1. WITH APPLICATIONS TO MECHANICS
OFFICE OF THEORETICAL AND APPLIED MECHANICS
ACADEMY OF ATHENS
NATIONAL TECHNICAL UNIVERSITY SCHOOL OF CIVIL ENGINEERING
ATHENS , GREECE
NUMERICAL SOLUTION OF DISTRIBUTED ORDER FRACTIONAL DIFFERENTIAL EQUATIONS
WITH APPLICATIONS TO MECHANICS
Prof. John T. Katsikadelis, Dr. Eng., PhD, Dr. h.c.
First Greek-Russian Symposium on Mechanics
Xanthi, Greece, October 10-13, 2011

2. Outline of the presentation
Introduction
Fractional differential equations (FDEs)
Distributed order fractional differential equations
Numerical solution of distributed order FDEs
Application to mechanics.- The distributed order fractional oscillator
Conclusions

3. I. Introduction

4. Fractional differential equations (FDEs) appear more and more frequently in different research
areas and engineering applications. Various physical phenomena are successfully described by
ordinary or partial FDEs. Among of them
Diffusion problems
Viscoelastic response
Dispersion of solutes in natural porous media
Image processing
Modelling and behaviour of elastic and viscoelastic materials
Modelling of damping
Bioengineering
Control theory
Mathematical psychology
Surveys or collections of such applications can also be found in:
Nonnenmacher and Metzler (1995)
Mainardi (1997) (2010)
Gorenflo and Mainardi (1997)
Matignon andMontseny (1998)
Podlubny (1999)
Hilfer [94] (2000)
Uchaikin (2002)
Le Mehaute et al (2005)
Sabatier et al (2007)
Tas et al. (2007)
Klages et al] (2008)
Baleanu et al. (2012)

5. From the mathematical point of view The related literature on fractional differential equations
is very rich, e.g
Podlubny, I. (1999). Fractional Differential Equations. Academic Press, San Diego
Oldham, K.B., Spanier, J. (1974)The Fractional Calculus. Academic, New York
Kilbas A.A., Srivastava, H.M Trujillo J.J. (2006), “Theory and Applications of Fractional Differential Equations”, Elsevier, Amsterdam.
Diethelm Kai, 2010, The Analysis, of Fractional Differential Equations, Springer-Verlag Berlin Heidelberg
More recently, several problems in mathematical physics and engineering have been modelled via distributed order FDEs. An interesting application field of these equations appears in viscoelasticity by generalizing the multi-term fractional derivative viscoelastic model to a derivative model of distributed order.
As the realm of the distributed order FDEs describing the real life response of physical systems grows, the demand for numerical solutions to treat these equations becomes more pronounced in order to overcome the mathematical complexity of analytical solutions. Therefore, the development of effective and easy-to-use numerical schemes for solving such equations acquires an increasing interest.

6. II. Fractional Differential Equations

7. 1. The fractional derivative and its definitions
Fractional (noninteger order) Derivatives are as old as Calculus.
Theory of derivatives of non integer order goes back to G. W. Leibnitz.
After Leibnitz defined the derivative
n: integer,
L’ Hospitale asked: “What if n is a fraction , say
Leibnitz answered (30 September 1695)
“It will be a paradox”
and added prophetically
“From this apparent paradox, one day useful consequences will be drawn”
For 3 centuries the fractional derivative inspired pure theoretical mathematical developments useful only for mathematicians
In the last few decays, however, many authors pointed out that fractional derivatives are very suitable for the description of real materials.
Thus fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes.,

8. Definition of the Fractional Derivative
There are several definitions of the Fractional derivative:
Riemann-Liouville derivative,
Grünwald-Letnikov derivative,
Caputo derivative
All are of integro-differential form
Podlubny, I. (1999). Fractional Differential Equations.
Diethelm, K.(2010). The Analysis of Fractional Differential Equations.
Definition of the Caputo derivative (1967)
order
Caputo

9. What is the Physical meaning of the FD
What is the Physical meaning of the FD? Long discussion (Podlubny, 2002)

10. Why the Caputo Derivative?
Laplace transform of the Caputo derivative
For example:
If m=1, k=0
If m=2, k=1
It is Suitable to apply initial conditions with physical meaning
The description of physical laws (constitutive equations) via fractional derivatives leads to differential equations involving derivatives of noninteger order or Fractional Differential Equations

