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Acta mathematicae Applicatae Sinica 2016年第4期32卷 793-812页

作者： elsayed m.e.zayed Yaser A.AmeR Abdul-Ghani AL-NOWeHY mathematics Department Faculty of SciencesZagazig University mathematics Department Faculty of EducationAin Shams University

In this article, the fractional derivatives in the sense of the modified Riemann-Liouville derivatives together with the modified simple equation method and the multiple exp-function method are employed for constructing the exact solutions and the solitary wave solutions for the nonlinear time fractional Sharma-Tasso- Olver equation. With help of maple, we can get exact explicit l-wave, 2-wave and 3-wave solutions, which include l-soliton, 2-soliton and 3-soliton type solutions if we use the multiple exp-function method while we can get only exact explicit l-wave solution including l-soliton type solution if we use the modified simple equation method. Two cases with specific values of the involved parameters are plotted for each 2-wave and 3-wave solutions.

关键词： modified simple equation method multiple exp-function method nonlinear fractional partialdifferential equations exact solution solitary wave solutions 1-wave solution 2-wave solution 3-wave solution

New sub-equation method to construct solitons and other solutions for perturbed nonlinear Schrödinger equation with Kerr law nonlinearity in optical fiber materials

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Journal of Ocean engineering and Science 2019年第1期4卷 14-23页

作者： elsayed m.e.zayed Abdul-Ghani Al-Nowehy Reham m.A.Shohib Department of mathematics Faculty of ScienceZagazig UniversityZagazigEgypt Department of mathematics Faculty of Education and ScienceTaiz UniversityTaizYemen

In a previous work,zayed and Al-Nowehy have applied the Riccati equation mapping method combined with the generalized extended(G/G)-expansion method and found new exact solutions of the nonlinear KPP *** the present article,we propose a different method,namely,a new sub-equation method consists of the Riccati equation mapping method and the(G/G,1/G)-expansion method to find new exact solutions of the perturbed nonlinear Schrödinger equation with Kerr law nonlinearity in optical fiber *** proposed method is not found ***,trigonometric and rational function solutions are *** solutions of the generalized Riccati equation are presented for the first time which are not reported *** solutions of the given nonlinear equation can be applied in ocean engineering for calculating the height of tides in the ocean.

关键词： New sub-equation method (G/G,1/G)-expansion method Generalized Riccati equation mapping method Perturbed nonlinear Schrödinger equation exact solutions.

The 3D Inverse Problem of the Wave equation for a General multi-connected Vibrating membrane with a Finite Number of Piecewise Smooth Boundary Conditions

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Acta mathematica Sinica,english Series 2005年第4期21卷 733-752页

作者： e.m.e.zayed mathematics Department Faculty of ScienceZagazig UniversityZagazig.Egypt

The trace of the wave kernel μ（t） =∑ω=1^∞ exp（-iteω^1/2）, where {eω}ω^∞=1 are the eigenvalues of the negative Laplacian -△↓2 = -∑k^3=1 （δ/δxk）^2 in the （x^1, x^2, x^3）-space, is studied for a variety of bounded domains, where -∞ 〈 t 〈 ∞ and i= √-1. The dependence of μ （t） on the connectivity of bounded domains and the Dirichlet, Neumann and Robin boundary conditions are analyzed. Particular attention is given for a multi-connected vibrating membrane Ω in Ra surrounded by simply connected bounded domains Ω j with smooth bounding surfaces S j （j = 1,……, n）, where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Si^＊（i = 1 ＋ kj-1,……, kj） of the bounding surfaces S j are considered, such that S j = Ui-1＋kj-1^kj Si^＊, where k0=0. The basic problem is to extract information on the geometry Ω by using the wave equation approach from a complete knowledge of its eigenvalues. Some geometrical quantities of Ω （e.g. the volume, the surface area, the mean curvuture and the Gaussian curvature） are determined from the asymptotic expansion ofexpansion of μ（t） for small │t│.

关键词： Inverse problem Wave kernal eigenvalues Greens function multi-connected vibrating membrane Negative Laplacian Heat kernel

INVeRSe PROBLemS FOR A GeNeRAL mULTI-CONNeCTeD BOUNDeD DRUm WITH APPLICATIONS IN PHYSICS

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Acta mathematica Scientia 2003年第1期23卷 104-116页

作者： e.m.e.zayed mathematics Department Faculty of Science Zagazig University Zagazig Egypt

This paper studies the influence of a finite container on an ideal *** trace of the heat kernel (t) =exp, where are the eigenvalues of the negative Laplacian -in Rn(n = 2 or 3), is studied for a general multi-connected bounded drum ft which is surrounded by simply connected bounded domains Ωi with smooth boundaries Ωi(i = 1,… ,m) where the Dirichlet, Neumann and Robin boundary conditions on Ωi(i = 1,…,m) are considered. Some geometrical properties of Ω are determined. The thermodynamic quantities for an ideal gas enclosed in Ω are examined by using the asymptotic expansions of (t) for short-time t. It is shown that the ideal gas can not feel the shape of its container Ω, although it can feel some geometrical properties of it.

