الفهرس 
I HISTORICAL INTRODUCTION TO FRACTIONAL DIFFERENTIAL EQUATIONS AND LIE groupOnly 14 pages are availabe for public view
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Abstract
It is an introduction to the optimal system of subalgebra. Also it introduces fractional differential equations (FDE) including some definitions related Riemann-Liouvill, Capito and their properties. This chapter also includes an introduction to the main concepts of the group analysis, the main concepts of the invariant solutions, the theorems and definitions needed for the study of the invariance properties of differential equations.
Bäcklund transformations (BTs) are also introduced in this chapter.
Chapter II
This chapter is concerned with finding the exact solutions by using Bäcklund transformations (BTs) for the following equations:
the stable fractional nonlinear Schrödinger equation, a Perturbed fractional Nonlinear Schrödinger equation in two cases, the time fractional Korteweg-de Vries (KdV) equation, the time fractional generalized Korteweg-de Vries equation and the time fractional Modified Korteweg-de Vries (FmKdV) equation.
Bäcklund transformations help us to get new solutions for the previous equations in terms of well-known solutions. These known solutions are presented as the constant, the simple and the travelling wave solutions.
Chapter III
This chapter discusses the fractional analysis through studing some definitions and properties which are relevant to it as the definition of Riemann-Liouvill, Capito and their aspects. This chapter also includes several examples to find the symmetry of FDE and using them to obtain exact solutions of equations.
This chapter also highlights the nonlocal symmetry and applying Lie group analysis methods to the class of fractional differential equations containing fractional derivatives of a function with respect to another function.
H. A. Zedan, S. S. Tantawy and A. R. Abdel-Malek, ”Invariance of the nonlinear generalized NLS equation under the Lie group of scaling transformations”, Nonlinear Dynamics Journal, 82:2001--2005; 2015.
Chapter IV
This chapter pays attention to the Q-symmetry for the equations: the space-time fractional diffusion-wave equation, the space-time fractional nonlinear generalized Schrödinger equation (FNLS) and the space-time fractional Drinfeld’s Sokolov-Wilson (FDSW) to obtain the Lie point transformation generators. The application of one-parameter group reduces the number of independent variables consequently these mentioned equations are reduced to set of fractional ordinary differential equations (FODEs) which are solved analytically.
In additition to that, we are interested in the conditional Q-symmetry for the systems mentioned above to transformation them into fractional ordinary differential equations by using symmerty properties. Also, we got new solutions to the previously mentioned systems through conservation laws.
H. A. Zedan, S. S. Tantawy and A. R. Abdel-Malek, ”Conservation laws for the space-time fractional of classical Drinfeld’s Sokolov-Wilson (FDSW) system”, Journal of Fractional Calculus and Applications, 10(2):207-215; 2019.
Chapter V
This chapter aims at determining the optimal subalgebras to fractional differential equations. We transformed the equations: the time fractional generalized Burger’s equation and the fractional two-dimensional coupled Burger’s equations to fractional ordinary differential equations. Solving these fractional ordinary differential equations, we got new variant solutions the studied equations.
H. A. Zedan, S. S. Tantawy and A. R. Abdel-Malek, ”Optimal System of Subalgebras for the Time Fractional Generalized Burgers Equation”, Asian Research Journal of Mathematics, 11(4):1-23; 2018.