The Asymptotic Behavior of Solution for the Singularly Perturbed Initial Boundary Value Problems of the Reaction Diffusion Equations in a Part of Domain

Ｔｈｅａｕｔｈｏｒｓｓｔｕｄｉｅｄａｃｌａｓｓｏｆｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓｉｎ [1 ] -[7] .Ｎｏｗｗｅｒａｉｓｅａｃｌａｓｓｏｆｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓｏｎａｐａｒｔｏｆｄｏｍａｉｎ .Ｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｉｎｉｔｉａｌｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｆｏｒｔｈｅｒｅａｃｔｉｏｎｄｉｆｆｕｓｉｏｎｅｑｕａｔｉｏｎｓ　　ｕｔ-λε(ｘ) ( μ(ｕ)ｕｘ) ｘ Ｋｘ(ｕ) ｆ(ｘ ,ｔ,ｕ) =0 ,(ｔ,ｘ) ∈ ( 0 ,Ｔ)× ( ( 0 ,α) ∪ (…     

Singularly perturbed solution for weakly nonlinear equations with two parameters

A class of singularly perturbed boundary value problems of weakly non- linear equation for fourth order on the interval[a,b]with two parameters is considered. Under suitable conditions,firstly,the reduced solution and formal outer solution are con- structed using the expansion method of power series.Secondly,using the transformation of stretched variable,the first boundary layer corrective term near x=a is constructed which possesses exponential attenuation behavior.Then,using the stronger transfor- mation of stretched variable,the second boundary layer corrective term near x=a is constructed,which also possesses exponential attenuation behavior.The thickness of second boundary layer is smaller than the first one and forms a cover layer near x=a. Finally,using the theory of differential inequalities,the existence,uniform validity in the whole interval[a,b]and asymptotic behavior of solution for the original boundary value problem are proved.Satisfying results are obtained.     

Singularly perturbed solution to semilinear reaction diffusion equations with two parameters

A class of singularly perturbed initial boundary value problems for semilinear reaction diffusion equations with two parameters is considered, Under suitable conditions and using the theory of differential inequalities, the existence and the asymptotic behavior of the solution to the initial boundary value problem are studied.     

Boundary value problem for a singularly perturbed nonlinear system

I.Intr0ducti0n'Manyresultshavebeeng0tinstudyingthesingu1arperturbation0fboundaryvalueproblemfornonlinearsystemeg#=j(t,g,y',8)(l.l)bythemethodandthetechniqueofdiagonalizationl'~'l.Thepapersarebasedonsucha.conditionthattheeigenvaluesofJacobimatrixlu'offwit…     

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…     

The nonlinear nonlocal singularly perturbed problems for reaction diffusion equations

A class of nonlinear nonlocal for singularly perturbed Robin initial boundary value problems for reaction diffusion equations is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed, finally, using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems are studied and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation is discussed. Foundation items: the National Natural Science Foundation of China (10071048); the “Hunfred Talents Project” by Chinese Academy of Sciences Biography: Mo Jia-qi (1937−)     

NUMERICAL SOLUTION OF THE SINGULARLY PERTURBED PROBLEM WITH NONLOCAL BOUNDARY CONDITION

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｉｓｐａｐｅｒｉｓｃｏｎｃｅｒｎｅｄｗｉｔｈｔｈｅｎｕｍｅｒｉｃａｌｓｏｌｕｔｉｏｎ ,ｂｙｆｉｎｉｔｅｄｉｆｆｅｒｅｎｃｅｍｅｔｈｏｄ ,ｏｆｔｈｅｆｏｌｌｏｗｉｎｇｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｎｏｎｌｏｃａｌｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍ :Ｌｕ≡-εｕ″ +ａ(ｘ)ｕ=ｆ(ｘ)　　 ( 0 <ｘ<ｌ) ,( 1 )Ｌ0 ｕ≡-εｕ′( 0 ) +γｕ( 0 ) =μ0 , ( 2 )Ｌ1ｕ≡ｕ(ｌ) -δｕ(ｄ) =μｌ　　 ( 0 <ｄ<ｌ) ,( 3 )ｗｈｅｒｅεｉｓａｓｍａｌｌｐｏｓｉｔｉｖｅｐａｒａｍｅｔｅｒ,…     

