On a generalized taylor theorem: A rational proof of the validity of the homotopy analysis method

IntroductionIn 1 7thcenturyIsaacNewton[1]gavesuchabinomialexpressionforfractionalandnegativeexponents(1 +t) α,i.e.,(1 +t) α =1 + +∞k=1α(α-1 ) (α-2 )… (α -k+ 1 )k !tk　　 (α≠ 0 ,1 ,2 ,… ) ,(1 )whoseconvergenceradiusisone.Furthermore ,theclassicalTaylorseries (seeRef.[2 ] )limm→+∞ mk=0f(k) (z0 )k !(z-z0 ) k (2 )ofacomplexfunctionf(z)atz=z0 isvalidmostlyinarestrictedconvergenceregion｜z-z0 ｜相似文献   

SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR SEMI-LINEAR RETARDED DIFFERENTIAL EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

IntroductionInthispaper,westudiedakindofboundaryvalueproblems (BVPs)forsemi_linearretardeddifferentialequationwithnonlinearboundarycondition :　　　　εx″(t) =f(t,x(t) ,x(t-ε) ,ε) ,　　t∈(0 ,1 ) ,(1 )　　　　x(t) =φ(t,ε) ,　t∈[-ε0 ,0 ] ,h(x(1 ) ,x′(1 ) ,ε) =A(ε) ,(2 )whereε>0isasmallparameterandε0 isasufficientlysmallpositiveconstant.ThereweremanyresultsofstudyingonsingularlyperturbedboundaryvalueproblemforretardeddifferentialequationinRefs.[1～5] .Butthosestudiespossessedanesse…     

Periodicity and strict oscillation for generalized lyness equations

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｄｅｌａｙｄｉｆｆｅｒｅｎｃｅｅｑｕａｔｉｏｎｘｎ 1=ｘｎ(ａ ｂｘｎ)ｘｎ- 1,　　ｎ=0 ,1 ,2 ,… ,(1 )ｗｈｅｒｅａ ,ｂ∈ [0 ,∞ )ｗｉｔｈａ ｂ>0 (2 )ａｎｄｗｈｅｒｅｔｈｅｉｎｉｔｉａｌｖａｌｕｅｓｘ- 1ａｎｄｘ0 ａｒｅａｒｂｉｔｒａｒｙｐｏｓｉｔｉｖｅｎｕｍｂｅｒｓ.Ｅｑ .(1 )ｉｓｒｅｇａｒｄｅｄａｓａｇｅｎｅｒａｌｉｚｅｄＬｙｎｅｓｓｅｑｕａｔｉｏｎｂｙＧ .Ｌａｄａｓｉｎ [1 ] .Ｏｂｖｉｏｕｓｌｙ ,ｕｎｄｅｒｃｏｎｄｉｔｉｏｎ (2 ) ,…     

What are the separatrix values named by Leontovich on homoclinic bifurcation

Introduction LetVλ(x),x∈R2,λ∈RkbeafamilyofplanarC∞(oranalytic)vectorfields.Suppose thatforλ=λ0,Vλ0hasahyperbolicsaddlepointattheorigin(0,0)inthephaseplaneandthere isahomoclinicorbit(oraseparatrixloop)Γ0totheorigin.Thehyperbolicityratioattheorigin isr(λ)=-λ1/λ2,whereλ1<0<λ2arethetwoeigenvaluesofthelinearizedsystematthe origin.Generally,whenλ≠λ0and｜λ-λ0｜issmallenough,nearΓ0limitcycleswillbe created.Thisso_calledhomoclinicbifurcationhasbeenstudiedbyalotofauthors[1-3].Deno…     

A CLASS OF ALTERNATING GROUP METHOD OF BURGERS' EQUATION

IntroductionWeconsiderthefollowingnonlinearBurgers’equation : u t u u x=ε 2 u x2 ,　　 0 0 ,( 1 )withtheinitialandtheboundaryconditions　　　　　u(x,0 ) =f(x) ,　　 0 相似文献   

Global linear and quadratic one-step smoothing newton method for vertical linear complementarity problems

IntroductionLetN ∈Rm0 ×n(m0 ≥n)beaverticalblockmatrixoftype(m1,… ,mn) ,q∈Rm0 beaconstantvectorpartitionedconformablywithN ,thatareN =N1Nn,　q =q1qn,　m0 = ni=1mi,whereNi ∈Rmi×n,qi ∈Rmi,i ∈I =1 ,… ,n .WeconsidertheverticallinearomplementarityproblemVLCP(N ,q)associatedwith (N ,q)offindingavectorx∈Rnsuchthatx≥ 0 ,　si(x) :=Nix+qi ≥ 0 ,　xi∏mij=1sij(x) =0　　 (i∈I) ,wherexi,sij(x)denotetheithcomponentofxandthejthcomponentofsi(x) ,respectively ;∏ mj=1ajdenotesa1…am.Iti…     