关键词： Inverse problem heat kernel eigenvalues an ideal gas multi-connected bounded domain

Asymptotic expansions of the Heat Kernel of the Laplacian for General Annular Bounded Domains with Robin Boundary Conditions：Further Results

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Acta mathematica Sinica,english Series 2003年第4期19卷 679-694页

作者： e.m.e.zayed Departmentofmathematics FacultyofScienceZagazigUniversityZagazigEgypt

The asymptotic expansions of the trace of the heat kernel Θ(t) = Σ_(v =1)~∞exp(-tλ_v) for small positive t, where {λ_v} are the eigenvalues of the negative Laplacian-△_n = -Σ_(i = 1)~n(partial deriv/(partial deriv)x^i)~2 in R^n (n = 2 or 3), are studied for ageneral annular bounded domain Ω with a smooth inner boundary (partial deriv)Ω_1 and a smoothouter boundary (partial deriv)Ω_2, where a finite number of piecewise smooth Robin boundaryconditions (partial deriv/(partial deriv)n_j + γ_j)φ = 0 on the components Γ_j(j = 1, …, k) of(partial deriv)Ω_1 and on the components Γ_j(j = k + 1, …, m) of (partial deriv)Ω_2 areconsidered such that (partial deriv)Ω_1 = ∪_(j = 1)~kΓ_j and (partial deriv)Ω_2 = ∪_(j = k +1)~mΓ_j and where the coefficients γ_j(j = 1, …, m) are piecewise smooth positive functions. Someapplications of Θ(t) for an ideal gas enclosed in the general annular bounded domain Ω are *** results are also obtained.

关键词： inverse problem heat kernel eigenvalues Robin boundary conditions classical ideal gas

The Asymptotics of the Two-Dimensional Wave equation for a General multi-Connected Vibrating membrane with Piecewise Smooth Robin Boundary Conditions

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Acta mathematica Sinica,english Series 2004年第2期20卷 209-222页

作者： e.m.e.zayed Department of mathematics Faculty of ScienceZagazig UniversityZagazigEgypt

The asymptotic expansion for small |t| of the trace of the wave kernel , where and are the eigenvalues of the negative Laplacian in the (x 2,x 2)-plane, is studied for a multi-connected vibrating membrane ? in R 2 surrounded by simply connected bounded domains ? j with smooth boundaries ?? j (j = 1, ..., n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components Γ i (i = 1+k j?1, ..., k j ) of the boundaries ?? j are considered, such that and k 0 = 0. The basic problem is to extract information on the geometry of ? using the wave equation approach. Some geometric quantities of ? (e. g. the area of ?, the total lengths of its boundary, the curvature of its boundary, the number of the holes of ?, etc.) are determined from the asymptotic expansion of the trace of the wave kernel for small |t|.

关键词： Inverse problem Wave kernel eigenvalues Robin boundary conditions Vibrating membrane Hearing the shape of a drum

DNA Dynamics Studied Using the Homogeneous Balance method

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Chinese Physics Letters 2012年第8期29卷 9-11页

作者： e.m.e.zayed A.H.Arnous mathematics Department Faculty of ScienceZagazig UniversityP.O.Box 44519ZagazigEgypt

We employ the homogeneous balance method to construct the traveling waves of the nonlinear vibrational dynamics modeling of *** new explicit forms of traveling waves are *** is shown that this method provides us with a powerful mathematical tool for solving nonlinear evolution equations in mathematical *** and weaknesses of the proposed method are discussed.

关键词： traveling nonlinear method

The Asymptotics of the Heat Semigroup for a General Bounded Domain with mixed Boundary Conditions

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Acta mathematica Sinica,english Series 2000年第4期16卷 627-636页

作者： e. m. e. zayed mathematics Department. Faculty of Science. Zagazig University. Zagazig. egypt

Small-time asymptotics of the trace of the heat semigroup θ（t）=Σv=1x exp（-tμv）. where {μv} are the eigenvalues of the uegative Laplacian -Δ= -Σβ=12（/xβ）2 in the （x1, x2）-plane. is studied for a general bounded domain Ω with a smooth boundary Ω. where a finite number of Dirichlet. Neumann and Robin boundary conditions, on the piecewise smooth parts Γi（i=1, ..., n） of )Ω such that)Ω=∪i=1nΓ are considered. Some geometrical properties associated with Ω are determined

关键词： Heat semigroup Inverse problem mixed boundary conditions eigenvalues

SOme OSCILLATION CRITeRIA FOR SeCOND ORDeR NONLINeAR FUNCTIONAL ORDINARY DIFFeReNTIAL eQUATIONS

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Acta mathematica Scientia 2007年第3期27卷 602-610页

作者： e.m.e.zayed m.A.el-moneam mathematics Department Faculty of ScienceZagazig UniversityZagazigEgypt

The main objective of this article is to study the oscillatory behavior of the solutions of the following nonlinear functional differential equations（a（t）x＇（t））＇＋δ1p（t）x＇（t）＋δ2q（t）f（x（g（t））） = 0,for 0 ≤ to≤ t, where 51 = ：el and δ±1. The functions p,q,g ： [t0, ∞） → R, f ： R → are continuous, a（t）〉 0,p（t） ≥0,q（t）〉 0 for t ≥ to,lirn g（t） = ∞, and q is not identically zero on any subinterval of [to, ∞）. moreover, the functions q（t）, g（t）, and a（t） are continuously differentiable.

关键词： Oscillatory and nonoscillatory solutions nonlinear functional differential equations

exact Solutions of Nonlinear evolution equations in mathematical Physics Using the modified Simple equation method

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Chinese Physics Letters 2012年第6期29卷 1-4页

作者： e.m.e.zayed S.A.Hoda Ibrahim Department of mathematics Faculty of ScienceZagazig UniversityP.O.Box 44519ZagazigEgypt

The modified simple equation method is employed to construct the exact solutions involving parameters of nonlinear evolution equations via the (1+1)-dimensional modified KdV equation,and the (1+1)-dimensional reaction-diffusion *** these parameters are taken to be special values,the solitary wave solutions are derived from the exact *** is shown that the proposed method provides a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

关键词： equation. exact equation