Bogoljub Stanković

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

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.

Dynamics of a Viscoelastic Rod of Fractional Derivative Type

We study dynamics of a viscoelastic rod with fractional derivative type of dissipation under time dependent loading. First the moment curvature relation for the rod from the generalized Zener model of a standard linear solid is derived. Then, we show that the dynamics of the problem is governed by a single linear differential equation with fractional derivative. The existence and uniqueness of solutions for this equation is studied in detail. The form of the solution for the case of periodic compressive force is also obtained.

Structural Problems of the Asymptotic Behaviour of Generalized Functions

Development of the theory of generalized functions and their applications in solving different mathematical models have pointed at the need of the adequate theory of the asymptotic behaviour of distributions. Let us point at integral transforms of distributions, Abelian and Tauberian type theorems, asymptotic behaviour of solutions of partial differential equations,... These and many other problems have pushed on the elaboration of a theory of asymptotic behaviours of generalized functions.

Cauchy problems for some classes of linear fractional differential equations

Cauchy problems for a class of linear differential equations with constant coefficients and Riemann-Liouville derivatives of real orders, are analyzed and solved in cases when some of the real orders are irrational numbers and when all real orders appearing in the derivatives are rational numbers. Our analysis is motivated by a forced linear oscillator with fractional damping. We pay special attention to the case when the leading term is an integer order derivative. A new form of solution, in terms of Wright’s function for the case of equations of rational order, is presented. An example is treated in detail.

Distribution-valued functions and their applications

Some properties of the Laplace transform of distribution-valued functions are proved. The mathematical model of an elastic, simply supported axially loaded rod on a viscoelastic foundation is considered and the uniqueness of the solution is proved.

Wiener-Type Tauberian Theorems for Fourier Hyperfunctions

Two Wiener-type Tauberian theorems concerning Fourier hyperfunctions are proved and commented. It is shownt that the shift asymptotics (S-asymptotics) of a hyperfunction f is determined by the ordinary asymptotics of (f ∗ K)(x) as x → ∞, where K is Hörmaner’s kernel. Moreover, Wiener-type theorems are used for the asympthotic analysis of solutions to some (pseudo-)differential equations.

Differential Equation with Fractional Derivative and Nonconstant Coefficients

For a differential equation of the second order, with fractional derivative and nonconstant coefficients we find a solution for the initial value problem, using Laplace transform of hyperfunctions.

The structure of a convergent family of fourier hyperfunctions

We give the structural properties of a family of Fourier hyper-functions which converges as . The main result is also given in [5] but here we give an extended version with all the details of the proofs.

S-Asymptotic of Distributions

In the last thirty years many definitions of the asymptotoic behaviour of distributions have been presented. We can roughly divide them in two sets. To the first one belong those definitions which directly use the classical definition of the asymptotic behaviour of a numerical function. The distribution T has to be equal to a numerical function f or to a derivative, in the sense of distributions, of a numerical functions, DPf, in a neighbourhood of infinity. The behaviour of the distribution at infinity is in reality the behaviour of the function f or corrected by p. All of these definitions are basically given in the one dimensional case.