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Fractional Calculus and its Applications to Science and Engineering

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.

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Fractional Calculus and its Applications to Science and Engineering

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  1. Fractional Calculus and itsApplications 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 15:40-17.30  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.

  3. Derivative and integral as inverse operations 28.8.2014 3

  4. If the lower limit is different from zero 28.8.2014 4

  5. nth derivative can be written as 5

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  7. Successive integrals 28.8.2014 7

  8. For n successive integrals we write 28.8.2014 8

  9. Comparing the two expressions 28.8.2014 9

  10. Finally, 28.8.2014 10

  11. Grünwald-Letnikov definition of Differintegrals for all q For positive integer n this satisfies 28.8.2014 11

  12. Differintegrals via the Cauchy integral formula We first write 28.8.2014 12

  13. Riemann-Liouville definition 28.8.2014 13

  14. Differintegral of a constant 28.8.2014 14

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  16. Some commonly encountered semi-derivatives and integrals 28.8.2014 16

  17. Special functions as differintegrals 28.8.2014 17

  18. Applications to Science and Engineering • Laplace transform of a Differintegral 28.8.2014 18

  19. Caputo derivative 28.8.2014 19

  20. Relation betwee the R-L and the Caputo derivative 28.8.2014 20

  21. Summary of the R-L and the Caputo derivatives 28.8.2014 21

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  23. Fractional evolution equation 28.8.2014 23

  24. Mittag-Leffler function 28.8.2014 24

  25. Euler equation y’(t)=iω y(t) We can write the solution of the following extra-ordinary differential equation: 28.8.2014 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 28.8.2014 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. 28.8.2014 33

  34. Thermal motion of atoms 28.8.2014 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) 28.8.2014 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 28.8.2014 43

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  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. 28.8.2014 48

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