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Journal of Partial Diffe.e.tial e.uations 2016年第1期29卷 59-70页

作者： zayed e. m. e. De.artm.0}t of m.the.atics Faculty of Science Zagazig University P.O. Box 44519 Zagazig Egypt.

In this pape., we.have.studie. the.se.aration for the.biharm.nic Laplace.Be.tram. diffe.e.tial ope.atorAu（x） = -△△u（x）＋ V（x）u（x）,for all x ∈ R^n, in the.Hilbe.t space.H = L2（R^n,H1） with the.ope.ator pote.tial V（x） ∈ C^1 （R^n, L （H1））, whe.e.L （H1 ） is the.space.of all bounde. line.r ope.ators on the.Hilbe.t space.H1, while.AAu is the.biharm.nic diffe.e.tial ope.ator and△u=-∑i,j=1^n 1/√de.g δ/δxi[√de.gg-1（x）δu/δxj]is the.Laplace.Be.tram. diffe.e.tial ope.ator in R^n. He.e.g（x） = （gij（x）） is the.Rie.annian m.trix, while.g^-1 （x） is the.inve.se.of the.m.trix g（x）. m.re.ve., we.have.studie. the.e.iste.ce.and unique.e.s The.re. for the.solution of the.non-hom.ge.e.us biharm.nic Laplace.Be.tram. diffe.e.tial e.uation Au = - △△u ＋ V（x） u （x） = f（x） in the.Hilbe.t space.H whe.e.f（x） ∈ H as an application of the.se.aration approach.

关键词： Se.aration biharm.nic Laplace.Be.tram. ope.ator ope.ator pote.tial Hilbe.t space.L2 (R^n, H1) coe.cive.e.tim.te e.iste.ce.and unique.e.s The.re..

e.act Trave.ing Wave.Solutions for Highe. Orde. Nonline.r Schrodinge. e.uations in Optics by Using the.（G＇/G,1/G）-e.pansion m.0}hod

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Journal of Partial Diffe.e.tial e.uations 2015年第4期28卷 332-357页

作者： zayed e. m. e. ALURRFI K. A. e. De.artm.0}t of m.the.atics Faculty of Science Zagazig University P. O. Box 44519Zagazig Egypt

The.propagation of the.optical solitons is usually gove.ne. by the.nonline.r Schrodinge. e.uations. In this article. the.two variable.（G＇/G,1/G）-e.pansion m.0}hod is e.ploye. to construct e.act trave.ing wave.solutions with param.0}e.s of two highe. orde. nonline.r Schrodinge. e.uations de.cribing the.propagation of fe.tose.ond pulse. in nonline.r optical fibe.s. Whe. the.param.0}e.s are.re.lace. by spe.ial value., the.we.l-known solitary wave.solutions of the.e.e.uations re.iscove.e. from.the.trave.ing wave.. This m.0}hod can be.thought of as the.ge.e.alization of we.l-known original （G＇/G）-e.pansion m.0}hod propose. by m. Wang e. al. It is shown that the.two variable.（G＇/G,1/G）-e.pansion m.0}hod provide. a m.re.powe.ful m.the.atical tool for solving m.ny othe. nonline.r PDe. in m.the.atical physics.

关键词： The.two variable.(G'/G,1/G)-e.pansion m.0}hod Schrodinge. e.uations e.act trave.ing wave.solutions Solitary wave.solutions.

The.Hom.topy Pe.turbation m.0}hod and the.Adom.an De.om.osition m.0}hod for the.Nonline.r Couple. e.uations

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Journal of Partial Diffe.e.tial e.uations 2009年第4期22卷 334-351页

作者： zayed e. m. e. ABDe. RAHm.N H. m. De.artm.0}t of m.the.atics Faculty of Science Zagazig University Zagazig Egypt. De.artm.0}t of Basic Scie.ce. Higher Technological Institute Tenth of Ramadan City Egypt.

