On the boundedness and the stability properties of solution of certain fourth order differential equations

This paper investigates equation(1)in two cases:(i)P≡0,(ii)P(≠O)satisfies|P(t,x,y,z,ω)|≤(A |y| |z| |ω|)q(t),where q(t)is a nonnegative function of t.For case(i)the asymptotic stability in the large of the trivial solution x=0 is investigatedand for case(ii)the boundedness result is obtained for solutions of equation(1).Theseresults improve and include several well-known results.     

BOUNDARY INTEGRAL FORMULA OF ELASTIC PROBLEMS IN CIRCLE PLANE

IntroductionConcerningtheelasticplaneprobleminaunitcircle ,ZhengShenzhouandZhengXueliangdevelopedaboundaryintegralformulaofthestressfunction[1]:Φ(r,θ) =-( 1 -r2 ) 24π ∫2π0ν( φ)1 -2rcos(θ-φ) r2 dφ　　 12π∫2π011 -2rcos(θ-ω) r2 dω∫2π0μ( φ)1 -cos(ω-φ) dφ　　 1 -r22π∫2π0μ( φ)1 -2rcos(θ -φ) r2 dφ　　 ( 0 ≤r <1 ) ,( 1 )whereμ(θ) =Φ(r,θ) ｜r=1,ν(θ) = Φ n r=1= Φ r r=1.Intheformula ( 1 )theseconditemisastrongsingularintegral,itshouldbeunderstoodasanintegra…     

Asymptotic theory of initial value problems for nonlinear perturbed Klein-Gordon equations

Introduction InthispaperasymptotictheoryofthefollowinginitialvalueproblemforanonlinearKlein Gordonequationisconsidered.tt-Δ =εF(t,x,,ε),t>0,x∈R2,(0,x,ε)=0(x,ε),t(0,x,ε)=1(x,ε),x∈R2,(1)where(t,x)isarealvaluedunknownfunction,Δ=2i     

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…     

CONCENTRATION OF COUPLED CUBIC NONLINEAR SCHRDINGER EQUATIONS

IntroductionThe following coupled cubic nonlinear Schr dinger equations inR2are considered:it-Δ=(||2-|ψ|2),t>0,x∈R2,iψt-Δψ=(-||2 |ψ|2)ψ,t>0,x∈R2,(1)where=(t,x):[0,T)×R2→C,ψ=ψ(t,x):[0,T)×R2→C,Δis the Laplaceoperator onR2.The e     

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   

广义惯性力的广义非惯性势

相对于惯性系,非惯性系Oxyz 坐标原点O 的加速度和Oxyz 的转动角速度分别α_θ和ω,α_θ=α_θ(t)和ω=ω(t)均为时间t 的巳知函数而与其它因素无关,则对于在Oxyz 中运动的由N 个质点P_i(i=1,     

广义惯性力的广义非惯性势

<正> 相对于惯性系,非惯性系Oxyz 坐标原点O 的加速度和Oxyz 的转动角速度分别α_θ和ω,α_θ=α_θ(t)和ω=ω(t)均为时间t 的巳知函数而与其它因素无关,则对于在Oxyz 中运动的由N 个质点P_i(i=1,     

On the existence of periodic solutions for nonlinear system with multiple delays

IntroductionInthispaper,westudyT_periodicsolutionsofthefollowingnonlinearsystemwithmultipledelays x(t) =f(t,x(t) ,x(t-τ1(t) ) ,… ,x(t -τm(t) ) ) ,(1 )wherex(t) ∈C(R ,R) ,fiscontinuous,f(t+T ,·) =f(t,·) ,τi(t) (i=1 ,2 ,… ,m)arecontinuousperiodicfunctionsofperiodT .AlemmaisintroducedfordiscussingtheexistenceofT_periodicsolutionofsystem (1 ) .LetXbeaBanachSpace ,considerthefollowingoperatorequation :Lx =λNx　　 (λ∈ [0 ,1 ] ) ,whereL :DomL∩X→Xisalinearoperator,λ∈ [0 ,1 ]isapa…     

Singular perturbation of nonlinear vector boundary value problem

In this paper we study the perturbed boundary value problem of the form dx/dt=f(x,y,t;ε), εdy/dt=g(x,y,t;ε), a_1(ε)x(0,ε)+a_2(ε)y(0,ε)=a(ε) b_1(ε)x(1,ε)+εb_2(ε)y(1,ε)=β(ε)in whichx,f,β∈E~m, y,g,a∈E~n, 0<ε(?)1and a_1(ε), a_2(ε), b_2(ε)and b_2(ε) are matrices of the appropriate size. Under the condition that g_y(t) is nonsingular and other suitable restrictions, the existence of the solution is proved, the asymptotic expansion of solution of order n is constructed, and the remainder term is estimated.     

