الفهرس 
Definition of the Legendre PolynomialsOnly 14 pages are availabe for public view
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Abstract
Fractional calculus is [90] ”the theory of integrals and derivatives of arbitrary
order, which unify and generalize the notions of integer-order differentiation
and n-fold integration”. The idea of generalizing differential operators
to a non-integer order, in particular to the order 1
2 , first appears in the
correspondence of Leibniz with L’Hopital (1695), Johann Bernoulli (1695),
and John Wallis (1697) as a mere question or maybe even play of thoughts.
In the following three hundred years a lot of mathematicians contributed to
the fractional calculus: Laplace (1812), Lacroix (1812), Fourier (1822), Abel
(1823-1826), Liouville (1832-1837), Riemann (1847), Grunwald (1867-1872),
Letnikov (1868-1872), Sonin (1869), Laurent (1884), Heaviside (1892-1912),
Weyl (1917), Davis (1936), Erdelyi (1939-1965), Gelfand and Shilov (1959-
1964), Dzherbashian (1966), Caputo (1969), and many others. Yet, it is only
after the First Conference on Fractional Calculus and its applications that
the fractional calculus becomes one of the most intensively developing areas
of mathematical analysis. Its fields of application range from biology through
physics and electrochemistry to economics, probability theory and statistics.
On behalf of the nature of their definition the fractional derivatives provide an
excellent instrument for the modeling of memory and hereditary properties of
various materials and processes. Half-order derivatives and integrals prove to
be more useful for the formulation of certain electrochemical problems than
the classical methods. Fractional differentiation and integration operators are
also used for extensions of the diffusion and wave equations and, recently, of
the temperature field problem in oil strata.
This thesis consists of five chapters.