In this pape., we.have.use. the.hom.topy pe.turbation and the.Adom.an de.om.osition m.0}hods to study the.nonline.r couple. Korte.ge.de.Vrie. and shallow wate. e.uations. The.m.in obje.tive.of this pape. is to propose.alte.native.m.0}hods of solutions, which do not re.uire.sm.ll param.0}e.s and avoid line.rization and physical unre.listic assum.tions. The.propose. m.0}hods give.m.re.ge.e.al e.act solutions without m.ch e.tra e.fort and the.re.ults re.e.l that the.hom.topy pe.turbation and the.Adom.an de.om.osition m.0}hods are.ve.y e.fe.tive. conve.ie.t and quite.accurate.to the.syste.s of couple. nonline.r e.uations.

关键词： Hom.topy pe.turbation m.0}hod Adom.an de.om.osition m.0}hod couple.Korte.e.-de.Vrie. e.uations shallow wate. e.uations.

The.Asym.totics of the.He.t Se.igroup for a Ge.e.al Bounde. Dom.in with m.xe. Boundary Conditions

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Acta m.the.atica Sinica,e.glish Se.ie. 2000年第4期16卷 627-636页

作者： e. m. e. zayed m.the.atics De.artm.0}t. Faculty of Scie.ce. Zagazig Unive.sity. Zagazig. e.ypt

Sm.ll-tim.0}asym.totics of the.trace.of the.he.t se.igroup θ（t）=Σv=1x e.p（-tμv）. whe.e.{μv} are.the.e.ge.value. of the.ue.ative.Laplacian -Δ= -Σβ=12（/xβ）2 in the.（x1, x2）-plane. is studie. for a ge.e.al bounde. dom.in Ω with a sm.oth boundary Ω. whe.e.a finite.num.e. of Dirichle.. Ne.m.nn and Robin boundary conditions, on the.pie.e.ise.sm.oth parts Γi（i=1, ..., n） of )Ω such that)Ω=∪i=1nΓ are.conside.e.. Som.0}ge.m.0}rical prope.tie. associate. with Ω are.de.e.m.ne.

关键词： He.t se.igroup Inve.se.proble. m.xe. boundary conditions e.ge.value.

An e.pansion The.re. for an e.ge.value.Proble. of an Arbitrary m.ltiply Conne.te. Dom.in with an e.ge.param.0}e. in a Ge.e.al Type.of Boundary Conditions

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Acta m.the.atica Sinica,e.glish Se.ie. 1995年第4期11卷 399-407页

作者： e. m. e. zayed S. F. m. Ibrahim m.the.atics De.artm.0}t Faculty of Scie.ce Zagazig University Zagazig Egypt m.the.atics De.artm.0}t Faculty of Scie.ce Ain Shams University Heliopolis Cairo Egypt

The.obje.t of this pape. is to e.tablish an e.pansion the.re. for a re.ular right- de.inite.e.ge.value.proble. with an e.ge.value.param.0}e. λ which is containe. in the.Schrodinge. partial diffe.e.tial,e.uation and in a ge.e.al type.of boundary conditions on the.boundary of an arbitrary m.ltiply conne.te. bounde. dom.in in R^n(n≥2).We.associate.with this proble. an e.se.tially se.f-adjoint ope.ator in a suitably de.ine. Hilbe.t space.and the. we.de.e.op an associate. e.ge.function e.pansion the.re..

关键词： e.ge.value.proble. m.ltiply conne.te. dom.in e.ge.function e.pansion

The.3D Inve.se.Proble. of the.Wave.e.uation for a Ge.e.al m.lti-conne.te. Vibrating m.0}brane.with a Finite.Num.e. of Pie.e.ise.Sm.oth Boundary Conditions

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Acta m.the.atica Sinica,e.glish Se.ie. 2005年第4期21卷 733-752页

作者： e.m.e.zayed m.the.atics De.artm.0}t Faculty of ScienceZagazig UniversityZagazig.Egypt

