Dušan Zorica

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.

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.

Existence and calculation of the solution to the time distributed order diffusion equation

The aim of this paper is to prove the existence of the solution to the Cauchy problem for the time distributed order diffusion equation as well as to calculate it. The existence is proved in this paper by reducing the Cauchy problem to an abstract Volterra equation in the case where the weight distribution in the distributed order derivative is a finite sum of Dirac distributions. Calculation of the solution is done by the use of Fourier and Laplace transformations in the case where the weight distribution (or function) is not specified. The solution is expressed in terms of heat potential kernel. The solutions for several special cases of the weight distribution, including the case of a finite sum of Dirac distributions, are presented as well.

Thermodynamical Restrictions and Wave Propagation for a Class of Fractional Order Viscoelastic Rods

We discuss thermodynamical restrictions for a linear constitutive equation containing fractional derivatives of stress and strain of different orders. Such an equation generalizes several known models. The restrictions on coefficients are derived by using entropy inequality for isothermal processes. In addition, we study waves in a rod of finite length modelled by a linear fractional constitutive equation. In particular, we examine stress relaxation and creep and compare results with the quasistatic analysis.

Waves in viscoelastic media described by a linear fractional model

Recently, the classical wave equation has been generalized for the case of viscoelastic media described by the fractional Zener model (cf. [S. Konjik, Lj. Oparnica, and D. Zorica, Waves in fractional Zener type viscoelastic media, J. Math. Anal. Appl. (2009), doi:10.1016/j.jma.2009.10.043]). In this article, we use a more general fractional model for a viscoelastic body to describe the wave equation for viscoelastic infinite media, and prove existence and uniqueness of distributional solutions to the corresponding generalized Cauchy problem.

An expansion formula for fractional derivatives of variable order

In this work we extend our previous results and derive an expansion formula for fractional derivatives of variable order. The formula is used to determine fractional derivatives of variable order of two elementary functions. Also we propose a constitutive equation describing a solidifying material and determine the corresponding stress relaxation function.

Complex order fractional derivatives in viscoelasticity

We introduce complex order fractional derivatives in models that describe viscoelastic materials. This cannot be carried out unrestrictedly, and therefore we derive, for the first time, real valued compatibility constraints, as well as physical constraints that lead to acceptable models. As a result, we introduce a new form of complex order fractional derivative. Also, we consider a fractional differential equation with complex derivatives, and study its solvability. Results obtained for stress relaxation and creep are illustrated by several numerical examples.