الفهرس 
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Abstract
This work is an investigation of fractional order applications in engineering specially
in circuits and systems. Fractional calculus is a field of mathematics that studies the
effects of using non-integer order differential and integral operator in system modeling
and control. The term fractional is a misnomer as the order can be real or even complex.
Fractional calculus has been proved by numerous papers superior to integer order calculus
in describing non-local and unconventional physical phenomena due to the extra degrees
of freedom it provides that are not available in the integer order subspace.
Based on this motivation, this work investigates the basics of fractional calculus,
fractional differential equations and some of its engineering applications: control, bioengineering and viscoelasticity. Then an investigation of various analog and digital approaches
to approximate fractional order operators in circuits and systems is carried out. The
fractional order Taylor series method is used to solve fractional Hermite, Legendre and
Chebyshev differential equations. The solutions are further investigated to obtain fractional
versions of the well known polynomials after some proper scaling in preparation for use
in filter magnitude response.
A review of fractional order filters in literature is then carried out followed by three
new approaches to generalize Chebyshev low-pass filter. The first approach is based on
fractional power Chebyshev-like polynomials and a generalization of pole generating
formula. The second is based on the integer order poles of the traditional low-pass
filter which were used to construct the fractional order transfer functions having the
same poles in the s-plane. The third approach uses the polynomials obtained before
to construct the fractional order complex filter. All filters are realized using passive
circuits or active circuits or both and the ADS circuit simulation results are always
compared to Matlab numerical simulation of the transfer functions to prove the reliability
of the approaches developed. An investigation is carried out discussing the effect of nonzero initial conditions on the transient response of RC α circuit modeled using RiemannLiouville and Caputo fractional operator.