EXPONENTIAL ATTRACTOR FOR THE GENERALIZED SYMMETRIC REGULARIZED LONG WAVE EQUATION WITH DAMPING TERM

IntroductionAsymmetricversionofregularizedlongwaveSRLWequationsuxxt-ut =ρx uux, ( 1 )ρt ux =0 ( 2 )havebeenproposedasamodelforpropagationofweaklynonlinearionacousticandspace_chargewaves[1].Thehyperbolicsecantsquaredsolitarywavesolutions,thefourinvariantsandsomenumericalresultshavebeenobtainedinRef.[1 ] .Obviously ,eliminatingρfromEqs.( 1 )and( 2 ) ,wegetasymmetricregularizedlongwaveequation (SRLWE)utt-uxx 12 (u2 ) xt-uxxtt =0 . ( 3 )TheSRLWequation ( 3 )isexplicitlysymmetricinthexand…     

Analysis of chebyshev pseudospectral method for multi-dimensional generalized srlw equations

IntroductionAsymmetricregularizedlongwaveequation (SRLWE) 2 x2 -1 u t = x ρ+ 12 u2 ,　 ρ t+ u x=0 (1 )hasbeeninvestigatedinRef.[1 ] .Thesystem (1 )ofequationsisshowntodescribeweaklynonlinearion_acousticwaveandspace_chargewaves.Thehuperbolicsecantsquaredsolitarywaves ,thefourconservationlaws,andsomenumericalresultshavebeenobtainedinRef.[1 ] .Obviously ,eliminatingρin (1 ) ,weobtainaclassofregularizedlongwaveequationutt-uxx+ 12 u2xt-uxxtt =0 . (2 )TheSRLWequationisexplicitlysymmetric…     

APPLIED MATHEMATICS AND MECHANICS Vol.25 2004 INDEX

AN .Aktar,F .RahmanandS .K .SenStokesFlowDuetoFundamentalSingularitiesBeforeaPlaneBoundary 7(799)……………A .S .J .Al_SaifandZHUZheng_you (朱正佑 )UpwindLocalDifferentialQuadratureMethodforSolvingCoupledViscousFlowandHeatTransferEquations 1 0 (1 1 3 0 )…………………………………………………BBAIRui_xiang (白瑞祥 )andCHENHao_ran (陈浩然 )NumericalAnalysisofDelaminationGrowthforStiffenedCompositeLaminatedPlates 4(40 5)………………………………………………………………BIWei_pi…     

GLOBAL ATTRACTIVITY FOR A CLASS OF NONLINEARDELAY DIFFERENCE EQUATIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｇｌｏｂａｌａｔｔｒａｃｔｉｖｉｔｙｏｆｔｈｅｎｏｎｌｉｎｅａｒｄｅｌａｙｄｉｆｆｅｒｅｎｃｅｅｑｕａｔｉｏｎΔｘｎ+ａｎｘｎ+ｆ(ｎ ,∑0ｓ=-ｋ ｑｓ,ｎｘｓ+ｎ) =0　 (ｎ=0 ,1 ,2 ,… ) . ( 1 )ＴｈｅｉｎｉｔｉａｌｖａｌｕｅｏｆＥｑ .( 1 )ｉｓｘｓ =φｓ　 (ｓ=-ｋ ,-ｋ+ 1 ,… ,-1 ,0 ) . ( 2 )Ｉｎｔｈｅｓｅｑｕｅｌ,ｗｅａｌｗａｙｓｓｕｐｐｏｓｅｔｈａｔＤ1) ａｎ ｉｓａｎｏｎｎｅｇａｔｉｖｅｒｅａｌｓｅｑｕｅｎｃｅ ,0 ≤ａｎ <1ａｎｄｋｉｓａｐｏｓｉｔｉｖｅ…     

Explicit square conserving schemes of Landau-Lifshitz equation with Gilbert component

