Ahmed M. A. El-Sayed

Fractional-order diffusion-wave equation

The fractional-order diffusion-wave equation is an evolution equation of order α ε (0, 2] which continues to the diffusion equation when α → 1 and to the wave equation when α → 2. We prove some properties of its solution and give some examples. We define a new fractional calculus (negative-direction fractional calculus) and study some of its properties. We study the existence, uniqueness, and properties of the solution of the negative-direction fractional diffusion-wave problem.

On the fractional-order logistic equation

Abstract The topic of fractional calculus (derivatives and integrals of arbitrary orders) is enjoying growing interest not only among mathematicians, but also among physicists and engineers (see [E.M. El-Mesiry, A.M.A. El-Sayed, H.A.A. El-Saka, Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl. Math. Comput. 160 (3) (2005) 683–699; A.M.A. El-Sayed, Fractional differential–difference equations, J. Fract. Calc. 10 (1996) 101–106; A.M.A. El-Sayed, Nonlinear functional differential equations of arbitrary orders, Nonlinear Anal. 33 (2) (1998) 181–186; A.M.A. El-Sayed, F.M. Gaafar, Fractional order differential equations with memory and fractional-order relaxation–oscillation model, (PU.M.A) Pure Math. Appl. 12 (2001); A.M.A. El-Sayed, E.M. El-Mesiry, H.A.A. El-Saka, Numerical solution for multi-term fractional (arbitrary) orders differential equations, Comput. Appl. Math. 23 (1) (2004) 33–54; A.M.A. El-Sayed, F.M. Gaafar, H.H. Hashem, On the maximal and minimal solutions of arbitrary orders nonlinear functional integral and differential equations, Math. Sci. Res. J. 8 (11) (2004) 336–348; R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien, 1997, pp. 223–276; D. Matignon, Stability results for fractional differential equations with applications to control processing, in: Computational Engineering in System Application, vol. 2, Lille, France, 1996, p. 963; I. Podlubny, A.M.A. El-Sayed, On Two Definitions of Fractional Calculus, Solvak Academy of science-institute of experimental phys, ISBN: 80-7099-252-2, 1996. UEF-03-96; I. Podlubny, Fractional Differential Equations, Academic Press, 1999] for example). In this work we are concerned with the fractional-order logistic equation. We study here the stability, existence, uniqueness and numerical solution of the fractional-order logistic equation.

Multivalued fractional differential equations

In this paper we study the Cauchy problem of the multivalued fractional differential equation dδx(t)dtδ ∈ F(t,x(t)) a.e. onI = [O,T], δ ∈ R+ as a consequent result of the study of the Cauchy problem of the fractional differential equation dδx(t)dtδ = ƒ(t)), t ∈ I, δ ∈ R+ in the Banach space E, where F(t, x(t)) is a set-valued function defined on I × E. The existence and some other properties of the solution will be proved. Continuation of the problem to the Cauchy problem of the multivalued differential equation dnx(t)/dtn ϵ F(t, x(t)) a.e. on I, and n = 1,2,…, will be established.

Numerical methods for multi-term fractional (arbitrary) orders differential equations

Our main concern here is to give a numerical scheme to solve a nonlinear multi-term fractional (arbitrary) orders differential equation.

Adomian's decomposition method for solving an intermediate fractional advection-dispersion equation

This paper is concerned with a model that describes the intermediate process between advection and dispersion via fractional derivative in the Caputo sense. Adomians decomposition method is used for solving this model. The solution is obtained as an infinite series which always converges to the exact solution.

Exact Solutions of Fractional-Order Biological Population Model

In this paper, the Adomians decomposition method (ADM) is presented for finding the exact solutions of a more general biological population models. A new solution is constructed in power series. The fractional derivatives are described in the Caputo sense. To illustrate the reliability of the method, some examples are provided.

On the fractional differential equations

In this paper we are concerned with the semilinear differential equation d^@ax(t)dt^@a=f(t,x(t)), t>;0 where @a is any positive real number. In [3] the author has proved the existence, uniqueness, and some properties of the solution of this equation when 0<@a<1. Here we mainly study (besides the other properties) the continuation of the solution of this equation to the solution of the corresponding initial value problem when [@a]=k, k=1,2,3,... Applications of singular integro-differential equations are considered.

Circuit realization, bifurcations, chaos and hyperchaos in a new 4D system

Abstract This paper is devoted to introduce a new four-dimensional hyperchaotic system. Existence and uniqueness of the solution of the proposed system are proved. Continuous dependence on initial conditions of the system’s solution and some stability conditions of system’s equilibrium points are studied. The existence of pitchfork bifurcation is demonstrated by using the center manifold theorem and the local bifurcation theory. The Hopf bifurcation is examined in double parameters bifurcation diagrams along with degenerate types of Hopf bifurcations. The rich dynamical behaviors of the system are explored, then circuit implementation of the system is proposed. Numerical simulations are carried out to verify theoretical analysis.

Set-valued integral equations of fractional-orders

The topic of fractional calculus (derivative and integral of arbitrary orders) is enjoying growing interest not only among Mathematicians, but also among physicists and engineers (see [6-16, 18-20, 23-25]). The set-valued integral equations (integral inclusions) arises in the study of control system (see [21, 22, 26]). In this paper we prove the existence of locally bounded variation solution of a Volterra type set-valued integral equation of arbitrary (not necessarily integer) order. The proof will be based on the measure of weak noncompactness and the existence of Caratheodory selectors. As a consequence we study the initial value problem for some set-valued differential and integro-differential equations. The corresponding single-valued problems will be firstly considered.

Linear differential equations of fractional orders

Abstract We are concerned here with the linear differential equations of fractional orders. We will obtain some results concerning the existence, uniqueness and some other properties of the solution analogous (and more general) to that obtained for the linear ordinary differential equations.