Mohamed Abd El-Moneam - Academia.edu (original) (raw)

Papers by Mohamed Abd El-Moneam

Research paper thumbnail of Bifurcation and chaos in a discrete activator-inhibitor system

AIMS Mathematics

In this paper, we explore local dynamic characteristics, bifurcations and control in the discrete... more In this paper, we explore local dynamic characteristics, bifurcations and control in the discrete activator-inhibitor system. More specifically, it is proved that discrete-time activator-inhibitor system has an interior equilibrium solution. Then, by using linear stability theory, local dynamics with different topological classifications for the interior equilibrium solution are investigated. It is investigated that for the interior equilibrium solution, discrete activator-inhibitor system undergoes Neimark-Sacker and flip bifurcations. Further chaos control is studied by the feedback control method. Finally, numerical simulations are presented to validate the obtained theoretical results.

Research paper thumbnail of A General Scheme for Solving Systems of Linear First-Order Differential Equations Based on the Differential Transform Method

Journal of Mathematics

In this study, we develop the differential transform method in a new scheme to solve systems of f... more In this study, we develop the differential transform method in a new scheme to solve systems of first-order differential equations. The differential transform method is a procedure to obtain the coefficients of the Taylor series of the solution of differential and integral equations. So, one can obtain the Taylor series of the solution of an arbitrary order, and hence, the solution of the given equation can be obtained with required accuracy. Here, we first give some basic definitions and properties of the differential transform method, and then, we prove some theorems for solving the linear systems of first order. Then, these theorems of our system are converted to a system of linear algebraic equations whose unknowns are the coefficients of the Taylor series of the solution. Finally, we give some examples to show the accuracy and efficiency of the presented method.

Research paper thumbnail of On the rational difference equation y n + 1 = α 0 y n + α 1 y n − p + α 2 y n − q + α 3 y n − r + α 4 y n − s β 0 y n + β 1 y n − p + β 2 y n − q + β 3 y n − r + β 4 y n − s

Journal of Nonlinear Sciences and Applications, 2017

In this paper, we examine and explore the boundedness, periodicity, and global stability of the p... more In this paper, we examine and explore the boundedness, periodicity, and global stability of the positive solutions of the rational difference equation y n+1 = α

Research paper thumbnail of On the periodicity of the solution of a rational difference equation

Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, 2019

In this paper, some cases on the periodicity of the rational di¤erence equation S n+1 = S n p aS ... more In this paper, some cases on the periodicity of the rational di¤erence equation S n+1 = S n p aS n q + bS n r + cS n s dS n q + eS n r + f S n s ; are investigated, where a, b; c, d, e, f 2 (0; 1). The initial conditions S p , S

Research paper thumbnail of On the dynamics of the nonlinear rational difference equation $ { x_{n+1}} = \frac{{\alpha {x_{n-m}}} \ \ + \ \ \delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}} \ \ { x_{n-l}} \ \ \left({{x_{n-k}} \ \ + \ \ {x_{n-l}}} \ \ \right) }} $

AIMS mathematics, 2022

In this paper, we discuss some qualitative properties of the positive solutions to the following ... more In this paper, we discuss some qualitative properties of the positive solutions to the following rational nonlinear difference equation $ { x_{n+1}} = \frac{{\alpha {x_{n-m}}} \ \ + \ \ \delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}} \ \ { x_{n-l}} \ \ \left({{x_{n-k}} \ \ + \ \ {x_{n-l}}} \ \ \right) }} ,, , n = 0, 1, 2, ... $ where the parameters $ \alpha, \beta, \gamma, \delta \in (0, \infty) ,while, while ,while m, k, l $ are positive integers, such that $ m < k < l. $ The initial conditions $ {x_{-m}}, ..., {x_{-k}}, ..., {x_{-l}}, ..., {x_{-1}}, ..., {x_{0}} $ are arbitrary positive real numbers. We will give some numerical examples to illustrate our results.

