Wavelet Multi-Resolution Interpolation Galerkin Method for Linear Singularly Perturbed Boundary Value Problems

In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be readily extended to special node generation techniques,such as the Shishkin node.Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients.All the shape functions possess the Kronecker delta property,making the imposition of boundary conditions as easy as that in the finite element method.Four numerical examples are studied to demonstrate the validity and accuracy of the proposedwavelet method.The results showthat the use ofmodified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer.Compared with many other methods,the proposed method possesses satisfactory accuracy and efficiency.The theoretical and numerical results demonstrate that the order of theε-uniform convergence of this wavelet method can reach 5.

The Singularly Perturbed Boundary Value Problems for Elliptic Equation with Turning Point

The singularly perturbed elliptic equation boundary value problem with turning point is considered. Using the method of multiple scales and the comparison theorem, the asymptotic behavior of solution for the boundary value problem is studied.

MULTIPLE POSITIVE SOLUTIONS FOR SUPERLINEAR SECOND ORDER SINGULAR BOUNDARY VALUE PROBLEMS WITH DERIVATIVE DEPENDENCE

The existence of at least two positive solutions is presented for the singular second-order boundary value problem{1/p（t）（ p（t）x′（t））′＋Φ（t）f（t,x（t）,p（t）x′（t））=0,0〈t〈1, limt→0 p（t）x′（t）=0,x（1）=0by using the fixed point index, where f may be singular at x = 0 and px ′= 0.

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

A boundary value problems for functional differenatial equations, with nonlinear boundary condition, is studied by the theorem of differential inequality. Using new method to construct the upper solution and lower solution, sufficient conditions for the existence of the problems' solution are established. A uniformly valid asymptotic expansions of the solution is also given.

A UNIFORMLY DIFFERENCE SCHEME OF SINGULAR PERTURBATION PROBLEM FOR A SEMILINEAR ORDINARY DIFFERENTIAL EQUATION WITH MIXED BOUNDARY VALUE CONDITION

In the poper, the method of separating singularity is applied to study the uniformly difference scheme of a singular perturbation problem for a semilinear ordinary differential equation with mixed boundary value condition. The uniform convergence on small parameter ε of order one for an IVin type difference scheme constructed is proved. At the end of the paper, a numerical example is given. The computing results coincide with the theoretical analysis.

Existence of Boundary Value Problems for Impulsive Fractional Differential Equations with a Parameter

We investigate a class of boundary value problems for nonlinear impulsive fractional differential equations with a parameter.By the deduction of Altman’s theorem and Krasno-selskii’s fixed point theorem,the existence of this problem is proved.Examples are given to illustrate the effectiveness of our results.

A SINGULAR DIRICHLET PROBLEM FOR THE MONGE-AMPÈRE TYPE EQUATION

We consider the singular Dirichlet problem for the Monge-Ampère type equation■=0,whereΩis a strictly convex and bounded smooth domain in■is positive and strictly decreasing in(0,∞)with■is positive inΩ.We obtain the existence,nonexistence and global asymptotic behavior of the convex solution to such a problem for more general b and g.Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.

THE ASYMPTOTIC BEHAVIOR OF SOLUTION FOR THE SINGULARLY PERTURBED INITIAL BOUNDARY VALUE PROBLEMS OF THE REACTION DIFFUSION EQUATIONS IN A PART OF DOMAIN

A class of singularly perturbed initial boundary value problems for the reaction diffusion equations in a part of domain are considered. Using the operator theory the asymptotic behavior of solution for the problems is studied.

THE NONLINEAR BOUNDARY VALUE PROBLEM FOR k HOLOMORPHIC FUNCTIONS IN C^(2)

k holomorphic functions are a type of generation of holomorphic functions.In this paper,a nonlinear boundary value problem for k holomorphic functions is primarily discussed on generalized polycylinders in C^(2).The existence of the solution for the problem is studied in detail with the help of the boundary properties of Cauchy type singular integral operators with a k holomorphic kernel.Furthermore,the integral representation for the solution is obtained.

MULTIPLE POSITIVE SOLUTIONS OF SINGULAR THIRD-ORDER PERIODIC BOUNDARY VALUE PROBLEM

This paper deals with the singular nonlinear third-order periodic boundary value problem u'' + p(3)u = f (t, u), 0 less than or equal to t less than or equal to 2pi, with u((i)) (0) = u((i)) (2pi), i = 0, 1, 2, where p is an element of (Graphics) and f is singular at t = 0, t = 1 and u = 0. Under suitable weaker conditions than those of [1], it is proved by constructing a special cone in C[0, 2pi] and employing the fixed point index theory that the problem has at least one or at least two positive solutions.