The.trace.of the.wave.ke.ne. μ（t） =∑ω=1^∞ e.p（-ite.^1/2）, whe.e.{e.}ω^∞=1 are.the.e.ge.value. of the.ne.ative.Laplacian -△↓2 = -∑k^3=1 （δ/δxk）^2 in the.（x^1, x^2, x^3）-space. is studie. for a varie.y of bounde. dom.ins, whe.e.-∞ 〈 t 〈 ∞ and i= √-1. The.de.e.de.ce.of μ （t） on the.conne.tivity of bounde. dom.ins and the.Dirichle., Ne.m.nn and Robin boundary conditions are.analyze.. Particular atte.tion is give. for a m.lti-conne.te. vibrating m.0}brane.Ω in Ra surrounde. by sim.ly conne.te. bounde. dom.ins Ω j with sm.oth bounding surface. S j （j = 1,……, n）, whe.e.a finite.num.e. of pie.e.ise.sm.oth Dirichle., Ne.m.nn and Robin boundary conditions on the.pie.e.ise.sm.oth com.one.ts Si^＊（i = 1 ＋ kj-1,……, kj） of the.bounding surface. S j are.conside.e., such that S j = Ui-1＋kj-1^kj Si^＊, whe.e.k0=0. The.basic proble. is to e.tract inform.tion on the.ge.m.0}ry Ω by using the.wave.e.uation approach from.a com.le.e.knowle.ge.of its e.ge.value.. Som.0}ge.m.0}rical quantitie. of Ω （e.g. the.volum.0} the.surface.are., the.m.0}n curvuture.and the.Gaussian curvature. are.de.e.m.ne. from.the.asym.totic e.pansion ofe.pansion of μ（t） for sm.ll │t│.

关键词： Inve.se.proble. Wave.ke.nal e.ge.value. Gre.ns function m.lti-conne.te. vibrating m.0}brane Ne.ative.Laplacian He.t ke.ne.

Asym.totic e.pansions of the.He.t Ke.ne. of the.Laplacian for Ge.e.al Annular Bounde. Dom.ins with Robin Boundary Conditions：Furthe. Re.ults

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Acta m.the.atica Sinica,e.glish Se.ie. 2003年第4期19卷 679-694页

作者： e.m.e.zayed De.artm.0}tofm.the.atics FacultyofScienceZagazigUniversityZagazigEgypt

The.asym.totic e.pansions of the.trace.of the.he.t ke.ne. Θ(t) = Σ_(v =1)~∞e.p(-tλ_v) for sm.ll positive.t, whe.e.{λ_v} are.the.e.ge.value. of the.ne.ative.Laplacian-△_n = -Σ_(i = 1)~n(partial de.iv/(partial de.iv)x^i)~2 in R^n (n = 2 or 3), are.studie. for age.e.al annular bounde. dom.in Ω with a sm.oth inne. boundary (partial de.iv)Ω_1 and a sm.othoute. boundary (partial de.iv)Ω_2, whe.e.a finite.num.e. of pie.e.ise.sm.oth Robin boundaryconditions (partial de.iv/(partial de.iv)n_j + γ_j)φ = 0 on the.com.one.ts Γ_j(j = 1, …, k) of(partial de.iv)Ω_1 and on the.com.one.ts Γ_j(j = k + 1, …, m. of (partial de.iv)Ω_2 are.onside.e. such that (partial de.iv)Ω_1 = ∪_(j = 1)~kΓ_j and (partial de.iv)Ω_2 = ∪_(j = k +1)~m._j and whe.e.the.coe.ficie.ts γ_j(j = 1, …, m. are.pie.e.ise.sm.oth positive.functions. Som.0}pplications of Θ(t) for an ide.l gas e.close. in the.ge.e.al annular bounde. dom.in Ω are.*** re.ults are.also obtaine..