SINGULAR PERTURBATIONS FOR A CLASS OF BOUNDARY VALUE PROBLEMS OF HIGHER ORDER NONLINEAR DIFFERENTIAL EQUATIONS

ＳＩＮＧＵＬＡＲ　ＰＥＲＴＵＲＢＡＴＩＯＮＳ　ＦＯＲ　Ａ　ＣＬＡＳＳ　ＯＦ　ＢＯＵＮＤＡＲＹ　ＶＡＬＵＥ　ＰＲＯＢＬＥＭＳ　ＯＦ　ＨＩＧＨＥＲ　ＯＲＤＥＲ　ＮＯＮＬＩＮＥＡＲ　ＤＩＦＦＥＲＥＮＴＩＡＬ　ＥＱＵＡＴＩＯＮＳＳｈｉＹｕａｎｉｎｇ（史玉明）...     

A CLASS OF NONLINEAR NONLOCAL SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS WITH BOUNDARY PERTURBATION

A class of nonlinear nonlocal for singularly perturbed Robin initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions, first, the outer solution of the original problem was obtained. Secondly, using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer was constructed. Finally, using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems was studied, and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation were discussed.     

Numerical solution of a nonlinear reaction-diffusion equation

A nonlinear reaction-diffusion equation is studied numerically by a Petrov-Galerkinfinite element method,which has been proved to be2nd-order accurate in time and4th-orderin space.The comparison between the exact and numerical solutions of progressive wavesshows that this numerical scheme is quite accurate,stable and efficient.It is also shown thatany local disturbance will spread,have a full growth and finally form two progressive wavespropagating in both directions.The shape and the speed of the long term progressive wavesare determined by the system itself,and do not depend on the details of the initial values.     

The corner solution for quasilinear differential equation with two parameters

I.IntroductionWeconsiderinthispaperboundaryvalueproblemofquasilineardifferentialequationby# pi(x,y)y' g(x,y)=0,a相似文献   

Singular perturbation of second-order nonlinear system with boundary perturbation

Ｗｅｃｏｎｓｉｄｅｒｉｎｔｈｉｓｐａｐｅｒｔｈｅｓｉｎｇｕｌａｒｐｅｒｔｕｒｂａｔｉｏｎｏｆｓｅｃｏｎｄ＿ｏｒｄｅｒｎｏｎｌｉｎｅａｒｓｙｓｔｅｍｉｎｖｏｌｖｉｎｇｉｎｔｅｒｇｒａｌｏｐｅｒａｔｏｒεｙ″=ｆ(ｔ,ｙ,Ｔｙ,ε)ｙ′ ｇ(ｔ,ｙ,Ｔｙ,ε),(1)ｗｉｔｈｂｏｕｎｄａｒｙｐｅｒｔｕｒｂａｔｉｏｎｙ(ｔ,ε)｜ｔ=φ(ε)=α(ε),ｙ(ｔ,ε)｜ｔ=1 ψ(ε)=β(ε),(2)ｗｈｅｒｅε>0ｉｓａｓｍａｌｌｐａｒａｍｅｔｅｒ,ａｎｄφ(ε),ψ(ε)ａｒｅｂｏｔｈ,ｗｉｔｈｒｅｓｐｅｃｔｔｏε,ｓｕｆｆｉｃｉｅｎｔｌｙｓｍｏ…     

Asymptotic solution of a singularly perturbed nonlinear state regulator problem

I.1utroductlouWeconsidcrthenonlinearstateregulatorproblemsconsistingofthesystemontheinterval0't相似文献   

High order multiplication perturbation method for singular perturbation problems

This paper presents a high order multiplication perturbation method for sin- gularly perturbed two-point boundary value problems with the boundary layer at one end. By the theory of singular perturbations, the singularly perturbed two-point boundary value problems are first transformed into the singularly perturbed initial value problems. With the variable coefficient dimensional expanding, the non-homogeneous ordinary dif- ferential equations （ODEs） are transformed into the homogeneous ODEs, which are then solved by the high order multiplication perturbation method. Some linear and nonlinear numerical examples show that the proposed method has high precision.     

