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.

The Regularity of Solutions to Mixed Boundary Value Problems of Second-Order Elliptic Equations with Small Angles

This paper considers the regularity of solutions to mixed boundary value problems in small-angle regions for elliptic equations. By constructing a specific barrier function, we proved that under the assumption of sufficient regularity of boundary conditions and coefficients, as long as the angle is sufficiently small, the regularity of the solution to the mixed boundary value problem of the second-order elliptic equation can reach any order.

ASYMPTOTIC SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS WITH TURNING POINTS

Boundary value problem; for third-order ordinary differential equations with turning points are studied as follows : epsilon gamma ' ' + f(x ; epsilon) gamma ' + g(x ; epsilon) gamma ' +h(x ; epsilon) gamma = 0 (- a < x < b, 0 epsilon 1), where f(x ; 0) has several multiple zero points in ( - n, b). the necessary conditions for exhibiting resonance is given, and the uniformly valid asymptotic solutions and the estimations of remainder terms are obtained.

A class of boundary value problems for third-order differential equation with a turning point

A class of boundary value problems for a third-order differential equation with a turning point is considered. Using the method of multiple scales and others, the uniformly valid asymptotic expansion of solution for the boundary value problem is constructed.

SOME BOUNDARY VALUE PROBLEMS FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS OF SECOND ORDER

The present article deals with some boundary value problems for nonlinear elliptic equations with degenerate rank 0 including the oblique derivative problem. Firstly the formulation and estimates of solutions of the oblique derivative problem are given, and then by the above estimates and the method of parameter extension, the existence of solutions of the above problem is proved. In this article, the complex analytic method is used, namely the corresponding problem for degenerate elliptic complex equations of first order is firstly discussed, afterwards the above problem for the degenerate elliptic equations of second order is solved.

On Solvability of an Inverse Boundary Value Problem for a Fourth Order Elliptic Equation

In the paper an inverse boundary value problem for a fourth order elliptic equation with an integral condition of the first kind is investigated. First, the given problem is reduced to an equivalent problem in a certain sense. Then, using the Fourier method the equivalent problem is reduced to solving the system of integral equations. The existence and uniqueness of a solution to the system of integral equation is proved by the contraction mapping principle. This solution is also the unique solution to the equivalent problem. Finally, by equivalence, the theorem of existence and uniqueness of a classical solution to the given problem is proved.

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.

General Boundary Value Problems for Nonlinear Uniformly Elliptic Equations in Multiply Connected Infinite Domains

This article discusses the general boundary value problem for the nonlinear uniformly elliptic equation of second order in D (0.1) and the boundary condition,(0.2) in a multiply connected infinite domain D with the boundary T. The above boundary value problem is called Problem G. Problem G extends the work [8] in which the equation (0.1) includes a nonlinear lower term and the boundary condition (0.2) is more general. If the complex equation (0.1) and the boundary condition (0.2) meet certain assumptions, some solvability results for Problem G can be obtained. By using reduction to absurdity, we first discuss a priori estimates of solutions and solvability for a modified problem. Then we present results on solvability of Problem G.

THE BOUNDARY VALUE PROBLEMS OF A CLASS OF ELLIPTIC EQUATION WITH A CURVE OF TURNING POINT

The boundary value problems of a class of elliptic equation with a curve of turning point are considered. Using the method of multiple scales and the comparison theorem, the asymptotic behavior of solution for the boundary value problem is studied.

Existence of Positive Solutions of Three-point Boundary Value Problem for Higher Order Nonlinear Fractional Differential Equations

In this paper, we consider the positive solutions of fractional three-point boundary value problem of the form Dο^α＋u（t）＋f（t,u（t）,u＇（t）,…,u^（n-3）（5）,u^（n-2）（t））=0,u^（i）（0）=0,0≤i≤n-2,u^（n-2）（1）-βu^（n-2）（ξ）=0,where 0〈t〈1,n-1〈α≤n,n≥2,ξ Е（0,1）,βξ^a-n〈1. We first transform it into another equivalent boundary value problem. Then, we derive the Green＇s function for the equivalent boundary value problem and show that it satisfies certain properties. At last, by using some fixed-point theorems, we obtain the existence of positive solution for this problem. Example is given to illustrate the effectiveness of our result.

