# Mahmoud El-BoraiAlexandria University | AU · Department of Mathematics

Mahmoud El-Borai

Ph.D

## About

130

Publications

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Introduction

Mahmoud El-Borai currently works at the Department of Mathematics, Alexandria University. Mahmoud does research in Applied Mathematics. Their current project is 'Fractional boundary value problems'.

## Publications

Publications (130)

In this paper, we introduce the mild solution for a new class of noninstantaneous and nonlocal
impulsive Hilfer fractional stochastic integrodifferential equations with fractional Brownian motion and
Poisson jumps. The existence of the mild solution is derived for the considered system by using fractional
calculus, stochastic analysis and Sadovskii...

The aim of this paper is investigating and solvability of the nonlinear integral Equation due to Urysohn, in the space of p th Lebesgue integrable functions on R N , (L p (R N)). The Urysohn integral equations are enjoying interest among mathematicians , physicists and engineers. We try to assume the sufficient conditions under which the existence...

Using the technique of a suitable measure of non-compactness and the Darbo fixed point, we investigate the existence of nondecrasing solutions for a quadratic nonlinear -integral equation of convolution type. Our investigations take place in, the Banach space of real and continuous functions defined on , . An example is also discussed to indicate t...

Using the technique of a suitable measure of non-compactness and the Darbo fixed point theorem, we investigate the existence of a nonlinear functional integral equation of Urysohn type in the space of Lebesgue integrable functions L p (R N). In this space, we show that our functional-integral equation has at least one solution. Finally an example i...

We prove an existence theorem for a nonlinear quadratic integral equation of fractional order, in the Banach space of real functions defined and continuous on a closed interval. This equation contains as a special case numerous integral equation studied by other authors. Finally, we give an example for indicating the natural realizations of our abs...

We introduce the investigation of approximate controllability for a new class of nonlocal and noninstantaneous impulsive Hilfer fractional neutral stochastic integrodifferential equations with fractional Brownian motion. An appropriate set of sufficient conditions is derived for the considered system to be approximately controllable. For the main r...

Quantitative and qualitative analysis of the Averaging methods for the parabolic partial differential equation appears as an exciting field of the investigation. In this paper, we generalize some known results due to Krol on the averaging methods and use them to solve the parabolic partial differential equation.

Averaging method of the fractional general partial differential equations and a special case of these equations are studied, without any restrictions on the characteristic forms of the partial differential operators. We use the parabolic transform, existence and stability results can be obtained.

Existence and construction of the solutions of some Markov moment problems are discussed. Starting from the moments of a solution, one recalls a method of recovering this solution, also solving approximately related systems with infinite many nonlinear equations and infinite unknowns. This is the first aim of this review paper. Extension of linear...

Sobolev-type nonlocal fractional differential systems with Clarke’s subdifferential are studied. Sufficient conditions for controllability and constrained controllability for Sobolev-type nonlocal fractional differential systems with Clarke’s subdifferential are established, where the time fractional derivative is the Hilfer derivative. An example...

Sufficient conditions for exact null controllability of semilinear fractional stochastic delay integro-differential equations in Hilbert space are established. The required results are obtained based on fractional calculus, semigroup theory, Schauder’s fixed point theorem and stochastic analysis. In the end, an example is given to show the applicat...

By using stochastic analysis, fractional analysis, compact semigroups and the Schauder fixed-point theorem, we discuss the approximate boundary controllability of a nonlocal Hilfer fractional stochastic differential system with fractional Brownian motion and a Poisson jump. In addition, we establish the sufficient conditions for exact null controll...

In this paper, we discuss the existence of solutions for a stochastic initial value problem of Hyprid fractional dierential equations of Hadamard type given by where HD is the Hadamard fractional derivative, and is the Hadamard fractional integral and be such that are investigated. The fractional calculus and stochastic analysis techniques are used...

In this paper, the existence and uniqueness about the solution for a class of abstract stochastic fractional-order differential equations where in and are given functions, are investigated, where the fractional derivative is described in Caputo sense. The fractional calculus, stochastic analysis techniques and the standard $Picard's$ iteration meth...

In this paper, we shall discuss the uniqueness ”pathwise uniqueness” of the solutions of stochastic partial differential equations (SPDEs) with non-local initial condition,We shall use the Yamada-Watanabe condition for ”pathwise uniqueness” of the solutions of the stochastic differential equation; this condition is weaker than the usual Lipschitz c...

Fractional integro-differential equations arise in the mathematical modeling of various physical phenomena like heat conduction in materials with memory, diffusion processes, etc. In this manuscript, we prove the existence of mild solution for Sobolev type nonlinear impulsive delay integro-differential system with fractional order 1 < q < 2. We est...