Research paper thumbnail of On the dynamics of the nonlinear rational difference equation $ { x_{n+1}} = \frac{{\alpha {x_{n-m}}} \ \ + \ \ \delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}} \ \ { x_{n-l}} \ \ \left({{x_{n-k}} \ \ + \ \ {x_{n-l}}} \ \ \right) }} $

AIMS Mathematics, 2022

In this paper, we discuss some qualitative properties of the positive solutions to the following ... more In this paper, we discuss some qualitative properties of the positive solutions to the following rational nonlinear difference equation $ { x_{n+1}} = \frac{{\alpha {x_{n-m}}} \ \ + \ \ \delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}} \ \ { x_{n-l}} \ \ \left({{x_{n-k}} \ \ + \ \ {x_{n-l}}} \ \ \right) }} ,, , n = 0, 1, 2, ... $ where the parameters $ \alpha, \beta, \gamma, \delta \in (0, \infty) ,while, while ,while m, k, l $ are positive integers, such that $ m < k < l. $ The initial conditions $ {x_{-m}}, ..., {x_{-k}}, ..., {x_{-l}}, ..., {x_{-1}}, ..., {x_{0}} $ are arbitrary positive real numbers. We will give some numerical examples to illustrate our results.

Research paper thumbnail of On the asymptotic behavior of some nonlinear difference equations

Journal of Computational Analysis and Applications, 2019

Research paper thumbnail of Periodicity of the solution of a higher order difference equation

AIP Conference Proceedings, 2018

Research paper thumbnail of Stability of a fractional difference equation of high order

Journal of Nonlinear Sciences and Applications, 2018

In this paper we investigate the local stability, global stability, and boundedness of solutions ... more In this paper we investigate the local stability, global stability, and boundedness of solutions of the recursive sequence x n+1 = x n−p 2 x n−q + a x n−r x n−q + a x n−r , where x −q+k = −a x −r+k

Research paper thumbnail of Periodicity of the solution of difference equation

AIP Conference Proceedings, 2018

Research paper thumbnail of On the Solutions of a System of Third-Order Rational Difference Equations

Discrete Dynamics in Nature and Society, 2018

The structure of the solutions for the system nonlinear difference equations xn+1=ynyn-2/(xn-1+yn... more The structure of the solutions for the system nonlinear difference equations xn+1=ynyn-2/(xn-1+yn-2), yn+1=xnxn-2/(±yn-1±xn-2), n=0,1,…, is clarified in which the initial conditions x-2, x-1, x0, y-2, y-1, y0 are considered as arbitrary positive real numbers. To exemplify the theoretical discussion, some numerical examples are presented.

Research paper thumbnail of The particular solutions of some types of Euler-Cauchy ODE using the differential transform method

Journal of Nonlinear Sciences and Applications, 2018

In this paper, we apply the differential transform method to find the particular solutions of som... more In this paper, we apply the differential transform method to find the particular solutions of some types of Euler-Cauchy ordinary differential equations. The first model is a special case of the nonhomogeneous n th order ordinary differential equations of Euler-Cauchy equation. The second model under consideration in this paper is the nonhomogeneous second order differential equation of Euler-Cauchy equation with a bulge function. This study showed that this method is powerful and efficient in finding the particular solution for Euler-Cauchy ODE and capable of reducing the size of calculations comparing with other methods.

Research paper thumbnail of On the Dynamics of the Solutions of the Rational Recursive Sequences

British Journal of Mathematics & Computer Science, 2015

In this article, we study the periodicity, the boundedness and the global stability of the positi... more In this article, we study the periodicity, the boundedness and the global stability of the positive solutions of the following nonlinear difference equation xn+1 = Axn +Bxn−k + Cxn−l +Dxn−σ + bxnxn−kxn−l dxn−k − exn−l , n = 0, 1, 2, ..... where the coefficients A,B,C,D, b, d, e ∈ (0,∞), while k, l and σ are positive integers. The initial conditions x−σ,..., x−l,..., x−k, ..., x−1, x0 are arbitrary positive real numbers such that k < l < σ. Some numerical examples will be given to illustrate our results.