11. FRACTIONAL DIFFERENTIAL EQUATIONS (FDEs)
Damped oscillator with viscous damping
(new notation)
Damped oscillator with fractional damping
Experiments have shown that damping can be better approximated with fractional derivative dampers ,

12. Analytical solutions: Very few
Ordinary multi-term fractional differential equations
(a) Linear
(b) Nonlinear
From the mathematical point of view, the related literature on fractional differential equations
is very rich, e.g the books
Oldham, K.B., Spanier, J. (1974)
Podlubny, I. (1999
Kilbas A.A., Srivastava, H.M Trujillo J.J. (2006),
Diethelm K., 2010
Analytical solutions: Very few
Numerical solutions: Give answer to real life engineering problems
Katsikadelis, J.T. (2009), ZAMM

13. 2. Partial fractional differential equations
Numerical solution
Katsikadelis, J.T. (2011). The BEM for numerical solution of partial fractional differential equations, Comp. Math. Appl., 62,

14. Some solved Problems
Katsikadelis, J.T., Babouskos, N.G. (2010), “Post-buckling Analysis of Viscoelastic Plates
with Fractional Derivative Model,” Engineering Analysis with Boundary Elements, 34, 1038–1048.
Babouskos, N.G., Katsikadelis, J.T. (2010), “Nonlinear Vibrations of Viscoelastic Plates of Fractional Derivative Type: An AEM Solution,” The Open Mechanics Journal 4, 8-20.
Nerantaki, M.S and Babouskos N.G. (2010), “Analysis of inhomogeneous anisotropic viscoelastic bodies described with multi-term fractional differential models", Computers a7 Mathematics with Applications, 62 (3)
Katsikadelis, J.T. (2010), “Nonlinear Resonance of Viscoelastic Membranes of Fractional
Derivative Type, Proc. 9th HSTAM Conference, July 12-14, Limassol, Cyprus.
Katsikadelis J.T. (2009). “Nonlinear Vibrations of Viscoelastic Membranes of Fractional Derivative Type”, Proc. BeTeq’09, July 22-24, Athens, Greece.
Katsikadelis, J.T. (2009), “The fractional Diffusion-Wave Equation in Bounded Inhomogeneous Anisotropic Media”, In: Recent Advances in Boundary Elements, , Springer.
Katsikadelis, J.T and Babouskos, N.G. (2011). The BEM for buckling analysis of viscoelastic plates modelled with fractional derivatives, BEM/MRM 2011,, June 2011, New Forest, UK.

15. III. Distributed order FDEs

16. Fluid material with viscous response
Constitutive Equations of Linear Viscoelastic materials
The viscoelastic materials combine the elastic and the viscous fluid response
Elastic material
Fluid material with viscous response
The combination gives various viscoelastic models
Kelvin-Voigt
Three parameter

17. Fractional order differential viscoelastic models
Kelvin-Voigt model and limiting cases

18. Distributed order viscoelastic model (Atanackovic, 2002)
Generalized viscoelastic models
Integer order multi-parameter model
Fractional order multi-parameter model
Distributed order viscoelastic model (Atanackovic, 2002)

19. of distributed order FDEs
IV. Numerical solution
of distributed order FDEs

20. Solution procedure: Distributed order FDEs of the form
Approximating the integral with a finite sum using a simple quadrature rule. Thus the distributed order FDE is converted into a multi-term FDE and
Developing a numerical method to solve the resulting multi-term fractional equation.

21. 2. Approximation of the distributed ordr FDE
2. Linear equations
We shall consider first linear equations of the form
2. Approximation of the distributed ordr FDE
Trapezoidal rule with K intervals yields

22. 2. The Analog Equation Solution
Fig. 1. Discretization of the interval

25. Example1

27. V. The distributed order fractional oscillator

28. Governing equations
a. Free vibrations

29. b. Forced vibrations

30. c. Forced vibrations. Harmonic excitation

31. d. Forced vibrations. Harmonic excitation

32. e. Forced vibrations. Harmonic excitation. Resonance

33. VI. Conclusions
A numerical solution is developed for solving distributed order fractional differential equations.
Several example equations are solved to illustrate the efficiency and accuracy of the method.
It is applied to investigate the response of the oscillator with distributed order fractional viscoelastic damping.
The presented method offers an efficient computational tool for studying mechanical systems described with distributed order FDEs.