDs) symmetric and anti-symmetric w.r.t. the reflection symmetry in a finite interval. Arbitrary functions are split into parts with well defined reflection symmetry properties in a hierarchy of intervals [0, b/2m], m ∈ ℕ0. For these parts — [J]-projections of function, we derive the representation formulas for generalized fractional operators (GFOs) and examine integration properties. It appears that GFOs can be reduced to operators determined in subintervals [0, b/2m]. The results are applied in the derivation of Euler-Lagrange equations for action dependent on Riemann-Liouville type GFDs. We show that for Lagrangian being a sum (finite or not) of monomials, the obtained equations of motion can be localized in arbitrary short subinterval [0, b/2m].

On the existence of solutions to the fractional derivative equations d(alpha)u/dt(alpha)+Au = f, of relevance to diffusion in complex systems

Fractional derivative equations account for relaxation and diffusion processes in a large variety of condensed matter systems. For instance, diffusion of position probability density displayed by a random walker in complex systems – such as glassy materials – is often modeled by fractional derivative partial differential equations. This paper deals with the existence of solutions to the general fractional derivative equation dαu/dtα+Au = f for 0 < α < 1, with A a self-adjoint operator. The results are proved using the von Neumann–Dixmier theorem.

Generalized Multiparameters Fractional Variational Calculus

This paper builds upon our recent paper on generalized fractional variational calculus (FVC). Here, we briefly review some of the fractional derivatives (FDs) that we considered in the past to develop FVC. We first introduce new one parameter generalized fractional derivatives (GFDs) which depend on two functions, and show that many of the one-parameter FDs considered in the past are special cases of the proposed GFDs. We develop several parts of FVC in terms of one parameter GFDs. We point out how many other parts could be developed using the properties of the one-parameter GFDs. Subsequently, we introduce two new two- and three-parameter GFDs. We introduce some of their properties, and discuss how they can be used to develop FVC. In addition, we indicate how these formulations could be used in various fields, and how the generalizations presented here can be further extended.

An Efficient Method for Solving

This paper presents a new method for solving fractional partial differential equations (FPDE) which is called the polynomial approximation method based on the polynomial approximation uN(x,t) and on its general fractional derivative formula.By modifying the general fractional derivative formula of uN(x,t) and with the aid of the linear FPDE, another new formula can be found for the approximation uN(x,t). This is the basic idea of the proposed method. Furthermore, the mathematical proof of the convergence and stability of this method have been studied. Some numerical examples show that the proposed method exhibits a satisfactory results.

General Fractional Opial Type Inequalities

A large variety of Lp(p 0) form very general Opial type inequalities arepresented engaging different order generalized fractional derivatives of a function. These arebased on a generalization of Taylor’s formula for generalized fractional derivatives. In the finalresults of this work, a monotonicity property of the involved function/highest-order generalizedfractional derivative is used.

Fractional Derivatives and Expansion Formulas Involving H-Functions of One and More Variables

Certain general fractional derivatives formulas involving theH-function of one and more variables are established that generalize the corresponding results considered by Srivastava and Goyal. This leads us to an extension of the expansion formula for the Lauricella functionFD(r)given by Srivastava and Goyal. Various other analogous formulas are also established. These results are significantly different from Burchnall and Chaundy-type expansions.