A uniform high-order method for a singular perturbation problem in conservative form

A uniform high order method is presented for the numerical solution of a singular perturbation problem in conservative form. We firest replace the original second-order problem (1.1) by two equivalent first-order problems (1.4), i.e., the solution of (1.1) is a linear combination of the solutions of (1.4). Then we derive a uniformly O(h~m+1)accurate scheme for the first-order problems (1.4), where m is an arbitrary nonnegative integer, so we can get a uniformly O(h~m+1) accurate solution of the original problem (1.1) by relation (1.3). Some illustrative numerical results are also given.     

SINGULARLY PERTURBED NONLINEAR BOUNDARY VALUE PROBLEM FOR A KIND OF VOLTERRA TYPE FUNCTIONAL DIFFERENTIAL EQUATION

IntroductionThereweresomeresultsofstudyingonboundaryvalueproblemsforfunctionaldifferentialequation[1～6 ]byemployingthetoplolgicaldegreetheoryandsomefixedpointprinciplesinrecentyears.Buttheworktostudyboundaryvalueproblemsfordelaydifferentialequationwithsmallparameterbymeansofthetheoryofsingularperturbationrarelyappeared[7～11].Thereasonforitisthattheworktoconstructtheuppersolutionandlowersolutionforthecaseofdifferentialequationwithdelayisdifficult.Theauthorhasstudiedakindofboundaryvalueproblem…     

Coupling of high order multiplication perturbation method and reduction method for variable coefcient singular perturbation problems

Based on the precise integration method(PIM), a coupling technique of the high order multiplication perturbation method(HOMPM) and the reduction method is proposed to solve variable coefcient singularly perturbed two-point boundary value problems(TPBVPs) with one boundary layer. First, the inhomogeneous ordinary diferential equations(ODEs) are transformed into the homogeneous ODEs by variable coefcient dimensional expansion. Then, the whole interval is divided evenly, and the transfer matrix in each sub-interval is worked out through the HOMPM. Finally, a group of algebraic equations are given based on the relationship between the neighboring sub-intervals, which are solved by the reduction method. Numerical results show that the present method is highly efcient.     

Coupling of high order multiplication perturbation method and reduction method for variable coefficient singular perturbation problems

Based on the precise integration method （PIM）, a coupling technique of the high order multiplication perturbation method （HOMPM） and the reduction method is proposed to solve variable coefficient singularly perturbed two-point boundary value prob lems （TPBVPs） with one boundary layer. First, the inhomogeneous ordinary differential equations （ODEs） are transformed into the homogeneous ODEs by variable coefficient dimensional expansion. Then, the whole interval is divided evenly, and the transfer ma trix in each sub-interval is worked out through the HOMPM. Finally, a group of algebraic equations are given based on the relationship between the neighboring sub-intervals, which are solved by the reduction method. Numerical results show that the present method is highly efficient.     

INTERPOLATION PERTURBATION METHOD FOR SOLVING THE BOUNDARY LAYER TYPE PROBLEMS

ＹｕａｎＹｉｗｕ（袁镒吾）（ＲｅｃｅｉｖｅｄＯｃｔ．２，１９９４；ＣｏｍｍｕｎｉｃａｔｅｄｂｙＣｈｉｅｎＷｅｉｚａｎｇ）ＩＮＴＥＲＰＯＬＡＴＩＯＮＰＥＲＴＵＲＢＡＴＩＯＮＭＥＴＨＯＤＦＯＲＳＯＬＶＩＮＧＴＨＥＢＯＵＮＤＡＲＹＬＡＹＥＲＴＹＰＥＰＲＯＢ...     

Initial layer phenomena for a class of singular perturbed nonlinear system with slow variables

The initial layer phenomena for a class of singular perturbed nonlinear system with slow variables are studied. By introducing stretchy variables with different quantity levels and constructing the correction term of initial layer with different “ thickness“, the Norder approximate expansion of perturbed solution concerning small parameter is obtained, and the “ multiple layer“ phenomena of perturbed solutions are revealed. Using the fixed point theorem, the existence of perturbed solution is proved, and the uniformly valid asymptotic expansion of the solutions is given as well.