
Mahmoud Abul-EzSohag University · Department of Mathematics
Mahmoud Abul-Ez
Ph D.; Prof. of Complex and Clifford Analysis
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Publications (62)
Over the last two decades, special matrix functions have become a major area of study for mathematicians and physicists. The famous four Appell hypergeometric matrix functions have received considerable attention by many authors from different points of view. The present paper is devoted to provide further investigations on the two variables second...
In this paper, the coupled homotopy-variational approach (CHVA) based on combining homotopy with the variational approach is applied to solve the nonlinear Duffing equation, and new frequency- amplitude relationships are obtained. The coupled method works very well for the entire range of initial amplitudes and calculates higher-order approximation...
The theory of Bessel functions is a rich subject due to its essential role in providing solutions for differential equations associated with many applications. As fractional calculus has become an efficient and successful tool for analyzing various mathematical and physical problems, the so-called fractional Bessel functions were introduced and stu...
Based on the suggested parameter, a new analytical perturbation technique is presented to obtain highly ordered accurate analytical solutions for nonlinear Duffing oscillator with nonlinearity of high order. Comparing the obtained results with the numerical and other previously published results reveals the usefulness and correctness of the present...
In this work, we investigate and apply higher-order Hamiltonian approach (HA) as one of the novelty techniques to find out the approximate analytical solution for vibrating double-sided quintic nonlinear nano-torsional actuator. Periodic solutions are analytically verified, and consequently, the relationship between the initial amplitude and the na...
This article presents an exhaustive study on the conformable fractional Gauss hypergeometric function (CFGHF). We start by solving the conformable fractional Gauss hypergeometric differential equation (CFGHDE) about the fractional regular singular points x=1 and x=∞. Next, various generating functions of the CFGHF are established. We also develop s...
Along with the theory of bases in function spaces, the existence of a basis is not always guaranteed. The class of power series spaces contains many classical function spaces, and it is of interest to look for a criterion for this class to ensure the existence of bases which can be expressed in an easier form than in the classical case given by Can...
The main objective of this paper is to give a wide study on the conformable fractional Legendre polynomials (CFLPs). This study is assumed to be a generalization and refinement, in an easy way, of the scalar case into the context of the conformable fractional differentiation. We introduce the CFLPs via different generating functions and provide som...
This paper presents a somewhat exhaustive study on the conformable fractional Gauss hypergeometric function (CFGHF). We start by solving the conformable fractional Gauss hypergeometric equation (CFGHE) about the fractional regular singular points $x=1$ and $x=\infty$. Next, various generating functions of the CFGHF are established. We also develop...
Recently, special functions of fractional order calculus have had many applications in various areas of mathematical analysis, physics, probability theory, optimization theory, graph theory, control systems, earth sciences, and engineering. Very recently, Zayed et al. (Mathematics 8:136, 2020) introduced the shifted Legendre-type matrix polynomials...
The main objective of this paper is to give a wide study on the conformable fractional Legendre polynomials (CFLPs). This study is assumed to be a generalization and refinement, in an easy way, of the scalar case into the context of the conformable fractional differentiation. We introduce the CFLPs via different generating functions and provide som...
In this paper, we establish an explicit relation between the growth of the maximum modulus and the Taylor coefficients of entire functions in several complex matrix variables (FSCMVs) in hyperspherical regions. The obtained formulas enable us to compute the growth order and the growth type of some higher dimensional generalizations of the exponenti...
The generalization of Rodrigues’ formula for orthogonal matrix polynomials has attracted the attention of many researchers. This generalization provides new integral and differential representations in addition to new mathematical results that are useful in theoretical and numerical computations. Using a recently studied operational matrix for shif...
Abstract: The study presents an alternative analytical method called Newton Harmonic Balance Method (NHBM) to provide an analytical solution for two nonlinear differential equations that appear in specific dynamics. This method is based on combining Newton’s method and the harmonic balance method. Because the periodic solution is analytically prove...
The main purpose of this paper is to study questions concerning representations of Clifford valued functions by the product bases of Clifford polynomials. By the way we generalize several results from complex analysis to the setting of Clifford analysis.
