Third-order nonlinear singularly perturbed boundary value problem

ＩｎｔｒｏｄｕｃｔｉｏｎＳｉｎｃｅ 1 958,Ｌｅｖｓｉｏｎａｎｄｓｅｖｅｒａｌｏｔｈｅｒａｕｔｈｏｒｓ[1]- [4 ]ｈａｖｅｍａｄｅａｓｕｂｓｔａｎｔｉａｌｓｔｕｄｙｆｏｒｅｘｉｓｔｅｎｃｅａｎｄｅｓｔｉｍａｔｅｏｆｓｏｌｕｔｉｏｎｓｆｏｒｔｈｅｔｈｉｒｄ＿ｏｒｄｅｒｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｗｉｔｈｔｗｏｐｏｉｎｔｂｏｕｎｄａｒｙｃｏｎｄｉｔｉｏｎｓｘ( 0 ) =Ａ ,ｘ′( 0 ) =Ｂ ,ｘ′( 1 ) =Ｃ ,ｏｒｓｌｉｇｈｔｌｙｅｘｔｅｎｓｉｖｅｌｙｌｉｎｅａｒｂｏ…     

Contrast structure for singular singularly perturbed boundary value problem

The step-type contrast structure for a singular singularly perturbed problem is shown. By use of the method of boundary function, the formal asymptotic expansion is constructed. At the same time, based on sewing orbit smooth, the existence of the step- type solution and the uniform validity of the asymptotic expansion are proved. Finally, an example is given to demonstrate the effectiveness of the present results.     

Boundary value problems for second order singularly perturbed differential equations

In this paper the existence and uniqueness of solutions of boundary value problems (which contains the Robin's problem) is discussed by using the upper and lower solution. In addition, the asymptotic estimation of the solution is given as well.     

On local asymptotic expansions of solutions of singularly perturbed systems of differential equations with degeneration at a point

We propose an algorithm for the construction of asymptotic solutions of singularly perturbed systems of differential equations in the case where the characteristic equation has simple roots. In contrast to previous investigations, in which the matrix multiplying the derivatives becomes degenerate on the entire interval, we study the case where degeneration occurs at a single point.     

Uniformly convergent numerical method for singularly perturbed convection-diffusion type problems with nonlocal boundary condition

In this article, we consider a class of singularly perturbed differential equations of convection-diffusion type with nonlocal boundary conditions. A uniformly convergent numerical method is constructed via nonstandard finite difference and numerical integration methods to solve the problem. The nonlocal boundary condition is treated using numerical integration techniques. Maximum absolute errors and rates of convergence for different values of perturbation parameter and mesh sizes are tabulated for the numerical example considered. The method is shown to be ϵ -uniformly convergent.     

Nonlinear singularly perturbed problems of ultra parabolic equations

A class of nonlinear singularly perturbed problem of ultra parabolic equations are considered. Using the comparison theorem, the existence, uniqueness and its asymptotic behavior of solution for the problem are studied.     

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−)     

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.     

Critical case for singularly perturbed Noether boundary-value problems

We construct an asymptotic expansion of a solution for singularly perturbed linear systems of ordinary differential equations of the Noether type in the critical case. We successively determine all terms of the asymptotic expansion by the method of boundary functions and pseudoinverse matrices. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 1, pp. 45–54, January–March, 2007.     

On the construction of asymptotic solutions of two-point boundary-value problems for degenerate singularly perturbed systems of differential equations

We consider the problem of the existence of a solution of a two-point boundary-value problem for degenerate singularly perturbed linear systems of differential equations. We obtain asymptotic formulas for this solution.     

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…     

Boundary value problem for a singularly perturbed nonlinear system

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

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,…     

Construction of asymptotic formulas for solutions of singularly perturbed degenerate systems of differential equations

We obtain an asymptotic solution of the Cauchy problem for a singularly perturbed degenerate system of differential equations in the case of a singular limit pencil of matrices. Translated from Neliniini Kolyvannya, Vol. 11, No. 3, pp. 408–420, July–September, 2008.     

Shift of shock position for a class of nonlinear singularly perturbed problems

Thenonlinearsingularlyperturbedproblemisaveryattentiveobjectofstudyintheinternationalacademiccircles[1].Recently ,manyscholarsstudiedagreatdealofwork .Forexample,Boh啨[2 ],ButuzovandSmurov[3],O’Malley ,Jr.[4 ],Butuzov ,NefedovandSchneider[5 ]andKelley[6 ]andsoon .TheauthorconsideredaclassofsingularlyperturbednonlinearboundaryvalueproblemsfortheordinarydifferentialequationinRefs .[7～ 1 0 ] ,reactiondiffusionequationsinRefs.[1 1～ 1 3 ] ,singularlyperturbedproblemswithnonlocalboundarycond…     

On the asymptotic integration of a singularly perturbed system of linear differential equations with deviating argument

We prove the existence and infinite differentiability of a solution of a singularly perturbed linear system of differential equations with deviating arguments and a turning point.     

On singularly perturbed quasilinear systems

In this paper, the objective is to give sufficient conditions for the existence of solution of the nonlinear two-point boundary value problem(1.1). And we employ these results to consider the boundary layer phenomena of the quasilinear weakly coupled singularly perturbed system (DP)_q.     

Existence of solutions for a singularly perturbed boundary value problem for the third order semi-linear differential equation with a turning point

In this paper,by using the techniques of differential inequalities,we prove the existence of the solutions of a singularly perturbed boundary value problem for the third order semilinear differential equation with a turning point.     

The boundary value problems for thin coats

We formulate and solve three boundary value problems for the coats of finite and infinite domains in the plane.