Stevan Pilipović

Variational problems with fractional derivatives: Euler-Lagrange equations

We generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized. Also, for the standard case when these two bounds coincide, we derive a new form of Euler–Lagrange equations. We use approximations for fractional derivatives in the Lagrangian and obtain the Euler–Lagrange equations which approximate the initial Euler–Lagrange equations in a weak sense.

Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes

Preface ix Part 1. Mathematical Preliminaries, Definitions and Properties of Fractional Integrals and Derivatives 1 Chapter 1. Mathematical Preliminaries 3 Chapter 2. Basic Definitions and Properties of Fractional Integrals and Derivatives 17 Part 2. Mechanical Systems 49 Chapter 3. Restrictions Following from the Thermodynamics for Fractional Derivative Models of a Viscoelastic Body 51 Chapter 4. Vibrations with Fractional Dissipation 83 Chapter 5. Lateral Vibrations and Stability of Viscoelastic Rods 123 Chapter 6. Fractional Diffusion-Wave Equations 185 Chapter 7. Fractional Heat Conduction Equations 257 Bibliography 289 Index 311

A diffusion wave equation with two fractional derivatives of different order

We analyse a diffusion wave equation with two fractional derivatives of different order on bounded and unbounded spatial domains. Thus, our model represents a generalized telegraph equation. Solutions to signalling and Cauchy problems in terms of a series and integral representation are given. Classical wave and heat conduction equations are obtained as limiting cases.

Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients

We investigate microlocal properties of partial differential operators with generalized functions as coefficients. The main result is an extension of a corresponding (microlocalized) distribution theoretic result on operators with smooth hypoelliptic symbols. Methodological novelties and technical refinements appear embedded into classical strategies of proof in order to cope with most delicate interferences by non-smooth lower order terms. We include simplified conditions which are applicable in special cases of interest.

Time distributed-order diffusion-wave equation. I. Volterra-type equation

A single-order time-fractional diffusion-wave equation is generalized by introducing a time distributed-order fractional derivative and forcing term, while a Laplacian is replaced by a general linear multi-dimensional spatial differential operator. The obtained equation is (in the case of the Laplacian) called a time distributed-order diffusion-wave equation. We analyse a Cauchy problem for such an equation by means of the theory of an abstract Volterra equation. The weight distribution, occurring in the distributed-order fractional derivative, is specified as the sum of the Dirac distributions and the existence and uniqueness of solutions to the Cauchy problem, and the corresponding Volterra-type equation were proven for a general linear spatial differential operator, as well as in the special case when the operator is Laplacian.

On a fractional distributed-order oscillator

We consider a viscoelastic rod with a concentrated mass at its end. The mass is moving along the straight line that coincides with the rod axis. The mass is connected by a linear spring and a known active force is acting on it. We assume that the rod is light and described by fractional dissipation. The dynamics of such a system constitutes a problem of a fractional oscillator. In this paper, we shall study some properties of the solutions for the distributed-order fractional derivative viscoelastic rod.

Classes of degenerate elliptic operators in Gelfand-Shilov spaces

We propose a novel approach for the study of the uniform regularity and the decay at infinity for Shubin type pseudo-differential operators which are globally hypoelliptic but not necessarily globally and even locally elliptic. The basic idea is to use the special role of the Hermite functions for the characterization of inductive and projective Gelfand-Shilov spaces. In this way we transform the problem to infinite dimensional linear systems on S Banach spaces of sequences by using Fourier series expansion with respect to the Hermite functions. As applications of our general results we obtain new theorems for global hypoellipticity for classes of degenerate operators in tensorized generalizations of Shubin spaces and in inductive and projective Gelfand-Shilov spaces.

Distributed-order fractional wave equation on a finite domain. Stress relaxation in a rod

We study waves in a viscoelastic rod of finite length. Viscoelastic material is described by a constitutive equation of fractional distributed-order type with the special choice of weight functions. Prescribing boundary conditions on displacement, we obtain displacement and stress in a stress relaxation test. We use the Laplace transformation method in the time domain as a tool for solving system of differential and integro-differential equations, that describe the motion of the rod.

Fully fractional anisotropic diffusion for image denoising

This paper introduces a novel Fully Fractional Anisotropic Diffusion Equation for noise removal which contains spatial as well as time fractional derivatives. It is a generalization of a method proposed by Cuesta which interpolates between the heat and the wave equation by the use of time fractional derivatives, and the method proposed by Bai and Feng, which interpolates between the second and the fourth order anisotropic diffusion equation by the use of spatial fractional derivatives. This equation has the benefits of both of these methods. For the construction of a numerical scheme, the proposed partial differential equation (PDE) has been treated as a spatially discretized Fractional Ordinary Differential Equation (FODE) model, and then the Fractional Linear Multistep Method (FLMM) combined with the discrete Fourier transform (DFT) is used. We prove that the analytical solution to the proposed FODE has certain regularity properties which are sufficient to apply a convergent and stable fractional numerical procedure. Experimental results confirm that our model manages to preserve edges, especially highly oscillatory regions, more efficiently than the baseline parabolic diffusion models.

Microlocal analysis of Colombeau’s generalized functions: Propagation of singularities

AbstractWe introduce the notion of pointwise regularity (