The book contains mathematical preliminaries in which basic definitions of fractional derivatives and spaces which are presented. The central part of a book contains various applications in classical mechanics including such field as: viscoelasticity, heat conduction, wave propagation and variational principles of Hamilton's type. In application the mathematical rigor will be observed. There is no single book on fractional calculus that presents application of fractional derivatives of real order on problems of mechanics. Recently, there are a large number of publications on the application of fractional calculus. The authors' intention is to formulate the problems arising in mechanics in both classical and distributional setting.

1. INTRODUCTION

2. PRELIMINARIES

2.1. Notation and definitions

2.2. Fractional integrals and derivatives

2.2.1. On (0,b), b ?T ?V

2.2.2. On R

2.2.3. Caputo derivatives on (0,b), b ?T ?V

2.3. Laplace transform on a bounded interval

2.4. Weak solutions to equations with fractional derivatives

2.5. Wright´s function

3. VARIATIONAL PROBLEMS WITH FRACTIONAL DERIVATIVES

INTRODUCTION

3.1. Euler-Lagrange equations

3.2. Constrained variational problems; optimal control

3.3. Approximation Euler-Lagrange equations

3.4. Nöthe’s theorem

3.5. Generalized Hamilton´s principle

3.5.1. Formulation of the problem

3.5.2. Optimality conditions

3.5.3. Equivalent problems

3.6. Examples

4. EQUATIONS WITH LEFT AND WRIGHT FRACTIONAL DERIVATIVES

INTRODUCTION

4.1. Some special cases to equation (5.1)

4.1.1. Case p=1, q=1, k=n, 0 ?? ?? ?T ?? ??1, C(x) is a constant

4.1.2. p=1, q=1, k=n+1, 0 ?? ?? ?? ???? 1

4.2. General form of equation (5.1) with kp?dnq+1

4.2.1. Space D´mL(??0,b??)

4.2.2. Solutions to equation (5.2)

4.3. Equations defined on R

4.3.1. Some special cases

4.3.2. Weak solutions

4.4. Examples

5. GENERAL METHODS OF SOLVING LINEAR EQUATIONS WITH LEFT FRACTIONAL DERIVATIVES WITH CONSTANT COEFFICIENTS

INTRODUCTION

5.1. A proposition on Laplace transform

5.2. Solution to equation (4.1) with ??i , i=1,...,m, rational numbers

5.2.1. The form of the solution

5.2.2. Properties of the solutions

5.3. Solution to equation (4.1) with ??i ?? 0 , i=1,...,m

5.4. Generalized and week solutions to (4.1)

5.5. Solutions to (4.1) with „initial conditions“

5.6. Solutions to equation (4.l) in the space of tempered distributions

5.7. Solutions to (4.1) on R

5.8. Examples

6. SOME NONLINEAR EQUATIONS WITH FRACTIONAL DERIVATIVES

INTRODUCTION

6.1. A semilinear differential equation

6.2. Continuous solutions to equations of type ( tDb?? 0Dt?? y)(t) = ?? y(t) + g(t), 0 < ?? < 1

6.3. Solutions to equation (tDb?? 0Dt??y)(x)+A1(0Dt??y)(x)+A2(tDb??y)(x)+B(x)y(x)=C(x), 0<??<1, 0<x<b

6.3.1. Continuous solutions

6.3.2. Weak solutions

7. APPLICATION OF FRACTIONAL CALCULUS TO SPECIFIC PROBLEMS OF MECHANICS

7.1. Stability and creep of a fractional derivative order viscoelastic rod

7.1.1. Formulation of the problem

7.1.2. Solution of obtained mathematical model

7.1.3. Numerical results

7.1.4. Discussion

7.2. Lateral vibrations of a fractional derivative type of viscoelastic rod

7.2.1. Construction of the mathematical model

7.2.2. The space of Laplace hyper functions

7.2.3. Solutions to system (20)

7.2.4. Special cases

7.2.5. Asymptotic behavior of the solution

7.3. Generalized Kelvin-Voight body

7.3.1. Generalized solution to the non-homogeneous equation

7.3.2. Solutions and generalized solutions to homogeneous equation

7.4. Dynamics of a rod made of generalized Kelvin-Voight viscoelastic material

7.4.1. Mathematical model

7.4.2. Construction of the solution

7.4.3. Properties of the constructed solution

