S. Z. Rida

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.

Fractional modeling dynamics of HIV and CD4 + T-cells during primary infection

In this paper, we introduce fractional-order into a model of HIV-1 infection of CD4+ T cells. We study the effect of the changing the average number of viral particles N with different sets of initial conditions on the dynamics of the presented model. Generalized Euler method (GEM) will be used to find a numerical solution of the HIV-1 infection fractional order model.

Homotopy Analysis Method for Solving Biological Population Model

In this paper, the homotopy analysis method (HAM) is applied to solve generalized biological population models. The fractional derivatives are described by Caputos sense. The method introduces a significant improvement in this field over existing techniques. Results obtained using the scheme presented here agree well with the analytical solutions and the numerical results presented in Ref. [6]. However, the fundamental solutions of these equations still exhibit useful scaling properties that make them attractive for applications.

NONCOMMUTATIVE BELL POLYNOMIALS

The recursive definition for the sequence of the Bell polynomials is generalized to noncommutative variables and then explicitly solved. As applications, we present formulas for the powers of a first-order matrix-valued differential operator, of the “substantial derivative” to a dynamical system, and for the Taylor coefficients of the time-ordered exponential integral.

New Method for Solving Linear Fractional Differential Equations

We develop a new application of the Mittag-Leffler Function method that will extend the application of the method to linear differential equations with fractional order. 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. The results reveal that the technique introduced here is very effective and convenient for solving linear differential equations of fractional order.

Solving nonlinear fractional differential equation by generalized Mittag-Leffler function method.

In this paper, we use Mittag—Leffler function method for solving some nonlinear fractional differential equations. A new solution is constructed in power series. The fractional derivatives are described by Caputos sense. To illustrate the reliability of the method, some examples are provided.

Explicit formulae for the powers of a Schrödinger-like ordinary differential operator

We call a Schrodinger-like differential operator and derive some explicit formulae for the powers or iterations of it.

Numerical solutions for some generalized coupled nonlinear evolution equations

Abstract In this paper, the Adomian decomposition method (ADM) is presented for the numerical solutions of the coupled evolution equations of fractional order. The ADM in applied mathematics can be used as an alternative method for obtaining analytic and approximate solutions for different types of fractional differential equations. The fractional derivatives are described in the Caputo sense. The given solutions are compared with the traveling wave solutions. The results obtained are then graphically represented. Finally, numerical results demonstrate the accuracy, efficiency and high rate of convergence of this method.

Authenticated quantum secret sharing with quantum dialogue based on Bell states

This work proposes a scheme that combines the advantages of a quantum secret sharing procedure and quantum dialogue. The proposed scheme enables the participants to simultaneously make mutual identity authentications, in a simulated scenario where the boss, Alice, shares a secret with her two agents Bob and Charlie. The secret is protected by checking photons to keep untrustworthy agents and outer attacks from getting useful information. Before the two agents cooperate to recover Alices secret, they must authenticate their identity using parts of a pre-shared key. In addition, the whole pre-shared key is reused as part of recovering the secret data to avoid any leaks of information. In comparison with previous schemes, the proposed method can efficiently detect eavesdropping and it is free from information leaks. Furthermore, the proposed scheme proved to be secure against man-in-the-middle attacks, impersonation attacks, entangled-and-measure attacks, participant attacks, modification attacks and Trojan-horse attacks.

Explicit formulae for the powers of a Schrödinger-like ordinary differential operator II

Abstract We derive some new formulae in the Schur pseudodifferential operator calculus with respect to one real variable x , namely for (i) the inverse of a Schrodinger-like operator D r + u , where D =d/dx, r is a positive integer, and u = u ( x ) is a smooth function; (ii) arbitrary integer powers of D + u ; (iii) the square root of the Schrodinger operator D 2 + u . The result (iii) is closely connected with the theory of the KdV hierarchy.