Research paper thumbnail of On the Dynamics of the Higher Order Nonlinear Rational Difference Equation

Mathematical Sciences Letters, 2014

In this article, we study the periodicity, the boundedness and the global stability of the positi... more In this article, we study the periodicity, the boundedness and the global stability of the positive solutions of the following nonlinear difference equation x n+1 = Ax n + Bx n−k +Cx n−l + Dx n−σ + bx n−k [dx n−k − ex n−l ] , n = 0, 1, 2, ....., where the coefficients A, B,C, D, b, d, e ∈ (0, ∞), while k, l and σ are positive integers. The initial conditions x −σ ,..., x −l ,..., x −k , ..., x −1 , x 0 are arbitrary positive real numbers such that k < l < σ. Some numerical examples will be given to illustrate our results.

Research paper thumbnail of The Solutions of Legendre’s and Chebyshev’s Differential Equations by Using the Differential Transform Method

Mathematical Problems in Engineering

Chebyshev’s and Legendre’s differential equations’ solutions are solved employing the differentia... more Chebyshev’s and Legendre’s differential equations’ solutions are solved employing the differential transform method (DTM) and the power series method (PSM) in this study. This research shows that this method is efficient and effective in discovering Chebyshev’s and Legendre’s differential equation (DE) series solutions and that it can reduce calculation size when compared to other methods.

Research paper thumbnail of Closed-form solution of a rational difference equation $x_{n+1}=\frac{x_{n-(9 t+8)}}{1+\prod_{k=0}^{6} x_{n-(t+1) k-t} }

In this paper, we obtain the solutions of the difference equation \begin{equation*} x_{n+1}=\frac... more In this paper, we obtain the solutions of the difference equation \begin{equation*} x_{n+1}=\frac{x_{n-(9 t+8)}}{1+\prod_{k=0}^{6} x_{n-(t+1) k-t}}, \end{equation*} where the initials are positive real numbers.

Research paper thumbnail of Mathematica Bohemica

Research paper thumbnail of On the Rational Recursive Sequence xn+1=(A+∑i=0kαixn−i)/(B+∑i=0kβixn−i)

Research paper thumbnail of On the rational recursive sequence x n+1 =A + ∑ i=0 k α i x n-i /∑ i=0 k β i x n-i

Mathematica Bohemica

ABSTRACT

Research paper thumbnail of On the rational recursive sequence x n+1 =(D+αx n +βx n-1 +γc n-2 )/(Ax n +Bx n-1 +Cx n-2 )

Communications on Applied Nonlinear Analysis

Research paper thumbnail of Bifurcation and chaos in a discrete activator-inhibitor system

AIMS Mathematics

In this paper, we explore local dynamic characteristics, bifurcations and control in the discrete... more In this paper, we explore local dynamic characteristics, bifurcations and control in the discrete activator-inhibitor system. More specifically, it is proved that discrete-time activator-inhibitor system has an interior equilibrium solution. Then, by using linear stability theory, local dynamics with different topological classifications for the interior equilibrium solution are investigated. It is investigated that for the interior equilibrium solution, discrete activator-inhibitor system undergoes Neimark-Sacker and flip bifurcations. Further chaos control is studied by the feedback control method. Finally, numerical simulations are presented to validate the obtained theoretical results.

Research paper thumbnail of A General Scheme for Solving Systems of Linear First-Order Differential Equations Based on the Differential Transform Method

Journal of Mathematics

In this study, we develop the differential transform method in a new scheme to solve systems of f... more In this study, we develop the differential transform method in a new scheme to solve systems of first-order differential equations. The differential transform method is a procedure to obtain the coefficients of the Taylor series of the solution of differential and integral equations. So, one can obtain the Taylor series of the solution of an arbitrary order, and hence, the solution of the given equation can be obtained with required accuracy. Here, we first give some basic definitions and properties of the differential transform method, and then, we prove some theorems for solving the linear systems of first order. Then, these theorems of our system are converted to a system of linear algebraic equations whose unknowns are the coefficients of the Taylor series of the solution. Finally, we give some examples to show the accuracy and efficiency of the presented method.

