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 -- 2.1.2. 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 -- 2.2. 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. Restrictions Following from the Thermodynamics for Fractional Derivative Models of a Viscoelastic Body -- 3.1. Method based on the Fourier transform -- 3.1.1. Linear fractional model -- 3.1.2. Distributed-order fractional model -- 3.1.3. Constitutive equations for rod bending -- 3.1.4. Stress relaxation and creep for two special cases of viscoelastic bodies -- 3.1.5. Variable-order fractional derivative: application to stress relaxation problem.

3.1.6. Linear constitutive equation with fractional derivatives of complex order -- 3.2. Thermodynamical restrictions via the internal variable theory -- 3.2.1. Case I -- 3.2.2. Case II -- Chapter 4. Vibrations with Fractional Dissipation -- 4.1. Linear vibrations with fractional dissipation -- 4.1.1. Linear vibrations with the single fractional dissipation term -- 4.1.2. Fractional derivative-type creeping motion -- 4.1.3. Linear vibrations with the multiterm fractional dissipation -- 4.1.4. Linear fractional two-compartmental model with fractional derivatives of different order -- 4.2. Bagley-Torvik equation -- 4.2.1. Solution procedure -- 4.2.2. Numerical examples -- 4.3. Nonlinear vibrations with symmetrized fractional dissipation -- 4.3.1. Solvability and dissipativity of [4.58] -- 4.3.2. Stability of the solution -- 4.4. Nonlinear vibrations with distributed-order fractional dissipation -- 4.4.1. Existence of solutions -- 4.4.2. Uniqueness of solutions -- 4.4.3. Nonlinear vibrations with single term of fractional dissipation -- Chapter 5. Lateral Vibrations and Stability of Viscoelastic Rods -- 5.1. Lateral vibrations and creep of a fractional type viscoelastic rod -- 5.1.1. Rod made of fractional Kelvin-Voigt-type material -- 5.1.2. Rod made of fractional Zener-type material -- 5.1.3. Viscoelastic rod with two different fractional derivatives of strain -- 5.2. Stability of Beck's column on viscoelastic foundation -- 5.2.1. Solution to systems [5.130]-[5.133] -- 5.2.2. Properties of functions T and V -- 5.3. Compressible elastic rod on a viscoelastic foundation -- 5.3.1. Zeros of Dk -- 5.3.2. Existence of Tk and Vk -- 5.3.3. Asymptotic behavior of Tk -- 5.3.4. Summary of the stability analysis -- Chapter 6. Fractional Diffusion-Wave Equations.

6.1. Nonlinear fractional diffusion-wave equation and fractional Burgers/Korteweg-de Vries equation -- 6.1.1. Nonlinear fractional diffusion-wave equation -- 6.1.2. Fractional Burgers/Korteweg-de Vries equation -- 6.1.3. Exact solutions of the nonlinear fractional diffusion-wave equation -- 6.1.4. Numerical solutions to [6.19] and [6.27] -- 6.2. Fractional telegraph equation -- 6.2.1. Dirichlet problem -- 6.2.2. Signaling problem -- 6.2.3. Cauchy problem -- 6.2.4. Numerical results -- 6.3. Distributed-order diffusion-wave equation -- 6.3.1. Existence of a solution to Cauchy problems ([6.81] and [6.82]) -- 6.3.2. Solution to the Cauchy problem -- 6.4. Maximum principle for fractional diffusion-wave type equations -- 6.4.1. Maximum principle for fractional telegraph equation -- 6.4.2. Maximum principle for distributed-order diffusion equation -- Chapter 7. Fractional Heat Conduction Equations -- 7.1. Cattaneo-type space-time fractional heat conduction equation -- 7.1.1. Existence and uniqueness of a solution -- 7.1.2. Explicit form of the solution -- 7.1.3. Numerical examples -- 7.2. Fractional Jeffreys-type heat conduction equation -- 7.2.1. Solution to the Cauchy problem -- 7.2.2. Numerical examples -- Bibliography -- Index.