We introduce a new analytical-approximate method based on the global residue harmonic balance method (GRHBM) to study the oscillation of nonlinear nanoelectromechanical resonators. The proposed method is efficiently implemented and evaluated numerically. An illustrative example demonstrates the validity and applicability of the method, further disc...
We investigate a class of HIV infection models with two kinds of target cells: CD4+ T cells and macrophages. We incorporate three distributed time delays into the models. Moreover, we consider the effect of humoral immunity on the dynamical behavior of the HIV. The viruses are produced from four types of infected cells: short-lived infected CD4+T c...
New relations for the first Appell hypergeometric matrix function are obtained including generating matrix functions, contiguous relations, recursion formulas, differentiations and series.
In recent years, much attention has been paid to the role of classical Special Functions of a real or complex variable in mathematical physics, especially in boundary value problems. In the present paper, we propose a higher-dimensional analogue of the Generalized Bessel Polynomials within Clifford Analysis via a special set of monogenic polynomial...
The main goal of this paper is to investigate the convergence properties of the inverse base of axially monogenic polynomials. These convergence properties proceed from the investigation of the relation between the effectiveness in closed balls, open balls as well as effectiveness for integral functions. The obtained results are the natural general...
In this article, we estimated upper bounds for the growth order and growth type of generalized Hadamard product entire axially monogenic functions. Also, some results concerning the linear substitution are discussed. The obtained results are the natural generalizations of those given in complex setting of one variable to higher dimensions of more t...
A class of HIV infection models are proposed and analyzed. The models incorporate three types of immune cells, CD4\(^{+}\) T cells, macrophages and B cells. We consider also two types of intracellular discrete delays to describe the latent period from the virus contacts an uninfected target cell and the production of new HIV particles. The infectio...
In this article, we obtain an explicit expression of the difference and sum of the unit base in the Clifford analysis case by using the analogy with the complex one. These formulae then lead to the introduction of a class of Bernoulli polynomials in the Clifford analysis setting which are linked to the sum base. The newly introduced Bernoulli polyn...
This paper is devoted to the study of reverse generalized Bessel matrix polynomials (RGBMPs) within complex analysis. This study is assumed to be a generalization and improvement of the scalar case into the matrix setting. We give a definition of the reverse generalized Bessel matrix polynomials Θn(A; B; z), , for parameter (square) matrices A and...
In this paper, the L’Hospital rule for evaluating limits of complex matrix functions is introduced. We present some specific examples on certain matrix functions showing the applicability of our approach.
For entire axially monogenic functions, which are monogenic in the whole space, the lower order and type are defined, as in the complex case, in terms of the maximum modulus of the functions and the Taylor coefficients. The study carried out in this paper bears certain novelty to the familiar literature concerning the Clifford valued functions.
The classical Hadamard three-circles theorem (1896) gives a relation between the maximum absolute values of an analytic function on three concentric circles. More precisely, it asserts that if $f$ is an analytic function in the annulus $\{z \in \mathbb{C}: r_1 < |z| < r_2\}$, $0 < r_1 < r < r_2 < \infty$, and if $M(r_1)$, $M(r_2)$, and $M(r)$ are t...
In the present paper the authors treat two different problems. They start by answering the question posed in [5] concerning the structure of derivative bases, then investigate the convergence properties (the effectiveness) of the generalized derivative and primitive of a given base of polynomials with values in a Clifford algebra. Finally, they stu...
The present paper is a continuation of our paper [4]. The problem to be studied is the non-impact of algebraic property on the convergence properties of the μ-th root base of special monogenic polynomials. These convergence properties are the investigation of the relation between the effectiveness in closed balls of a given base and that of its μ-t...
In this paper, we deals with the study of analytic functions of complex matrices and a standard method for evaluating the order and type of the entire function of complex matrices is given independently of the scalar entire function of complex variables associated with it.
We present some results on the asymptotic growth behavior of entire special monogenic functions. A generalization of the classical Valiron inequality for this class of functions and some basic properties related to the lower order are discussed.
