الفهرس 
Fundamentals of Fractional CalculusOnly 14 pages are availabe for public view
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Abstract
Fractional Calculus is a branch of mathematical analysis that studies the non-integer-order differentiation
and integration. The application of fractional calculus in the field of electromagnetics has been given
much attention in the last few decades. Indeed, imposing fractional calculus in electromagnetic problems
enables us to study the intermediate behavior between the various integer-order responses thus leading
to new ideas, promising results and better understanding of such intricate engineering problems.
In this thesis, we focus on investigating the effect of imposing fractional-order time derivatives on the
conventional Maxwell’s equations that control the behavior of electromagnetic fields. We study in the
beginning some special cases of Maxwell’s equations in the fractional-order time domain and illustrate
that the integer-order case is considered as a special case. We apply the modified formulas of Maxwell’s
equations on an example of a rectangular waveguide showing that imposing the fractional parameters
gives some degrees of freedom to control the characteristics of the waveguide, such as the cutoff
frequency and the intrinsic impedance which is shown to be complex in value.
The concept of the fractional curl operator introduced a few years ago provides additional intermediate
solutions to an electromagnetic problem rather than the canonical solutions. In this thesis, this concept
of fractional curl operator is restudied, taking into consideration the fractional-order time derivatives in
Maxwell’s equations. The modified analysis adds two extra fractional parameters onto the conventional
curl concept which increases the number of degrees of freedom to control or optimize the design
problem. As an example, the modified form of the fractional curl operator is applied to the operation of
a parallel-plate waveguide in the TM mode where several special cases are illustrated, showing the
importance of the added parameters. For example, it is shown that the added parameters introduce a
power loss term that can be used to model the frequency-dependent losses, although all resistive elements
are neglected during the study.
Moreover, this thesis demonstrates some fundamentals concerning the study of the Fractional-order
Transmission Line (FTL) operation. Whereby the application of fractional calculus is shown to
outperform the conventional domain.
The transmission-line modeling (TLM) is a space- and time-discretizing method for the computation of
electromagnetic fields based on the analogy between the electromagnetic field and a mesh of
transmission lines. This analogy enables us to solve a network problem rather than a field problem. This
thesis also restudies the TLM in the fractional order sense showing how to utilize it to solve some
fractional-order differential equations. In all the work in this thesis, it is verified that the analysis of any
problem in the conventional domain is a special case from the fractional-order study when all the
fractional-orders are equal to “1”.