Existence and controllability results for nonlinear Hilfer fractional differential equations are studied. Sufficient conditions for existence and approximate controllability for Sobolev-type impulsive fractional differential equations are established, where the time fractional derivative is the Hilfer derivative. An example for Sobolev-type Hilfer...

In this paper, a nonlinear fractional parabolic stochastic integro-partial differential equations with nonlocal effects driven by a fractional Brownian motion is considered. In particular, first we have formulated the suitable solution form for the fractional partial differential equations with nonlocal effects driven by fractional Brownian motion...

In this paper, we investigate the existence of mild solutions of Hilfer fractional stochastic integro-differential equations with nonlocal conditions. The main results are obtained by using fractional calculus, semigroups and Sadovskii fixed point theorem. In the end, an example is given to illustrate the obtained results.

In this paper, the existence of the solution of the parabolic partial fractional differential equation is studied and the solution bound estimate which are used to prove the existence of the solution of the optimal control problem in a Banach space is also studied, then apply the classical control theory to parabolic partial differential equation i...

Nonlinear Schrödinger type equations arise from a wide variety of fields, such as fluids, nonlinear optics, the theory of deep water waves, plasma physics, and so on. In this paper, two integration schemes are employed to obtain solitons, periodic waves and other forms of solutions of the coupled nonlinear Schrödinger type equations. The two scheme...

We present a necessary optimality conditions for a class of optimal control problems. The dynamical control system involves integer and fractional order derivatives and the final time is free. Optimality conditions are obtained. Feedback control laws for linear dynamic system are obtained.

In this paper, the Adomian decomposition method (ADM) is applied to obtain the approximate solution of a mathematical model of carcinogenesis which is Riccati differential equation. The numerical solution obtained by this way have been compared with the exact solution. This comparison show that the (ADM) is a powerful method for solving this differ...

Cantor ternary set, which was named after George Cantor (1845-1918) although he was not the first to discover it, has gained much popularity in the mathematical community for it possesses riveting properties and is regarded as a gateway to pause and reflect upon some important concepts in mathematical analysis. The aim of this five-chapter thesis i...

A nonlinear transmission line (NLTL) is comprised of a transmission line periodically loaded with varactors, where the capacitance nonlinearity arises from the variable depletion layer width, which depends both on the DC and AC voltages of the propagating wave. An equivalent circuit model of NLTL is discussed analytically, in this article, and diff...

This paper obtains optical soliton solutions to the governing nonlinear Schrödinger's equation that is studied with spatio-temporal dispersion. The integration algorithm that is employed in this paper is the modified simple equation method. This leads to dark and singular soliton solutions that are valuable in the field of optoelectronics. The soli...

In this paper, we apply the extended Kudryashov method to a nonlinear Schrödinger type equation called the Kundu–Eckhaus equation or the Eckhaus equation which was independently introduced by Wiktor Eckhaus and by Anjan Kundu in 1984–1985 to model the propagation of waves in dispersive media. The proposed method is direct, effective and takes full...

It is well known that the Cauchy problem for elliptic partial differential equations is ill-posed. The question, which arises, how a priori knowledge about solutions can bring about stability? A parabolic transform is defined to discuss the stability of the Cauchy problem for some stochastic partial differential equations under a priori knowledge a...

We have investigated the motion of the time-independent flow of a viscous incompressible fluid passing a rectangular plate. The cross-section of this plate is considered to be in the form of a rectangle with dimensions 2W transverse to the flow and T along the flow. The fluid is assumed to be steady flow of water with incident velocity
equals V0exp...

5-Amino-4-heterylazo-3-phenyl-1H-pyrazoles (2a-d) were diazotized and coupled with malononitrile to give pyrazoloazo malononitrile which by heating in glacial acetic acid gave novel pyrazolo[5,1-c][1,2,4]triazine dyes (3a-d). Also, some diazopyrazolyl pyrazolone dyes (4a-h) were synthesized by diazotization of 2a-d and coupled with some pyrazolone...

Three series of mono and disazo disperse dyes were synthesized from 2-amino-4-(pyridin-3-yl) thiazole. The structure of the synthesized dyes was confirmed by elemental analysis, ultraviolet-visible, infrared, proton and carbon nuclear magnetic resonance and mass spectroscopy. The dyeing parameters, perspiration, wash and light fastness of eighteen...

Let A be a linear closed operator defined on a dense set in a Banach space E to E. In this note it is supposed that A is the generator of α − times integrated semi group, where α is a positive number. The abstract Cauchy problem of the fractional differential equation: d β u(τ) dt β == Au t + F t , With the initial conditionu 0 ∈ E, is studied, whe...