The main goal of this article is to study special polynomial power series expansions for entire axially monogenic functions. In relation to the difference and sum bases of monogenic polynomials, which have link with Bernoulli polynomials, the best possible bounds for the orders of such bases are determined. Moreover the T ρ-property of such differe...
Classical results on the expansion of complex functions in a series of special polynomials (namely inverse similar sets of polynomials) are extended to the Clifford setting. This expansion is shown to be valid in closed balls.
In this paper, we define a similar transposed base of special monogenic polynomials. The convergence properties in several domains in the higher dimensional space of that similar transposed base of monogenic polynomials are investigated. Certain inevitable normalizing conditions have been formulated to be undergone by the given base to ensure the e...
In the present paper the problem of taking the power of a base of special monogenic polynomials is studied, thus leading to a number of re-sults under some additional conditions of associated infinite matrices, related essentially to the so-called algebraicness and Boes condition of these matrices. The obtained results are the extent of generalizat...
In the present article we introduce the similar bases of special monogenic polynomials. The convergence properties and mode of increase of these similar bases are treated.
In this paper, it is shown that certain classes of special monogenic functions cannot be represented by the basic series in the whole space. New definitions for the order of basis of special monogenic polynomials are given together with theorems on representation of classes of special monogenic functions in certain balls and at a point.
In this paper, it is shown that certain classes of special monogenic functions cannot
be represented by the basic series in the whole space. New definitions for the order of basis of
special monogenic polynomials are given together with theorems on representation of classes of
special monogenic functions in certain balls and at a point.
In this paper the problem of taking the square root of bases of special monogenic
polynomials is studied, thus leading to a number of results under some additional conditions of
associated infinite matrices, related essentially to the so-called algebraicness of these matrices.
It is shown that certain classes of special monogenic functions cannot
be represented by the basic series in the whole space. New definitions
for the order of basis of special monogenic polynomials are given,
together with theorems on the representation of classes of special
monogenic functions in certain balls and at a point.
In this paper we present a topology for the spaces of entire functions having finite growth. This topology stems naturally from the order – and – type formalism and gives tools to turn the spaces considered into metrisable spaces. Further results are that they are nuclear Fréchet spaces as well as a basis transforms in such spaces are given. Finall...
This paper is concerned with the extension of the notion of Hadamard product of bases of polynomials in one complex variable to the setting of Clifford analysis. It is proposed to study the effect of the algebraic property of a Hadamard product base of special monogenic polynomials on its convergence properties. Unlike the complex case some of thes...
General methods of nuclear Fréchet spaces in conjunction with the theory of Köthe sequence spaces have been used to obtain a basis criterion for power sequence spaces. How this basis criterion is thus yielding a refinement of Cannon's criterion in the case of space of holomorphic functions is illustrated here. Hence, we give a characterization of t...
Effectiveness properties of basic sets in two complex variables in closed balls for the class of functions regular in these balls are discussed. Using a topological approach, some results given by B. Cannon and W. F. Newns are extended to nuclear Fréchet spaces.
According to the definition of the exponential function in Clifford analysis, the exponential base of special monogenic polynomials is introduced. With this in hand, some convergence properties of this base are obtained.
In the present paper we establish a result on teh representation of special monogenic functions by the product set of special monogenic polynomials. This generalises to the Clifford setting the analogue result in the complex case. We derive also bounds for the order of the product set of special monogenic polynomials. The constrcted examples show t...
A criterion is proved for basis transforms in nuclear Fréchet spaces. Special attention goes to spaces which are isomorphic to power series spaces, because this class of spaces contains many classical function spaces and the criterion for this class can be expressed in an easier form than in the classical case. Also a criterion is given in which ca...
This paper is concerned with the extension of the theory of basic sets of polynomials in one complex variable, as introduced by J. M. Whittaker and B. Cannon, to the setting of Clifford analysis. This is the natural generalization of complex analysis to Euclidean space of dimension larger than two, where the regular functions have values in a Cliff...
Order and type of functions and sets of polynomials of two complex matrices are given by order and type of functions and sets of polynomials of original matrices when linear substitutions are carried out.