Cover -- Title Page -- Contents -- Preface -- PART 1. MATHEMATICAL PRELIMINARIES, DEFINITIONS AND PROPERTIES OF FRACTIONAL INTEGRALS AND DERIVATIVES -- Chapter 1. Mathematical Preliminaries -- 1.1. Notation and definitions -- 1.2. Laplace transform of a function -- 1.3. Spaces of distributions -- 1.4. Fundamental solution -- 1.5. Some special functions -- Chapter 2. Basic Definitions and Properties of Fractional Integrals and Derivatives -- 2.1. Definitions of fractional integrals and derivatives -- 2.1.1. Riemann-Liouville fractional integrals and derivatives -- 2.1.1.1. Laplace transform of Riemann-Liouville fractional integrals and derivatives -- 2.1.2. Riemann-Liouville fractional integrals and derivatives on the real half-axis -- 2.1.3. Caputo fractional derivatives -- 2.1.4. Riesz potentials and Riesz derivatives -- 2.1.5. Symmetrized Caputo derivative -- 2.1.6. Other types of fractional derivatives -- 2.1.6.1. Canavati fractional derivative -- 2.1.6.2. Marchaud fractional derivatives -- 2.1.6.3. Grünwald-Letnikov fractional derivatives -- 2.2. Some additional properties of fractional derivatives -- 2.2.1. Fermat theorem for fractional derivative -- 2.2.2. Taylor theorem for fractional derivatives -- 2.3. Fractional derivatives in distributional setting -- 2.3.1. Definition of the fractional integral and derivative -- 2.3.2. Dependence of fractional derivative on order -- 2.3.3. Distributed-order fractional derivative -- PART 2. MECHANICAL SYSTEMS -- Chapter 3. Waves in Viscoelastic Materials of Fractional-Order Type -- 3.1. Time-fractional wave equation on unbounded domain -- 3.1.1. Time-fractional Zener wave equation -- 3.1.2. Time-fractional general linear wave equation -- 3.1.3. Numerical examples -- 3.1.3.1. Case of time-fractional Zener wave equation.

3.1.3.2. Case of time-fractional general linear wave equation -- 3.2. Wave equation of the fractional Eringen-type -- 3.3. Space-fractional wave equation on unbounded domain -- 3.3.1. Solution to Cauchy problem for space-fractional wave equation -- 3.3.1.1. Limiting case ß -> 1 -- 3.3.1.2. Case u0(x)... -- 3.3.1.3. Case u0 (x)... -- 3.3.1.4. Case u0(x)... -- 3.3.2. Solution to Cauchy problem for fractionally damped space-fractional wave equation -- 3.4. Stress relaxation, creep and forced oscillations of a viscoelastic rod -- 3.4.1. Formal solution to systems [3.110]-[3.112], [3.113] and either [3.114] or [3.115] -- 3.4.1.1. Displacement of rod's end Υ is prescribed by [3.120] -- 3.4.1.2. Stress at rod's end Σ is prescribed by [3.121] -- 3.4.2. Case of solid-like viscoelastic body -- 3.4.2.1. Determination of the displacement u in a stress relaxation test -- 3.4.2.2. Case Υ = Υ0H + F -- 3.4.2.3. Determination of the stress s in a stress relaxation test -- 3.4.2.4. Determination of displacement u in the case of prescribed stress -- 3.4.2.5. Numerical examples -- 3.4.3. Case of fluid-like viscoelastic body -- 3.4.3.1. Determination of the displacement u in a stress relaxation test -- 3.4.3.2. Determination of the stress σ in a stress relaxation test -- 3.4.3.3. Determination of the displacement u in a creep test -- 3.4.3.4. Numerical examples -- Chapter 4. Forced Oscillations of a System: Viscoelastic Rod and Body -- 4.1. Heavy viscoelastic rod-body system -- 4.1.1. Formal solutions -- 4.1.2. Existence and uniqueness of the solution to [4.18]-[4.21] -- 4.1.2.1. Notation and assumptions -- 4.1.2.2. Theorems on the existence and uniqueness -- 4.1.2.3. Auxiliary results -- 4.1.2.4. Proofs of theorems 4.1 and 4.2 -- 4.1.2.5. The case of the elastic rod.

4.1.3. Distributed-order and fractional Zener model as special cases -- 4.1.4. Solid and fluid-like model of a rod. A parallel analysis of systems [4.18]-[4.21] -- 4.1.4.1. Auxiliary results -- 4.1.4.2. Existence theorems -- 4.1.5. Numerical examples -- 4.1.5.1. The case F = δ -- 4.1.5.2. The case F = H -- 4.2. Light viscoelastic rod - body system -- 4.2.1. The case of the solid-like viscoelastic rod -- 4.2.1.1. Analysis of the problem -- 4.2.1.2. Numerical examples -- 4.2.2. The case of the parallel connection of spring and viscoelastic rod -- 4.2.3. The case of the rod of distributed-order Kelvin-Voigt type -- 4.2.3.1. Study of the generalization of [4.216] -- 4.2.3.2. Two special cases of [4.217] -- Chapter 5. Impact of Viscoelastic Body Against the Rigid Wall -- 5.1. Rigid block with viscoelastic rod attached slides without friction -- 5.1.1. Analytical solution to [5.6] and [5.7] -- 5.1.1.1. Existence and uniqueness results -- 5.1.1.2. Qualitative properties of solutions -- 5.1.1.3. Iterative procedure -- 5.1.2. Numerical solution to [5.6] and [5.7] -- 5.2. Rigid block with viscoelastic rod attached slides in the presence of dry friction -- 5.2.1. Solution of system [5.42]-[5.44] -- 5.2.1.1. Approach phase -- 5.2.1.2. Rebound phase -- 5.2.2. Numerical examples -- Chapter 6. Variational Problems with Fractional Derivatives -- 6.1. Euler-Lagrange equations -- 6.2. Linear fractional differential equations as Euler-Lagrange equations -- 6.3. Constrained variational principles -- 6.4. Approximation of Euler-Lagrange equations -- 6.4.1. Approximation 1 -- 6.4.2. Approximation 2 -- 6.5. Invariance properties of J : Nöther's theorem -- 6.5.1. The case when a in aDαt u is not transformed -- 6.5.2. The case when a in aDαt u is transformed -- 6.5.3. Nöther's theorem.

6.5.4. Approximations of fractional derivatives in Nöther's theorem -- 6.6. Complementary fractional variational principles -- 6.6.1. Notation -- 6.6.2. Lagrangian depending on a vector function of scalar argument -- 6.6.2.1. An example with the quadratic potential Π -- 6.6.2.2. Complementary principles for the fractional and classical Painlevé equation -- 6.6.3. Lagrangian depending on a scalar function of vector argument -- 6.6.3.1. Example in the image regularization -- 6.7. Generalizations of Hamilton's principle -- 6.7.1. Generalized Hamilton's principle with constant-order fractional derivatives -- 6.7.1.1. Examples with Lagrangians linear in y -- 6.7.1.2. Examples with Lagrangians linear in 0Dat y -- 6.7.2. Generalized Hamilton's principle with variable-order fractional derivatives -- 6.7.2.1. Various definitions of variable-order fractional derivatives -- 6.7.2.2. Distributional setting of variable-order derivatives -- 6.7.2.3. Integration by parts formula -- 6.7.2.4. Stationarity conditions when function a is given in advance -- 6.7.2.5. Stationarity conditions when function a is a constant, not given in advance -- 6.7.2.6. Stationarity conditions when function a is not given in advance -- 6.7.2.7. Stationarity conditions when function a is given constitutively -- Bibliography -- Index.