1. Fractional Calculus and its Applications to Science and Engineering
Selçuk Bayın
Slides of the seminars
IAM-METU (21, Dec. 2010)
Feza Gürsey Institute (17, Feb. 2011) 1

2.  IAM-METU General Seminar: Fractional Calculus and its Applications            Prof. Dr. Selcuk Bayin  December 21, 2010, Tuesday 15:  The geometric interpretation of derivative as the slope and integral  as the area are so evident that one can hardly imagine that a  meaningful definition for the fractional derivatives and integrals can  be given. In 1695 in a letter to L’ Hopital, Leibniz mentions that he  has an expression that looks like the derivative of order 1/2, but  also adds that he doesn’t know what meaning or use it may have. Later,  Euler notices that due to his gamma function derivatives and integrals  of fractional orders may have a meaning. However, the first formal  development of the subject comes in nineteenth century with the  contributions of Riemann, Liouville, Grünwald and Letnikov, and since  than results have been accumulated in various branches of mathematics.  The situation on the applied side of this intriguing branch of  mathematics is now changing rapidly. Fractional versions of the well  known equations of applied mathematics, such as the growth equation,  diffusion equation, transport equation, Bloch equation. Schrödinger  equation, etc., have produced many interesting solutions along with  observable consequences. Applications to areas like economics, finance  and earthquake science are also active areas of research.  The talk will be for a general audience.

3. Derivative and integral as inverse operations 3

4. If the lower limit is different from zero 4

5. nth derivative can be written as 5

6. 6

7. Successive integrals 7

8. For n successive integrals we write 8

9. Comparing the two expressions 9

10. Finally, 10

11. Grünwald-Letnikov definition of Differintegrals
for all q
For positive integer n this satisfies 11

12. Differintegrals via the Cauchy integral formula
We first write 12

13. Riemann-Liouville definition 13

14. Differintegral of a constant 14

15. 15

16. Some commonly encountered semi-derivatives and integrals 16

17. Special functions as differintegrals 17

18. Applications to Science and Engineering
Laplace transform of a Differintegral 18

19. Caputo derivative 19

20. Relation betwee the R-L and the Caputo derivative 20

21. Summary of the R-L and the Caputo derivatives 21

22. 22

23. Fractional evolution equation 23

24. Mittag-Leffler function 24

25. Euler equation y’(t)=iω y(t)
We can write the solution of the following extra-ordinary differential equation: 25

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32. Other properties of Differintegrals:
Leibniz rule
Uniqueness and existence theorems
Techniques with differintegrals
Other definitions of fractional derivatives
Bayin (2006) and its supplements
Oldham and Spanier (1974)
Podlubny (1999)
Others 32

33. GAUSSIAN DISTRIBUTION
Gaussian distribution or the Bell curve is encountered
in many different branches of scince and engineering
Variation in peoples heights
Grades in an exam
Thermal velocities of atoms
Brownian motion
Diffusion processes
Etc.
can all be described statistically in terms of a Gaussian
distribution. 33

34. Thermal motion of atoms 34

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42. Classical and nonextensive information theory (Giraldi 2003)
Mittag-Leffler functions to pathway model to Tsallis statistics
(Mathai and Haubolt 2009) 42

43. Gaussian and the Brownian Motion
A Brownian particle moves under the influence of random
collisions with the evironment atoms.
Brownian motion (1828) (observation)
Einstein’s theory (1905)
Smoluchowski (1906)
In one dimension p(x) is the probability of a single particle making a
single jump of size x.
Maximizing entropy; S=
subject to the conditions
and
variance 43

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48. Constraint on the variance, through the central limit theorem, assures
Note:
Constraint on the variance, through the central limit theorem, assures
that any system with finite variance always tends to a Gaussian.
Such a distribution is called an
attractor. 48

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54. Memory
Initial condition
Probability density 54

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