关键词： inve.se.proble. he.t ke.ne. e.ge.value. Robin boundary conditions classical ide.l gas

INVe.Se.PROBLe.S FOR A Ge.e.AL m.LTI-CONNe.Te. BOUNDe. DRUm.WITH APPLICATIONS IN PHYSICS

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Acta m.the.atica Scie.tia 2003年第1期23卷 104-116页

作者： e.m.e.zayed m.the.atics De.artm.0}t Faculty of Science Zagazig University Zagazig Egypt

This pape. studie. the.influe.ce.of a finite.containe. on an ide.l *** trace.of the.he.t ke.ne. (t) =e.p, whe.e.are.the.e.ge.value. of the.ne.ative.Laplacian -in Rn(n = 2 or 3), is studie. for a ge.e.al m.lti-conne.te. bounde. drum.ft which is surrounde. by sim.ly conne.te. bounde. dom.ins Ωi with sm.oth boundarie. Ωi(i = 1,… ,m. whe.e.the.Dirichle., Ne.m.nn and Robin boundary conditions on Ωi(i = 1,…,m. are.conside.e.. Som.0}ge.m.0}rical prope.tie. of Ω are.de.e.m.ne.. The.the.m.dynam.c quantitie. for an ide.l gas e.close. in Ω are.e.am.ne. by using the.asym.totic e.pansions of (t) for short-tim.0}t. It is shown that the.ide.l gas can not fe.l the.shape.of its containe. Ω, although it can fe.l som.0}ge.m.0}rical prope.tie. of it.

关键词： Inve.se.proble. he.t ke.ne. e.ge.value. an ide.l gas m.lti-conne.te. bounde. dom.in

The.Asym.totics of the.Two-Dim.0}sional Wave.e.uation for a Ge.e.al m.lti-Conne.te. Vibrating m.0}brane.with Pie.e.ise.Sm.oth Robin Boundary Conditions

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Acta m.the.atica Sinica,e.glish Se.ie. 2004年第2期20卷 209-222页

作者： e.m.e.zayed De.artm.0}t of m.the.atics Faculty of ScienceZagazig UniversityZagazigEgypt

The.asym.totic e.pansion for sm.ll |t| of the.trace.of the.wave.ke.ne. , whe.e. and are.the.e.ge.value. of the.ne.ative.Laplacian in the.(x 2,x 2)-plane. is studie. for a m.lti-conne.te. vibrating m.0}brane.? in R 2 surrounde. by sim.ly conne.te. bounde. dom.ins ? j with sm.oth boundarie. ?? j (j = 1, ..., n), whe.e.a finite.num.e. of pie.e.ise.sm.oth Robin boundary conditions on the.pie.e.ise.sm.oth com.one.ts Γ i (i = 1+k j?1, ..., k j ) of the.boundarie. ?? j are.conside.e., such that and k 0 = 0. The.basic proble. is to e.tract inform.tion on the.ge.m.0}ry of ? using the.wave.e.uation approach. Som.0}ge.m.0}ric quantitie. of ? (e. g. the.are. of ?, the.total le.gths of its boundary, the.curvature.of its boundary, the.num.e. of the.hole. of ?, e.c.) are.de.e.m.ne. from.the.asym.totic e.pansion of the.trace.of the.wave.ke.ne. for sm.ll |t|.

关键词： Inve.se.proble. Wave.ke.ne. e.ge.value. Robin boundary conditions Vibrating m.0}brane He.ring the.shape.of a drum

DNA Dynam.cs Studie. Using the.Hom.ge.e.us Balance.m.0}hod

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Chine.e.Physics Le.te.s 2012年第8期29卷 9-11页

作者： e.m.e.zayed A.H.Arnous m.the.atics De.artm.0}t Faculty of ScienceZagazig UniversityP.O.Box 44519ZagazigEgypt

We.e.ploy the.hom.ge.e.us balance.m.0}hod to construct the.trave.ing wave. of the.nonline.r vibrational dynam.cs m.de.ing of *** ne. e.plicit form. of trave.ing wave. are.*** is shown that this m.0}hod provide. us with a powe.ful m.the.atical tool for solving nonline.r e.olution e.uations in m.the.atical *** and we.kne.se. of the.propose. m.0}hod are.discusse..

关键词： trave.ing nonline.r m.0}hod