Research paper thumbnail of On the rational difference equation y n + 1 = α 0 y n + α 1 y n − p + α 2 y n − q + α 3 y n − r + α 4 y n − s β 0 y n + β 1 y n − p + β 2 y n − q + β 3 y n − r + β 4 y n − s

Journal of Nonlinear Sciences and Applications, 2017

In this paper, we examine and explore the boundedness, periodicity, and global stability of the p... more In this paper, we examine and explore the boundedness, periodicity, and global stability of the positive solutions of the rational difference equation y n+1 = α

Research paper thumbnail of On the periodicity of the solution of a rational difference equation

Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, 2019

In this paper, some cases on the periodicity of the rational di¤erence equation S n+1 = S n p aS ... more In this paper, some cases on the periodicity of the rational di¤erence equation S n+1 = S n p aS n q + bS n r + cS n s dS n q + eS n r + f S n s ; are investigated, where a, b; c, d, e, f 2 (0; 1). The initial conditions S p , S

Research paper thumbnail of On the dynamics of the nonlinear rational difference equation $ { x_{n+1}} = \frac{{\alpha {x_{n-m}}} \ \ + \ \ \delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}} \ \ { x_{n-l}} \ \ \left({{x_{n-k}} \ \ + \ \ {x_{n-l}}} \ \ \right) }} $

AIMS mathematics, 2022

In this paper, we discuss some qualitative properties of the positive solutions to the following ... more In this paper, we discuss some qualitative properties of the positive solutions to the following rational nonlinear difference equation $ { x_{n+1}} = \frac{{\alpha {x_{n-m}}} \ \ + \ \ \delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}} \ \ { x_{n-l}} \ \ \left({{x_{n-k}} \ \ + \ \ {x_{n-l}}} \ \ \right) }} ,, , n = 0, 1, 2, ... $ where the parameters $ \alpha, \beta, \gamma, \delta \in (0, \infty) ,while, while ,while m, k, l $ are positive integers, such that $ m &lt; k &lt; l. $ The initial conditions $ {x_{-m}}, ..., {x_{-k}}, ..., {x_{-l}}, ..., {x_{-1}}, ..., {x_{0}} $ are arbitrary positive real numbers. We will give some numerical examples to illustrate our results.

Research paper thumbnail of On the dynamics of the nonlinear rational difference equation $ { x_{n+1}} = \frac{{\alpha {x_{n-m}}} \ \ + \ \ \delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}} \ \ { x_{n-l}} \ \ \left({{x_{n-k}} \ \ + \ \ {x_{n-l}}} \ \ \right) }} $

AIMS Mathematics, 2022

In this paper, we discuss some qualitative properties of the positive solutions to the following ... more In this paper, we discuss some qualitative properties of the positive solutions to the following rational nonlinear difference equation $ { x_{n+1}} = \frac{{\alpha {x_{n-m}}} \ \ + \ \ \delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}} \ \ { x_{n-l}} \ \ \left({{x_{n-k}} \ \ + \ \ {x_{n-l}}} \ \ \right) }} ,, , n = 0, 1, 2, ... $ where the parameters $ \alpha, \beta, \gamma, \delta \in (0, \infty) ,while, while ,while m, k, l $ are positive integers, such that $ m < k < l. $ The initial conditions $ {x_{-m}}, ..., {x_{-k}}, ..., {x_{-l}}, ..., {x_{-1}}, ..., {x_{0}} $ are arbitrary positive real numbers. We will give some numerical examples to illustrate our results.

Research paper thumbnail of On the asymptotic behavior of some nonlinear difference equations

Journal of Computational Analysis and Applications, 2019

Research paper thumbnail of Periodicity of the solution of a higher order difference equation

AIP Conference Proceedings, 2018

Research paper thumbnail of Stability of a fractional difference equation of high order

Journal of Nonlinear Sciences and Applications, 2018

In this paper we investigate the local stability, global stability, and boundedness of solutions ... more In this paper we investigate the local stability, global stability, and boundedness of solutions of the recursive sequence x n+1 = x n−p 2 x n−q + a x n−r x n−q + a x n−r , where x −q+k = −a x −r+k

Research paper thumbnail of Periodicity of the solution of difference equation

AIP Conference Proceedings, 2018

Research paper thumbnail of On the Solutions of a System of Third-Order Rational Difference Equations

Discrete Dynamics in Nature and Society, 2018

The structure of the solutions for the system nonlinear difference equations xn+1=ynyn-2/(xn-1+yn... more The structure of the solutions for the system nonlinear difference equations xn+1=ynyn-2/(xn-1+yn-2), yn+1=xnxn-2/(±yn-1±xn-2), n=0,1,…, is clarified in which the initial conditions x-2, x-1, x0, y-2, y-1, y0 are considered as arbitrary positive real numbers. To exemplify the theoretical discussion, some numerical examples are presented.

