الفهرس 
PreliminariesOnly 14 pages are availabe for public view
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Abstract
Fractional calculus is one of the oldest calculi invented. It dates back to Lipinze, in 1695 Lipinze asked L’Hopital: what about if we extend the order of derivative to half? which is fractional order. Lipinze said that if this question is answered, then it will open many consequences in differ-ent applications. Later, after about a hundred years, i.e., in 1819, Lacroui answered the question and initiated the following formula for every q ∈ Q:
dq Γ(m + 1)
xm = xm−q
dxq
Γ(m − q + 1)
We known that m!
dn
xm = xm−n
dxn (m − n)!
for every n ∈ N. But for n is fraction, the vectorial (m − n) become vecto-rial of fraction number but the vectorial is defined only for natural numbers. However, the gamma function is defined for fraction numbers and there is a relation between gamma function and vectorial so that he replace the vec-torial by gamma function.
Interesting point that the fractional derivative Dα for 1 is not equal to
zero but equal to x α (by using Riemann-Liouville definition of fractional
−α)
derivative), while Dn(1) = 0 for every n ∈ N
Subsequent mention of fractional derivatives was made, in some context or the other, by (for example) Euler in 1730, Lagrange in 1772, Laplace in 1812, Lacroix in 1819, Fourier in 1822, Liouville in 1832, Riemann in 1847, Greer in 1859, Holmgren in 1865, Griinwald in 1867, Letnikov in 1868, Sonin in 1869, Laurent in 1884, Nekrassov in 1888, Krug in 1890, and Weyl in 1917.
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CONTENTS 7
In addition, of course, to the theories of differential, integral, and integro-differential equations, and special functions of mathematical physics as well as their extensions and generalizations in one and more variables, some of the areas of present-day applications of fractional calculus include Fluid Flow, Rheology, Dynamical Processes in Self-Similar and Porous Structures, Diffusive Transport Akin to Diffusion, Electrical Networks, Probability and Statistics, Control Theory of Dynamical Systems, Viscoelasticity, Electro-chemistry of Corrosion, Chemical Physics, Optics and Signal Processing, and so on. So all science-topics based on integer derivative could be general-ized to the fractional calculus form. For an historical overview on fractional calculus, see [1, 26, 27].
There are a lot of numerical methods used for approximation, but most of these methods do not approximate all types of functions but deal with specific functions, for example: polynomials approximation deals with ana-lytic functions without singularities, and Fourier polynomials approximation is applicable to functions that are smooth and periodic on some domains. So, we follow a new direction of approximation theory ”Sinc method” in-troduced by Stenger in connection with convolution integrals [9]. The the-oretical background of the method can be found in [10]. We will use Sinc approximation because it solves problems with singularities, boundary layer problems, and problems over infinite and semi infinite domains. Further-more, the computer programs based on Sinc methods are usually considered shorter than the corresponding ones based on classical methods of approx-imation. Sinc methods are particularly powerful when the problem has singularities, that means, in case of the error of an n-point approximation
1
converges at an incredible O(e−cN 2 ) rate, whereas in such circumstances polynomial methods, at best, converge at the O(n−a) rate, where c and a are positive constant [10, 29], where
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CONTENTS 8
Notation Name Description
f(n) = O(g(n)) Big O |f| is bounded above by g asymptoti-
cally
f(n) = o(g(n)) Small o f is dominated by g asymptotically
f(n) = Θ(g(n)) Big Theta f is bounded both above and below by
g asymptotically
f(n) = Ω(g(n)) Big Omega in f is bounded below by g asymptotically
complexity theory
f(n) = Ω(g(n)) Big Omega in |f| is not dominated by g asymptoti-
number theory cally
f(n) = w(g(n)) Small Omega f dominates g asymptotically
f(n) ∼ (g(n)) On the order of f is equal to g asymptotically
Oftentimes the true value is unknown to us, especially in numerical com-puting. In this casewe will have to quantify errors using approximate values only. When an iterative method is used, we get an approximate value at the end of each iteration. The approximate error Eα is defined as the difference between the present approximate value and the pervious approximation (i.e. the change between the iterations).
Approximate error = Present approximation - Pervious approximation
Similarly, we can calculate the relative approximate error eα by dividing the approximate error by the present approximate value.
Relative approximate error = Approximate error / Present approximation.
This thesis deals with Sinc method via convolution integrals for solving fractional differential equations. We tested examples and compared them with exact solutions, so it is shown that Sinc method yields accurate results. Also we use the elliptic function procedure to derive a family of interpolat-ing rational approximations. The rationals which we shall construct have accuracy equivalent to that of Sinc approximation, in the spaces which we have shown Sinc methods to be effective. We may expect Thieles algorithm to yield an accurate approximation of a function G at a boundary point b of its region of analyticity. By accurate, we mean that using O(N) values
1
of G, we can approximate G(b) to be within an O(e−CN 2 ) error, with C a positive constant, depending for practical purposes neither on N, nor on the particular set of given data {wj , G(wj )} [1, 11].
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CONTENTS 9
The aim of this thesis is to introduce a new approach to represent frac-tional operators based on Sinc method. This approach has the advantage that we need only a single base of functions allowing us to represent approx-imations of functions by a truncation of a cardinal expansion [26, 29]. In addition, the derived approximation is not only numerically exponentially converging and can handle singularities but also provides symbolic access to the results which in combination with numeric delivers a hybrid repre-sentation of the approximation. We will discuss this approach on a specific selection of fractional differential operators.