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)
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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.
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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
y’(t)=iω y(t)
We can write the solution of the following extra-ordinary differential equation:
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Other properties of Differintegrals:
•
•
•
•
Leibniz rule
Uniqueness and existence theorems
Techniques with differintegrals
Other definitions of fractional derivatives
•
•
•
•
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Bayin (2006) and its supplements
Oldham and Spanier (1974)
Podlubny (1999)
Others
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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
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
Initial condition
Probability density
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