الفهرس 
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Abstract
This thesis introduces Maxwell’s equations in the fractional case where derivatives with respect to time are considered to be any number even complex. The conventional case is easily retrieved by setting all fractional orders to one. The fractional-order waveguide analysis in the case of TM mode under the condition α+β=2 is discussed. It is proved that, the conventional case is a special case where α=β=1 from the introduced fractional case. The effect of the fractional-order parameter on the magnetic fields is studied from both phase and magnitude prospective where an extra phase is added and the magnitude is multiplied by a nonlinear function of frequency. This function increases (decreases) exponentially as α decreases (increases). The general case for both α and β is also studied showing how imposing fractional derivatives in the operation of the rectangular waveguide gives us a wide degree of freedom to control its characteristics like the intrinsic impedance and cutoff frequency. However, for any conventional rectangular waveguide the cut-off frequency is fixed upon its dimensions, in the fractional case this frequency may be controlled by adjusting α and β which are different for each material, and similarly for its intrinsic impedance. Also, it was shown that the fractional derivatives may cause the wave to be no longer conventional as shown in the waveguide example.
By replacing the integer-order time derivatives by fractional ones, Maxwell’s curl equations are reconsidered. As such, the modified formulas of the fractional curl operator taking into account the fractional-order time domain Maxwell’s curl equations are introduced. Following this, an additional degree of freedom to control the characteristics of the fractional dual solutions is introduced due to the extra fractional parameter. Applying this work on a practical example of a parallel-plate waveguide with perfect electric conducting walls, there are mainly two fractional parameters ρ and β_1 affect its operation. As the fractional parameter ρ varies between zero and one, both the electric and magnetic fields are rotated by an angle ρπ/2 in the counter clockwise direction and the perfect electric conductor corresponding to ρ=0 changes to a perfect magnetic conductor at the limiting case ρ=1. By imposing the fractional parameter β_1 in the plots of the field lines, the regularity of the stream of the lines is lost giving a general expression for the frequency dependent losses of the wave although all the resistive elements are neglected in this study. Recovering of the conventional case is achieved by setting all fractional derivatives to unity. A hint is given to numerically determine the fractional dual solutions.
This thesis introduces some analysis in the fractional-order transmission line operation by studying the transient analysis which explains the abnormal diffusion of the voltage wave along the line. The fractional-order ABCD matrix in the fractional study is complex valued elements rather than the conventional study. By imposing fractional parameters, not only increase the degrees of freedom to control its characteristics but also gives more general behavior around the conventional case. For example, the frequency-dependent power loss can be modeled by the fractional-orders rather than a resistor as in the conventional case. The analogy between the electromagnetic field and a mesh of transmission lines was restudied considering the lumped components to be fractional elements which is more realistic. This analogy enables us to solve a network problem rather than a field problem known as transmission line modeling (TLM). Using numerical methods to solve the problem, there is an upper bound for the segment width to avoid wave dispersion. It was noticed that the scattering matrices in the fractional study of the TLM are complex valued elements rather than the conventional study; this gives more degrees of freedom in the study of TLM to control its characteristics. Moreover, fractional-order TLM is an efficient reliable method to solve some electromagnetic problems and can be extended to solve fractional-order differential equations.
7.2 Directions for Future Works
The following points are suggested as a future work
Restudying Maxwell’s equations in the space fractional domain.
Applying the modified curl operator on a rectangular waveguide.
Utilizing Fractional TLM to solve Fractional-order differential equations.