18.10.2015 1 fractional calculus and its applications to science and engineering selçuk bayın 1...
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Fractional Calculus Fractional Calculus and itsand its
Applications to Applications to Science and Science and EngineeringEngineering
Selçuk BayınSelçuk Bayın
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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)
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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.
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Derivative and integral as inverse operations
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If the lower limit is different from zero
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nth derivative can be written as
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Successive integrals
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For n successive integrals we write
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Comparing the two expressions
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Finally,
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Grünwald-Letnikov definition of Differintegrals
for all q
For positive integer n this satisfies
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Differintegrals via the Cauchy integral formula
We first write
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Riemann-Liouville definition
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Differintegral of a constant
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Some commonly encountered semi-derivatives and integrals
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Special functions as differintegrals
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Applications to Science and Engineering
• Laplace transform of a Differintegral
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Caputo derivative
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Relation betwee the R-L and the Caputo derivative
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Summary of the R-L and the Caputo derivatives
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Fractional evolution equation
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Mittag-Leffler function
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Euler equation
We can write the solution of the following extra-ordinary differential equation:
y’(t)=iω y(t)
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• 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:
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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.
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Thermal motion of atoms
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•Classical and nonextensive information theory (Giraldi 2003)
•Mittag-Leffler functions to pathway model to Tsallis statistics (Mathai and Haubolt 2009)
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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
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variance
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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.
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Memory
Probability density
Initial condition
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