MULTIPLE POSITIVE SOLUTIONS TO SINGULAR BOUNDARY VALUE PROBLEMS FOR SUPERLINEAR SECOND ORDER ODES

This paper deals with the existence of positive solutions to the singular boundary value problemwhere q(t) may be singular at t = 0 and t = 1, f(t,y) may be superlinear at y =∞ and singular, at y = 0.

Multiple positive solutions for a class of nonlinear four-point boundary value problem with p-Laplacian

This paper deals with the existence of multiple positive solutions for a class of nonlinear singular four-point boundary value problem with p-Laplacian：{（φ（u′））′＋a（t）f（u（t））=0, 0〈t〈1, αφ（u（0））-βφ（u′（ξ））=0,γφ（u（1））＋δφ（u′（η））0,where φ（x） = ｜x｜^p-2x,p 〉 1, a（t） may be singular at t = 0 and/or t = 1. By applying Leggett-Williams fixed point theorem and Schauder fixed point theorem, the sufficient conditions for the existence of multiple （at least three） positive solutions to the above four-point boundary value problem are provided. An example to illustrate the importance of the results obtained is also given.

A Class of Strongly Nonlinear Singular Perturbed Boundary Value Problems

In this paper, a class of strongly nonlinear singular perturbed boundary value problems are coasidered by the theory of differential inequalities and the correction of boundary layer, under which the existence of solution is proved and the uniformly valid asymptotic expansions is obtained as well.

EXISTENCE THEOREMS FOR A SECOND ORDER THREE-POINT BOUNDARY VALUE PROBLEM WITH IMPULSES

In this paper,two existence theorems are given concerning the following 3-point boundary value problem of second order differential systems with impulses[HL(2:1,1Z;2,1Z]x″(t)=f(t,x(t),x′(t)),t∈(0,1),t≠t_k,k=1,2,...,m, Δx|_~t=t_k =I_k(x(t_k)),k=1,2,...,m, Δx′|_~t=t_k =J_k(x(t_k),x′(t_k)),k=1,2,...,m, x(0)=0,x(1)=αx(η).

THE SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR HIGHER ORDER SEMILINEAR ELLIPTIC EQUATIONS

In this paper a singular perturbation of boundary value problem for elliptic partial differential equations of higher order is considered by using the differential inequalities. The uniformly valid asymptotic expansion in entire region is obtained.

The Shock Solution for a Class of Quasilinear Singularly Perturbed Boundary Value Problems

The existence and asymptotic behavior of solution for a class of quasilinear singularly perturbed boundary value problems are discussed under suitable conditions by the theory of differential inequalities and matching principle.

The Existence of Solutions of Three-point Boundary Value Problems for Second Order Impulsive Differential Equation

We study the existence of solutions to the second order three-point boundary value problem：{x″（t）＋f（t,x（t）,x′（t））=0,t≠ti,△x（ti）=Ii（x（ti）,x′（ti））,i=1,2,…,k,△x′（ti）=Ji（x（ti）,x′（t））,i=1,2,…,k,x（0）=0=x（1）-αx（η）,where 0〈η〈1,α∈R,and f：[0,1]×R×R→R,Ii：R×R→R,Ji：R×R→R（i=1,2,…,k）are continuous. Our results is new and different from previous results. In particular, we obtain the Green function of the problem, which makes the problem simpler.

A CLASS OF SINGULARLY PERTURBED NONLINEAR BOUNDARY VALUE PROBLEM

The singularly perturbed boundary value problem for the nonlinear boundary conditions is considered.Under suitable conditions,the asymptotic behavior of solution for the original problems is studied by using theory of differential inequalities.

TWO POSITIVE SOLUTIONS TO THREE-POINT SINGULAR BOUNDARY VALUE PROBLEMS

In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem： -u′′（t） = λf（t, u（t）） for all t ∈ （0, 1） subjecting to u（0） = 0 and αu（η） = u（1）, where η ∈ （0, 1）, α ∈ [0, 1）, and λ is a positive parameter. The nonlinear term f（t, u） is nonnegative, and may be singular at t = 0, t = 1, and u = 0. By the fixed point index theory and approximation method, we establish that there exists λ＊ ∈ （0, ＋∞], such that the above problem has at least two positive solutions for any λ ∈ （0, λ＊） under certain conditions on the nonlinear term f.

A Boundary Value Problem for Second Order Functional Differential Equations with Impulses

Aim To investigate the boundary value problem for second order functional differentiai equations with impulses. Methods The fixed point principle was used to establish our results. Results and Conclusion The results of the esistence, the uniqueness and the continuous dependence on aprameter of soiutions of the boundary value problems for second order functional differential equations with impulses are obtained.