Franklin

Fractional calculus with applications in mechanics : wave propagation, impact and variational principles / Teodor M. Atanacković [and three others] ; series editor, Noël Challamel.

Publication:
London ; Hoboken, New Jersey : ISTE : Wiley, 2014.
Format/Description:
Book
1 online resource (424 p.)
Series:
Focus series in mechanical engineering and solid mechanics.
Mechanical Engineering and Solid Mechanics Series
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Subjects:
Calculus.
Fractional calculus.
Viscoelasticity -- Mathematical models.
Waves -- Mathematical models.
Form/Genre:
Electronic books.
Language:
English
Summary:
The books Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes and Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles contain various applications of fractional calculus to the fields of classical mechanics. Namely, the books study problems in fields such as viscoelasticity of fractional order, lateral vibrations of a rod of fractional order type, lateral vibrations of a rod positioned on fractional order viscoelastic foundations, diffusion-wave phenomena, heat conduction, wave propagation, forced oscilla
Contents:
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 derivatives2.1.2. 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 derivatives2.3. 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 equation3.2. 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
Notes:
Description based upon print version of record.
Includes bibliographical references and index.
Description based on print version record.
Contributor:
Atanacković, Teodor M.
Challamel, Noël.
ISBN:
1-118-90913-5
1-118-90906-2
1-118-90901-1
OCLC:
876043683