Page 47 - Mathematical Techniques of Fractional Order Systems

Chapter 2

Nonlinear Fractional Order

Boundary-Value Problems With
Multiple Solutions

Mourad S. Semary 1, , Hany N. Hassan 1,2 and Ahmed G. Radwan 3,4
1 2
Benha Faculty of Engineering, Benha University, Benha, Egypt, Imam Abdulrahman Bin
3
Faisal University, Dammam, Saudi Arabia, Faculty of Engineering, Cairo University, Egypt,
4
Nanoelectronics Integrated Systems Center (NISC), Nile University, Cairo, Egypt
2.1 INTRODUCTION
The ordinary differential equations with boundary value problems play an
essential role in both theory and applications. They are used to describe a
large amount of biological, physic, heat transfer, circuits analysis, and chem-
ical phenomena. It is very important not to lose any solutions of nonlinear
differential equations with boundary conditions in engineering and physical
sciences. Many authors constructed approaches that are based on the semia-
nalytic methods for multiple solutions of nonlinear boundary value problems.
These approaches are based on the framework of some semianalytical meth-
ods such as homotopy analysis method (HAM) and its modifications
(Abbasbandy and Shivanian, 2010, 2011; Hassan and El-Tawil, 2011a; Liao,
2005, 2012; Soltani et al., 2017; Xu et al., 2010), variational iteration method
(VIM) (Semary and Hassan, 2015; Wazwaz, 2017), and Adomian decompo-
sition method (Wazwaz et al., 2016).
The idea of fractional calculus was proposed by L’Hopital and Leibniz in
1695. The first reasonable definitions for the fractional calculus were intro-
duced by Liouville, Riemann, and Grnwald in 1834, 1847, and 1867, respec-
tively (Podlubny, 1999). Recently, many theorems and generalized
fundamentals have been introduced using fractional-order domain such as
stability theorems, the generalized definition of fractional order resonance
conditions, Chaotic Systems, new mathematical formulations, fractional
order oscillators theorems, modeling of vegetables and fruits using

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Mathematical Techniques of Fractional Order Systems. DOI: https://doi.org/10.1016/B978-0-12-813592-1.00002-7
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