Research paper thumbnail of The particular solutions of some types of Euler-Cauchy ODE using the differential transform method

Journal of Nonlinear Sciences and Applications, 2018

In this paper, we apply the differential transform method to find the particular solutions of som... more In this paper, we apply the differential transform method to find the particular solutions of some types of Euler-Cauchy ordinary differential equations. The first model is a special case of the nonhomogeneous n th order ordinary differential equations of Euler-Cauchy equation. The second model under consideration in this paper is the nonhomogeneous second order differential equation of Euler-Cauchy equation with a bulge function. This study showed that this method is powerful and efficient in finding the particular solution for Euler-Cauchy ODE and capable of reducing the size of calculations comparing with other methods.

Research paper thumbnail of On the Dynamics of the Solutions of the Rational Recursive Sequences

British Journal of Mathematics & Computer Science, 2015

In this article, we study the periodicity, the boundedness and the global stability of the positi... more In this article, we study the periodicity, the boundedness and the global stability of the positive solutions of the following nonlinear difference equation xn+1 = Axn +Bxn−k + Cxn−l +Dxn−σ + bxnxn−kxn−l dxn−k − exn−l , n = 0, 1, 2, ..... where the coefficients A,B,C,D, b, d, e ∈ (0,∞), while k, l and σ are positive integers. The initial conditions x−σ,..., x−l,..., x−k, ..., x−1, x0 are arbitrary positive real numbers such that k < l < σ. Some numerical examples will be given to illustrate our results.

Research paper thumbnail of On the Dynamics of the Higher Order Nonlinear Rational Difference Equation

Mathematical Sciences Letters, 2014

In this article, we study the periodicity, the boundedness and the global stability of the positi... more In this article, we study the periodicity, the boundedness and the global stability of the positive solutions of the following nonlinear difference equation x n+1 = Ax n + Bx n−k +Cx n−l + Dx n−σ + bx n−k [dx n−k − ex n−l ] , n = 0, 1, 2, ....., where the coefficients A, B,C, D, b, d, e ∈ (0, ∞), while k, l and σ are positive integers. The initial conditions x −σ ,..., x −l ,..., x −k , ..., x −1 , x 0 are arbitrary positive real numbers such that k < l < σ. Some numerical examples will be given to illustrate our results.

Research paper thumbnail of The Solutions of Legendre’s and Chebyshev’s Differential Equations by Using the Differential Transform Method

Mathematical Problems in Engineering

Chebyshev’s and Legendre’s differential equations’ solutions are solved employing the differentia... more Chebyshev’s and Legendre’s differential equations’ solutions are solved employing the differential transform method (DTM) and the power series method (PSM) in this study. This research shows that this method is efficient and effective in discovering Chebyshev’s and Legendre’s differential equation (DE) series solutions and that it can reduce calculation size when compared to other methods.

Research paper thumbnail of Closed-form solution of a rational difference equation $x_{n+1}=\frac{x_{n-(9 t+8)}}{1+\prod_{k=0}^{6} x_{n-(t+1) k-t} }

In this paper, we obtain the solutions of the difference equation \begin{equation*} x_{n+1}=\frac... more In this paper, we obtain the solutions of the difference equation \begin{equation*} x_{n+1}=\frac{x_{n-(9 t+8)}}{1+\prod_{k=0}^{6} x_{n-(t+1) k-t}}, \end{equation*} where the initials are positive real numbers.

Research paper thumbnail of Mathematica Bohemica

Research paper thumbnail of On the Rational Recursive Sequence xn+1=(A+∑i=0kαixn−i)/(B+∑i=0kβixn−i)

Research paper thumbnail of On the rational recursive sequence x n+1 =A + ∑ i=0 k α i x n-i /∑ i=0 k β i x n-i

Mathematica Bohemica

ABSTRACT

Research paper thumbnail of On the rational recursive sequence x n+1 =(D+αx n +βx n-1 +γc n-2 )/(Ax n +Bx n-1 +Cx n-2 )

Communications on Applied Nonlinear Analysis