Abotract: This note is devoted to the study of an inverse Cauchy problem in a Hilbert space H for the abstract fractional differential equation of the form:), () () (=) (t g t f t u A dt t u d with the nonlocal initial condition:), (= (0) 1 = 0 k k p k t u c u u and the overdetermination condition:), (=)), ((t w v t u where (.,.) is the i...

Let A be a linear closed operator defined on a dense set in a Banach space E to E. In this note it is supposed that A is the generator of α − times integrated semi group, where α is a positive number. The abstract Cauchy problem of the fractional differential equation: d β u(τ) dt β == Au t + F t , With the initial conditionu 0 ∈ E, is studied, whe...

This note is devoted to the study of an inverse Cauchy problem in a Hubert space H for the abstract fractional differential equation of the form: dα u(t)/dtα = Au(t) +f(t) g(t), with the nonlocal initial condition: and the overdetermination condition: (u(t),v) = w(t), where (.,.) is the inner product in H, f is a real unknown function w is a given...

In this paper we are going to study the partial differential equation With the non-local condition Djtu(x,0)=fj(x); (j=,0),1,...,k-1 Where; • L is an elliptic partial differential operator, • Lij;j=1,..., is a family of partial differential operator with bounded operator coefficient in a suitable functional space.

In this paper, we prove the existence and uniqueness of a nonlinear perturbed stochastic fractional integro-differential equation of Volterra-Itô type involving nonlocal initial condition by using the theory of admissibility of integral operator and Banach fixed-point principle. Also the stability and boundedness of the second moments of the stocha...

Some classes of stochastic fractional integro-differential equations involving nonlocal initial condition are investigated. The theory of admissibility of integral operator and Banach fixed-point principle are used to establish the existence and uniqueness of stochastic solution. The boundedness and asymptotic behavior of the stochastic solution as...

Adomian decomposition method (ADM) is applied to approximately solve stochastic fractional integro-differential equations involving nonlocal initial condition. The convergence of the ADM for the considered problem is proved. The mean square error between approximate solution and accurate solution is also given.

An efficient one-pot synthesis in multi-component system (MRCs) for the preparation of pyrazolo[3,4-b]pyridine derivatives from the reaction of 5-amino-1-phenyl-3-(pyridin-3-yl)-1H-pyrazole with 4-anisaldehyde and p-substituted β-ketonitriles or with pyruvic acid and some aromatic aldehydes in acetic acid medium. The reactions were carried out by t...

Diazotized aryl amines were coupled with 3-substituted 5-amino pyrazoles to produce a series of novel 3-substituted 5-amino-4-arylazopyrazoles. Also, 3-substituted 5-amino-pyrazoles were diazotized and coupled with different phenols to give the corresponding novel 3-substituted 5-aryl azo pyrazoles. These dyes were characterized by elemental analys...

This paper deals with a class of backward stochastic differential equations (BSDE in short) under Lipschitz and monotonicity coefficients, the authors obtain the existence and uniqueness of solution to BSDE and estimate the difference between two solutions in terms of the difference between the data (Comparison theorem).

The tanh method is a powerful solution method; various extension forms of the tanh method have been developed with a computerized symbolic computation and is used for constructing the exact travelling wave solutions, of coupled nonlinear equations arising in physics. The obtained solutions include solitons, kinks and plane periodic solutions. First...

In this paper, the extended multiple Riccati equations expansion method has been used to construct a series of double soliton-like solutions, double triangular function solutions and complexiton soliton solutions for nonlinear partial differential equations , we obtain many new types of complexiton soliton solutions, i.e various combination of trig...

We consider the Cauchy problem for an abstract stochastic delay differential equation driven by fractional Brownian motion with the Hurst parameter H>12. We prove the existence and uniqueness for this problem, when the coefficients have enough regularity, the diffusion coefficient is bounded away from zero and the coefficients are smooth functions...

In this paper the solutions of some evolution equations with fractional orders in a Banach space are considered. Conditions are given which ensure the existence of a resolvent operator for an evolution equation in a Banach space.

The time-independent flow of a viscous, incompressible fluid past a rectangular plate is discussed. Accordingly, all time derivatives are set equal to zero in all equations. We have considered the case where the cross-section of this plate is a rectangle with dimensions 2 transverse to the flow and along the flow. The fluid is assumed to be steady...

We have investigated the motion of a stretched elastic circular membrane which is subjected to a restorative force proportional to the velocity. The wave equation which describes the vibrations in this case is written in polar form. The perturbation method is applied together with the usual method of separation of variables. The oscillations are th...

