Page 48 - Mathematical Techniques of Fractional Order Systems

38 Mathematical Techniques of Fractional Order Systems

Cole Cole model, generalized fractional order filters, and others in different
engineering applications (Radwan et al., 2009a,b, 2008a,b; Radwan, 2013;
Valerio et al., 2013; Odibat, 2010; Freeborn et al., 2013; Azar et al., 2017).
Also, the fractional theory was extended to belong to the memristive ele-
ments (Fouda and Radwan, 2015).
Moreover, the fractional differential equations are a main and important
subject in the area of physical sciences and engineering, as they have
become popular in modeling processes in physical science and engineering,
such as control theory, signal processing, bioengineering, circuit theory, and
viscoplasticity (Das and Pan, 2012; Monje et al., 2010; Petras, 2011; Semary
et al., 2016a,b). Therefore, the numerical methods for these equations have
undergone fast growth in recent years. But only a few methods are available
for solving the fractional order differential equations with multiple solutions
(Arqub et al., 2014; Alomari et al., 2013; Semary et al., 2017a,b). This chap-
ter introduces and discus the techniques based on the Picard method (Semary
et al., 2017a,b) and HAM (Alomari et al., 2013) to predict the multiplicity of
the solutions of fractional order differential equations with boundary condi-
tions. These techniques are not only capable of predicting but also calculat-
ing all branches of the solutions simultaneously. The applied approaches
detect multiple solutions to some of the practical models in fractional order
domain, the first problem arising in mixed convection flows in a vertical
channel (Abbasbandy and Shivanian, 2011; Alomari et al., 2013). The sec-
ond model appears in a number of applications such as the model of thermal
reaction process, the fuel ignition model of the thermal combustion theory,
the Chandrasekhar model of the expansion of the Universe, questions in
geometry and relativity about the Chandrasekhar model, nanotechnology,
radiative heat transfer and chemical reaction theory, and this model is called
the Bratu problem (Jacobsen and Schmitt, 2002; Jalilian, 2010; Wazwaz,
2005). The third problem describes the fractional order diffusion and reaction
model in porous catalysts and also steady state heat conduction of the tem-
perature distribution of a straight rectangular fin with a prescribed power-law
temperature dependent surface heat flux (Abbasbandy and Shivanian, 2010;
Semary et al., 2017a; Magyari, 2008). The fourth problem is a three-point
nonlinear boundary-value problem (Graef et al., 2003, 2004). The main goal
of this chapter is to illustrate the two methods procedures to predict and cal-
culate all branches of the problem solutions in the fractional order domain.
The advantages of these techniques are that they are very powerful for solv-
ing fractional order boundary value problems that admit multiple solutions.
And the obtained series solutions are converge to multiple solutions with
convergent region. The chapter symbols are listed in Table 2.1 and this
chapter is organized as follows: Section 2.2 introduces the mathematical
procedures for the used methods. Then, the numerical simulations for these
methods are discussed for different classes of fractional order differential
equations in Section 2.3. Finally, Section 2.4 concludes the presented work.