1　IntroductionandProblemsTotheordinarydifferentialequationsdY dt=A(t,Y)Y ,Y∈Rn,t≥ 0 ,( 1 )A(t,Y)isskewsymmetrymatrix .Fromd(YTY) dt=(Y′) TY YTY′=YT(AT A)Y =0 ,and‖Y( 0 )‖ =a ,aisapositivenumber,wecanget‖Y(t)‖ =a .Totheordinarydifferentialequationsofthesquareconservingproperty ( 1 ) ,classicalnumericalmethods,suchasexplicitRunge_Kuttamethods,multiplestepmethods,cannotpreservethesquareconservingproperty.Theimplicitmiddleschemecanpreservethesquareconservingproperty ,butitis…     

Oscillation criteria for second order nonlinear differential equation with damping

1IntroductionandProblem Thefollowingisthesecondordernonlineardifferentialequationwithdamping: [p(t)ψ(y)u(y′)]′ r(t)y′(t) q(t)f(y)g(y′)=0(t≥t0),(1) inwhich,p(t)∈C′([t0,∞),(0,∞)),r(t)∈C([t0,∞),(-∞,∞)),q(t)∈C([t0, ∞),[0,∞))withtheexistenceofT≥t0,q(t)≠0,t∈[T,∞),f(y),g(y),ψ(y),u(y) ∈C((-∞,∞),(-∞,∞)),andyf(y)>0,yu(y)>0,y≠0. Thesolutiony(t)ofEq.(1)iscallednormalsolutionify(t)isthenon_constantsolutionof Eq.(1)andsupt≥t0｜y(t)｜>0(refertoRef.[1]).Anormalsolutionisoscill…     

Positive Solutions of Boundary Value Problems for Second-Order Singular Nonlinear Differential Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ,ｗｅｓｈａｌｌｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｓｉｎｇｕｌａｒｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓ (ＢＶＰ)ｕ″ ｇ(ｔ)ｆ(ｕ) =0 ,　　 0 <ｔ<1 ,αｕ(0 ) -βｕ′(0 ) =0 ,　　γｕ(1 ) δｕ′(1 ) =0 ,(1 )ｗｈｅｒｅα ,β,γ ,δ≥ 0 ,ρ:=βγ αγ αδ>0 ,ｆ∈Ｃ([0 ,∞ ) ,[0 ,∞ ) ) ,ｇｍａｙｂｅｓｉｎｇｕｌａｒａｔｔ=0ａｎｄ/ｏｒｔ=1 .Ｔｈｉｓｐｒｏｂｌｅｍａｒｉｓｅｓｎａｔｕｒａｌｌｙｉｎｔｈｅｓｔｕｄｙｏｆｒａｄｉａｌｌｙｓｙｍｍｅｔ…     

ON THE PERTURBATIONAL GLOBAL ATTRACTIVITY OF NONAUTOMOUS DELAY DIFFERENTIAL EQUATIONS

I.Introductionillthispaper,wewillstudytheglobalattractivityofzerosolutionofnonautonomouslineardelaydit'ferentialequationswiththeperturbedtermft(t)=--a(t)x(t--r) F(t,x,),t30(1.l)whereF:[0,ac)xC[-6,0]~Riscontinuous,C[-a,0]isthespaceofcontilluousfunctions(di:[--8,0]-R),F(t,0)-0,t30.a(t)eC((0,co),(0,co)),andr>0.Fortheasylllptoticstabilityoflilleardiflbrentialequatiollwithperturbedtcrlns,mostoftheknownpapershaveonlybeenconcernedwiththelocalbehaviorll--sl.Evenfortheglobalbehavior,theauthorsonlyde…     

Positive solutions to a singular second order three-point boundary value problem

1　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＭａｉｎＲｅｓｕｌｔｓＴｈｅｅｘｉｓｔｅｎｃｅｏｆｐｏｓｉｔｉｖｅｓｏｌｕｔｉｏｎｓｈａｓｂｅｅｎｅｓｔａｂｌｉｓｈｅｄｆｏｒａｎｏｎｌｉｎｅａｒｓｅｃｏｎｄｏｒｄｅｒｔｈｒｅｅ＿ｐｏｉｎｔｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｏｆｔｈｅｆｏｒｍ-ｙ″ =Ｑ(ｘ)ｆ(ｙ)　　 ( 0 <ｘ<1 ) ,ｙ( 0 ) =0 ,ｙ( 1 ) =αｙ( η) ( 1 )ｏｎｌｙｖｅｒｙｒｅｃｅｎｔｌｙｉｎ [1 ] .Ｉｔｗａｓａｓｓｕｍｅｄｔｈｅｒｅｔｈａｔ 0 <η <1 ,0 <αη <1 ,Ｑ(ｘ) ∈Ｃ( [0 ,1 ] ;Ｒ+) ,ｆ(ｙ)∈Ｃ…     