THE LINEARIZATION OF CERTAIN CLASS OF NONLINEAR SIMULTANEOUS EQUATION SET CONTAINING AMPLITUDE AND PHASE FREQUENCY CHARACTERISTICS AND APPLICATION

A class of complex function of rational fraction type G(jω)=[1 a_1jω a_2(jω)~2 … a_m(jω)~n]/[b_0 b_1jω b_2(jω)~2 … b_n(jω)~n ] is frequently used to describe the dyna-mical properties of systems.It is however quite difficult to es-tablish a mathematical model of this type on the basis of ampli-tude and phase frequency data collected from experiments conductedon the related physical system.Since the erection of mathematicalmodel G(jω)would involve the solution of a set of nonlinear si-multaneous equations|G(jω_i)|=g_i∠G (jω_i)=θ_i i=1, 2,…,r. with the unknown coefficients a_isand b_is(i=0,1,…,m,…,n)in G(jω).Up to now,these nonlinear equa-tions have been considered to be very difficult to solve directly.In spite of the fact there are special computer programmes in cer-tain software packages available to tackle this problem,it is byno means an easy task due to the complex procedures involved inpicking up a set of initial values that should be close enough tothe exact solutions.This paper     

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…     

HIGHER ORDER MULTIVARIABLE NRLUND EULER-BERNOULLI POLYNOMIALS

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｋｔｈｏｒｄｅｒＮ ｒｌｕｎｄＥｕｌｅｒｐｏｌｙｎｏｍｉａｌｓＥ(ｋ)ｖ (ｘ｜ω1,… ,ωｋ)ａｎｄＢｅｒｎｏｕｌｌｉｐｏｌｙｎｏｍｉａｌｓＢ(ｋ)ｖ (ｘ｜ω1,… ,ωｋ)ａｒｅｇｉｖｅｎｂｙｔｈｅｆｏｌｌｏｗｉｎｇｅｘｐａｎｓｉｏｎｆｏｒｍｕｌａｒｅｓｐｅｃｔｉｖｅｌｙ[1]2 ｋｅｘｔ(ｅω1 ｔ+ 1 )… (ｅωｋｔ+ 1 ) = ∞ｖ=0Ｅ(ｋ)ｖ (ｘ｜ω1,… ,ωｋ) ｔｖｖ!,( 1 )ω1…ωｋｔｋｅｘｔ(ｅω1 ｔ-1 )… (ｅωｋｔ-1 ) = ∞ｖ=0Ｂ(ｋ)ｖ (ｘ｜ω1,… ,ωｋ) ｔｖｖ!. ( 2 )Ｃｌｅａｒｌｙｔ…     

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   

On the problem of preventing blowing-up and quenching for semilinear heat equation

In this paper,the global existence of solutions to the IVP=Δu+g(t)f(u) (t>0),u|_(t=0)=u_0(x)and the (?)PVPu_t=Δu-g(t,x)f(u)(t>0,x∈Ω),u|_(t-0)=u|_(?)(?)is investigated. As has been done in [6]the (?)duction of factor g(t) or g(t.x) innonlinear term is to prevent(?) occurrance of blowing-up or quenching of solutions.It isshown in this paper that most of the restrictions on f,g and u_0 in the theorems of[6] maybe cancelled or relaxed,that the smallness of g is required only for t large,and thatunder certain conditions controlling initial state can avoid blowing-up.     

Chaotic belt phenomena in nonlinear elastic beam

IntroductionThechaoticphenomenainsolidmechanicsfieldsbringmoreandmoreinterest.In 1 998,F .C .Moon[1]analyzedthechaoticbehaviorsofbeamsexperimentallyfirst.Thenhestudiedthedynamicsresponseoflinearelasticbeamsubjectedtransverseperiodicload .Thechaoticmotionsoflineardampingbeamshavebeenstudiedbymanyscholarsathomeandabroadinrecentyears[2 ,3].ThedynamicbehaviorsofnonlineardampingbeamssubjectedtotransverseloadP=δP0 (f+cosωt)sin(πx/l)arestudiedinthispaper.Thecriticconditionsthatchaosoccursinthes…     

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…     

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…     

GLOBAL DYNAMICS OF DELAYED BIDIRECTIONAL ASSOCIATIVE MEMORY （BAM） NEURAL NETWORKS

IntroductionConsiderthebidirectionalassociativememory (BAM )neuralnetworkswithconstanttransmissiondelaysdescribedbyasystemofdelaydifferentialequationsoftheform[1,2 ]:dxi(t)dt =-aixi(t) nj=1bijfj(yj(t-σij) ) Ii,　　i=1 ,2 ,… ,m ,dyj(t)dt =-cjyj(t) mi=1djigi(xi(t-τji) ) Jj,　　j=1 ,2 ,… ,n ,fort >0 .Thesystem ( 1 )consistsoftwosetsofneurons (orunits)arrangedontwolayers,namely ,I_layerandJ_layer.Inthesystem ( 1 ) ,xi( ·)andyj( ·)denotemembranepotentialoftheithneuronsfromtheI_laye…     

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…