We study nonlinear singular integral equations of Volterra type in the Banach space of real functions defined and continuous on a bounded and closed interval. Using a suitable measure of noncompactness we prove the existence of monotonic solutions of the considered equation and its generalization. We illustrate our existence results by numerical ex...

The existence and uniqueness solution of the nonlinear integral equation of Hammerstein type with discontinuous kernel are discussed. The normality and continuity of the integral operator are proved. Toeplitz matrix method is used, as a numerical method, to obtain a nonlinear system of algebraic equations. Also, many important theorems related to t...

In this article, we prove the existence of optimal mild solutions for linear fractional evolution equations with an analytic semigroup in a Banach space. As in [16], we use the Gelfand-Shilov principle to prove existence, and then the Bochner almost periodicity condition to show that solutions are weakly almost periodic. As an application, we study...

In this paper, our basic tools are the use of Gelfand-Shilov principle and fractional powers of operators to establish the existence and unique-ness of local mild then local classical solutions of a class of nonlinear fractional integrodifferential equations in Banach space with analytic semigroups, see Bahuguna [2]. As an application, we study non...

In this paper, which is continuation of [1], As in [5, 10], we use the theory of fractional calculus to establish the existence and uniqueness of almost periodic solutions of a class of nonlinear fractional differential equations with analytic semigroup in Banach space, and we prove under suitable conditions that their optimal mild solutions are al...

A method for solving some nonlinear fractional parabolic partial differential equations is considered.Using this method, new exact solutions are obtained. By introducing suitable transformation, we obtain a system of fractional differential equations. This system will generate the exact solutions. The efficiency of the considered method can be demo...

We consider the existence and uniqueness solution of nonlinear integral equation of the second kind of type Volterra-Hammerstein. Also, the normality and continuity of the integral operator are discussed. A numerical method is used to obtain a system of nonlinear integral equations in position. The solution is obtained, and many applications in one...

New pyridyl-5-one derivatives were synthesized by reactions of 3-(pyridin-3-yl)isoxazol-5(4//)-one (I) with various aryl aldehydes to yield 4-(arylidene-3-(pyridin-3-yl)isoxazole-5(4//)-ones (Ila-c). The reactions of these compounds with hydrazine hydrate gave new heterocyclic compounds (IIIa,b). Compound (I) reacted with phosphorus oxychloride to...

In this article, A numerical method is used to solve the two dimensionalFredholm integral equation of the second kind with weak singular kernel using the Toeplitz matrix and product Nystrom method. The numerical results given in this paper are computed using maple 8. The error, in each case, is computed. In this article, a numerical method is used...

Here, A nonlinear Volterra integral equation (NVIE) is considered, where the existence and uniqueness solution is discussed and proved. Using the product trapezoidal rule, the Toeplitz matrix and the product Nystrom method, the numerical solutions of the (NIE) is obtained. Also, we consider an application in the viscoelastic nonlinear material, whi...

Two fixed point theorems proved by B. C. Dhage [5], [6] are used to prove the ex-istence theorems for some integro-differential equations of fractional orders involving the mixed generalized Lipschitz and Carathéodory conditions.

In this paper, we shall study the approximate solutions of the Cauchy problem, (Dα 0+u)(x, t )= n

We study nonlinear singular integral equation of Volterra type in Banach space of real functions defined and continuous on a bounded and closed interval. Using a suitable measure of noncompactness we prove the existence of monotonic solutions. Also a generalized result is taken in the consideration

Schauder’s fixed point theorem is used to establish an existence result for an infinite system of singular integral equations of the form x i (t)=a i (t)+∫ 0 t (t-s) -α f i (s,x 1 (s),x 2 (s),...)ds, where i=1,2,..., α∈(0,1) and t∈I=[0,T]. The result obtained is applied to show the solvability of an infinite system of differential equation of fract...

Some classes of stochastic fractional integro-partial differential equations are in-vestigated. Mild solutions of the nonlocal Cauchy problem for the considered classes are studied. The Leray–Schauder principle is used to establish the exis-tence of stochastic solutions. The uniqueness of the solution of the considered problem is also studied under...

Using a suitable measure of noncompactness and Darbo’s Fixed
point theorem we establish an existence theorem of nonlinear quadratic integral equation of fractional order with supremum in Banach space of real functions defined and continuous on a bounded and closed interval (C [0, T]).

In this note the solutions of some integrodifferential equations with fractional orders in a Banach space are considered.Conditions are given which ensure the existence of a resolvent operator for an integrodiffer-ential equation in a Banach space.