Bifurcations of travelling wave solutions for a coupled nonlinear wave system

Introduction FangShaomeiandGuoBoling[1]consideredthefollowingtimeperiodicproblemof dampedcouplednonlinearwaveequations:ut f(u)x-αuxx βuxxx 2vvx=G1(u,v) h1(x),vt-γvxx 2(uv)x g(v)x=G2(u,v) h2(x),(1)whereα,β,γareconstants,andγ>0,β≠0.Undertheperiodicboundaryconditions,the authorsobtainedtheuniqueexistenceofstrongsolutionsfortheabovesystem.InthispaperweshallconsiderbifurcationbehaviorofthetravellingwavesolutionsofEq.(1)inthecaseGi(u,v)≡0,hi(u,v)≡0(i=1,2).Letξ=x-ct,u=u(x-ct),where cis…     

BIFURCATIONS OF SUBHARMONIC SOLUTIONS IN PERIODIC PERTURBATION OF A HYPERBOLIC LIMIT CYCLE

Ｃｏｎｓｉｄｅｒｐｌａｎａｒｐｅｒｉｏｄｉｃｐｅｒｔｕｒｂｅｄｓｙｓｔｅｍ ｘ=ｆ(ｘ) +εｇ(ｔ,ｘ,ε ,δ) ,　　ｘ∈Ｒ2 ,( 1 )ｗｈｅｒｅε∈Ｒ ,δ∈Ｄ ＲｎｗｉｔｈＤｃｏｍｐａｃｔ,ａｎｄｆ,ｇａｒｅＣ3ｆｕｎｃｔｉｏｎｓａｎｄｇｉｓＴ＿ｐｅｒｉｏｄｉｃｉｎｔ,Ｔ >0 .Ｌｅｔｓｙｓｔｅｍ ( 1 )ｈａｓａｌｉｍｉｔｃｙｃｌｅＬ0 :ｘ =ｕ(ｔ) ,0 ≤ｔ≤Ｔ0 ｆｏｒε =0ｗｉｔｈＴ0 ｔｈｅｐｅｒｉｏｄｏｆＬ0 .ＳｕｐｐｏｓｅＴ0 /Ｔｉｓｒａｔｉｏｎａｌ,ｔｈａｔｉｓＴ0 /Ｔ=ｍ/ｋ,　　 (ｍ ,ｋ) =1 . ( 2 )Ａｓｗｅｋｎｏ…     

DISCUSSIONS ON“AN ASYMPTOTIC ELASTIC PLASTIC ANALYSIS IN PLANE STRAIN DEFORMATION NEAR A CRACK TIP”

I.In his paper,prof.D.W.Hsueh derived the following functionf(θ)=-[(1 2n)/(1 n)~2-k_1~2]A_1/k_1θ-[(1 2n)/(1 n)~2-k_2~2]A_2/k_2-cosk_2θ(1)In calculation of coefficients A_1,A_2 a mistake occurred.Making use of f′(π)=0 andf(0)=0,     

Asymptotic solution of a singularly perturbed nonlinear state regulator problem

I.1utroductlouWeconsidcrthenonlinearstateregulatorproblemsconsistingofthesystemontheinterval0't相似文献   

THE POSITIVE SOLUTION OF CLASSICAL GELFAND MODEL WITH COEFFICIENT THAT CHANGE SIGN

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｎｏｎｌｉｎｅａｒｔｗｏ＿ｐｏｉｎｔｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍ (ＢＶＰ)(Ｐ) ｗ″(ｔ) +λｈ(ｔ)ｅｗ(ｔ) =0 ,　　 0 ≤ｔ≤ 1 ,ｗ( 0 ) =ｗ( 1 ) =0(λ>0 )ｈａｓｂｅｅｎｐｒｏｐｏｓｅｄｂｙＧｅｌｆａｎｄ[1]ｗｉｔｈｈ(ｔ) ≡ 1 ,ｔｈｉｓｉｓｔｈｅｒｅａｓｏｎｗｈｙｗｅｓａｙｔｈｅｐｒｏｂｌｅｍ (Ｐ)ｉｓａＧｅｌｆａｎｄｍｏｄｅｌ.Ｓｅｍｉｌｉｎｅａｒｐｒｏｂｌｅｍｓｏｆｔｈｉｓｔｙｐｅａｒｉｓｅｉｎａｖａｒｉｅｔｙｏｆｉｎｔｅｒｅｓｔｉｎｇａｐｐｌｉｃａｔｉｏ…