21.04.23 1

Fractional Calculus Fractional Calculus and itsand its

Applications to Applications to Science and Science and EngineeringEngineering

Selçuk BayınSelçuk Bayın

11

Slides of the seminarsSlides of the seminarsIAM-METU (21, Dec. 2010)IAM-METU (21, Dec. 2010)

Feza Gürsey Institute (17, Feb. 2011) Feza Gürsey Institute (17, Feb. 2011)

2

 IAM-METU General Seminar: Fractional Calculus and its Applications            Prof. Dr. Selcuk Bayin  December 21, 2010, Tuesday 15:40-17.30

 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.

21.04.23 3

Derivative and integral as inverse operations

21.04.23 3

21.04.23 4

If the lower limit is different from zero

21.04.23 4

21.04.23 5

nth derivative can be written as

5

21.04.23 621.04.23 6

21.04.23 7

Successive integrals

21.04.23 7

21.04.23 8

For n successive integrals we write

21.04.23 8

21.04.23 9

Comparing the two expressions

21.04.23 9

21.04.23 10

Finally,

21.04.23 10

21.04.23 11

Grünwald-Letnikov definition of Differintegrals

for all q

For positive integer n this satisfies

21.04.23 11

21.04.23 12

Differintegrals via the Cauchy integral formula

We first write

21.04.23 12

21.04.23 13

Riemann-Liouville definition

21.04.23 13

21.04.23 14

Differintegral of a constant

21.04.23 14

21.04.23 1521.04.23 15

21.04.23 16

Some commonly encountered semi-derivatives and integrals

21.04.23 16

21.04.23 17

Special functions as differintegrals

21.04.23 17

21.04.23 18

Applications to Science and Engineering

• Laplace transform of a Differintegral

21.04.23 18

21.04.23 19

Caputo derivative

21.04.23 19

21.04.23 20

Relation betwee the R-L and the Caputo derivative

21.04.23 20

21.04.23 21

Summary of the R-L and the Caputo derivatives

21.04.23 21

21.04.23 2221.04.23 22

21.04.23 23

Fractional evolution equation

21.04.23 23

21.04.23 24

Mittag-Leffler function

21.04.23 24

21.04.23 25

Euler equation

We can write the solution of the following extra-ordinary differential equation:

y’(t)=iω y(t)

21.04.23 25

21.04.23 2621.04.23 26

21.04.23 2721.04.23 27

21.04.23 2821.04.23 28

21.04.23 2921.04.23 29

21.04.23 3021.04.23 30

21.04.23 3121.04.23 31

21.04.23 3221.04.23 32

• 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

Other properties of Differintegrals:

21.04.23 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.

21.04.23 33

21.04.23 34

Thermal motion of atoms

21.04.23 34

21.04.23 3521.04.23 35

21.04.23 3621.04.23 36

21.04.23 3721.04.23 37

21.04.23 3821.04.23 38

21.04.23 3921.04.23 39

21.04.23 4021.04.23 40

21.04.23 4121.04.23 41

21.04.23 42

•Classical and nonextensive information theory (Giraldi 2003)

•Mittag-Leffler functions to pathway model to Tsallis statistics (Mathai and Haubolt 2009)

21.04.23 42

21.04.23 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

21.04.23 43

variance

21.04.23 4421.04.23 44

21.04.23 4521.04.23 45

21.04.23 4621.04.23 46

21.04.23 4721.04.23 47

21.04.23 48

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.

21.04.23 48

21.04.23 4921.04.23 49

21.04.23 505021.04.23

21.04.23 5121.04.23 51

21.04.23 5221.04.23 52

21.04.23 5321.04.23 53

21.04.23 54

Memory

Probability density

Initial condition

21.04.23 54

21.04.23 5521.04.23 55