7.4.4. Uniqueness of the solution to the initial value problem

7.4.5. Main theorem

7.5. On a distributed derivative model of a viscoelastic body

7.5.1. Mathematical model

7.5.2. Special cases of the constructed model

7.6. A modified Zener model of a viscoelastic body

7.6.1. Preliminaries

7.6.2. The internal variable theory

7.6.3. The fractional derivative model

7.6.4. Creep and stress relaxation

7.6.5. Discussions

7.7. A generalized model for the uni-axial isothermal deformation of a viscoelastic body

7.7.1. Preliminaries

7.7.2. New model

7.7.3. Two special cases

7.7.4. Discussions

7.8. On a system of differential equations with fractional derivatives arising in rod theory

7.8.1. Preliminaries

7.8.2. Mathematical model

7.8.3. Solution of the system

7.8.4. Properties of the solution

7.8.5. Interpretation of the solution and conclusion

7.9. On a fractional distributed-order oscillator

7.9.1. A viscoelastic rod with concentrated mass at the end

7.9.2. Some properties of the solution

7.9.3. Integral form of solutions

7.9.4. Discussions

7.10. Stability of an elastic rod on a fractional derivative type of foundations

7.10.1. Mathematical model

7.10.2. Solution to the system of equations

7.10.3. Asymptotic behavior of the solution

7.10.4. Discussions

7.11. On a viscoelastic rod with constitutive equation containing fractional derivatives of two different orders

7.11.1. Mathematical model of the lateral vibration of a viscoelastic rod

7.11.2. Laplace transform of tempered distributions

7.11.3. Solution of the system

7.11.4. Properties of solutions

7.11.5. Stability of solutions

7.11.6. Summary of the results

7.12. Diffusion-wave equations

7.12.1. Formulation of the problem

7.12.2. Solution to governing equations

7.12.3. Nonlinear fractional diffusion equation: the similarity transformation

7.12.4. Discussions

7.13. Fractional telegraph equation

7.12.1. Formulation of the problem

7.12.2. Maximum principle

7.12.3. Solution to governing equations

7.12.4. Discussions

7.14. Diffusion-wave equation of distributed order

7.14.1. Formulation of the problem

7.14.2. Maximum principle

7.14.3. Solution to governing equations

7.14.4. Discussions

7.15. Wave equation for Zener viscoelastic media

7.15.1. Formulation of the problem

7.15.2. Existence of the solution

7.15.3. Solution to governing equations

7.15.4. Numerical examples

7.15.5. Discussions

7.16. Waves in a viscoelastic rod of finite length: stress relaxation and creep

7.16.1. Formulation of the problem

7.16.2. Existence of the solution to governing equations

7.16.3. Numerical examples

7.16.4. Discussions

7.16.5. The rod with attached mass at the end

7.17. Generalized heat conduction of Cattaneo type

7.17.1. Formulation of the problem

7.17.2. Existence of the solution to governing equations

7.17.3. Numerical examples

7.17.4. Discussions

APPENDIX I: SPACES OF DISTRIBUTIONS AND OPERATIONS IN IT

1. Space of distributions

1.1. Definitions

1.2. Different topologies in D´

2. Operations in D´

2.1. Change of variables

2.2. Value at a point

2.3. Multiplication and division by a smooth function

2.4. Derivatives

2.5. Relation betwen classical and distributional derivatives

3. Convolution

4. Tempered distributions

4.1. Space S of rapidly decreasing functions

4.2. Space S´ of tempered distributions

5. Fourier and Laplace transform

5.1. Fourier transform on S´

5.2. Laplace transform of distributions defined on a cone

5.3. Laplace transform on an interval ??0,b), b<?V

6. Quasi-asymptotic behavior at 0 and at b<?V

APPENDIX II INTEGRAL EQUATIONS OF THE SECOND KIND

1. Fredholm´s integral equations

2. Weakly singular Fredholm equation

3. Volterra´s equations of the second kind

REFERENCES

INDEX