Teodor M. Atanackovic

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.

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.

Time distributed-order diffusion-wave equation. II. Applications of Laplace and Fourier transformations

A Cauchy problem for a time distributed-order multi-dimensional diffusion-wave equation containing a forcing term is reinterpreted in the space of tempered distributions, and a distributional diffusion-wave equation is obtained. The distributional equation is solved in the general case of weight function (or distribution). Solutions are given in terms of solution kernels (Greens functions), which are studied separately for two cases. The first case is when the order of the fractional derivative is in the interval [0, 1], while, in the second case, the order of the fractional derivative is in the interval [0, 2]. Solutions of fractional diffusion-wave and fractional telegraph equations are obtained as special cases. Numerical experiments are also performed. An analogue of the maximum principle is also presented.

A generalized model for the uniaxial isothermal deformation of a viscoelastic body

SummaryUsing the notion of a fractional derivative we formulate a new model for a uniaxial deformation of a visco-elastic body. The basic assumption is that all derivatives σ(γ) with respect to time of the stress depend (with specified weighting factor) on all derivatives ε(γ) with respect to time of the strain (multiplied with another weighting factor), for 0≤γ≤1. In this respect our model is a generalization of the Zener model, i.e., it is a Zener fractional model with infinitely many terms. The relation between stress and strain is given in explicit form. For two specific choices of parameters the behavior of the model under suddenly applied stress (creep) and suddenly applied strain (stress relaxation) are examined.

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.

A model for memory alloys in plane strain

Abstract A model for shape memory is presented which is capable of simulating the plane-strain-response of a polycrystalline body under biaxial loading. Numerical solutions to particular load histories are given.

On a system of differential equations with fractional derivatives arising in rod theory

We study a system of equations with fractional derivatives, that arises in the analysis of the lateral motion of an elastic column fixed at one end and loaded by a concentrated follower force at the other end. We assume that the column is positioned on a viscoelastic foundation described by a constitutive equation of fractional derivative type. The stability boundary is determined. It is shown that as in the case of an elastic (Winkler) type of foundation the stability boundary remains the same as for the column without a foundation! Thus, with the solution analysed here, the column exhibits the so-called Hermann–Smith paradox.

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.