Existence of solutions of nonlinear two-point boundary value problems for 4nth-order nonlinear differential equation

Studies the existence of solutions of nonlinear two point boundary value problems for nonlinear 4n-th-order differential equationy (4n)=f(t,y,y′,y″,...,y (4n-1))(a)with the boundary conditions g 2i(y (2i)(a),y (2i+1)(a))=0,h 2i(y (2i)(c),y (2i+1)(c))=0,(i=0,1,...,2n-1)(b) where the functions f, g i and h i are continuous with certain monotone properties. For the boundary value problems of nonlinear nth order differential equationy (n)=f(t,y,y′,y″,...,y (n-1))many results have been given at the present time. But the existence of solutions of boundary value problem (a),(b) studied in this paper has not been covered by the above researches. Moreover, the corollary of the important theorem in this paper, i.e. existence of solutions of the boundary value problem.y (4n)=f(t,y,y′,y″,...,y (4n-1)) a 2iy (2i)(a)+a 2i+1y (2i+1)(a)=b 2i,c 2iy (2i)(c)+c 2i+1y (2i+1)(c)=d 2i,(i=0,1,...2n-1)has not been dealt with in previous works.

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.

NUMERICAL SOLUTIONS OF DISCONTINUOUS BOUNDARY VALUE PROBLEMS FOR GENERAL ELLIPTIC COMPLEX EQUATIONS OF FIRST ORDER

In this paper, authors discuss the numerical methods of general discontinuous boundary value problems for elliptic complex equations of first order, They first give the well posedness of general discontinuous boundary value problems, reduce the discontinuous boundary value problems to a variation problem, and then find the numerical solutions of above problem by the finite element method. Finally authors give some error-estimates of the foregoing numerical solutions.

HIGH ACCURACY FINITE VOLUME ELEMENT METHOD FOR TWO-POINT BOUNDARY VALUE PROBLEM OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

In this paper, a high accuracy finite volume element method is presented for two-point boundary value problem of second order ordinary differential equation, which differs from the high order generalized difference methods. It is proved that the method has optimal order error estimate O(h3) in H1 norm. Finally, two examples show that the method is effective.

Analytical Solutions of Some Two-Point Non-Linear Elliptic Boundary Value Problems

Several problems arising in science and engineering are modeled by differential equations that involve conditions that are specified at more than one point. The non-linear two-point boundary value problem (TPBVP) (Bratu’s equation, Troesch’s problems) occurs engineering and science, including the modeling of chemical reactions diffusion processes and heat transfer. An analytical expression pertaining to the concentration of substrate is obtained using Homotopy perturbation method for all values of parameters. These approximate analytical results were found to be in good agreement with the simulation results.

Picard Boundary Value Problems of Second Order p-Laplacian Differential Equations

Suffcient conditions for the existence of at least one solution of two-point boundary value problems for second order nonlinear differential equations [φ(x(t))] + kx(t) + g(t,x(t)) = p(t),t ∈(0,π) x(0) = x(π) = 0 are established,where [φ(x)] =(|x |p-2x) with p > 1.Our result is new even when [φ(x)] = x in above problem,i.e.p = 2.Examples are presented to illustrate the effciency of the theorem in this paper.

On the Existence of Positive Solution for a Nonlinear Third-order Three-point Boundary Value Problem

An existence theorem of positive solution is established for a nonlinear third-order three-point boundary value problem. Here, we concentrated on the case that the nonlinear term is neither superlinear nor sublinear, and is not asymptotic at zero and infinity.

Positive Solutions of Four-order Dierence Equations with Two Point Boundary Value Problems

In this paper,by using the Krasnoselskii xed point theorem,we prove the existence one or multiple of positive solutions of fourth-ordernonlinear dierence equations with two point boundary value problem.

Solvability of Third-order Three-point Boundary Value Problems with Caratheodory Nonlinearity

A class of third-order three-point boundary value problems is considered, where the nonlinear term is a Caratheodory function. By introducing a height function and considering the imtegration of this height function, an existence theorem of solution is proved when the limit growth function exists. The main tools are the Lebesgue dominated convergence theorem and the Schauder fixed point theorem.

Existence of Solutions for Infinity-Point Nonlinear Fractional Boundary Value Problem at Resonance

A class of the boundary value problem for fractional order nonlinear differential equation with Riemann-Liouville fractional derivative on the half line was studied. By using the coincidence degree theory due to Mawhin and constructing the suitable operators,the existence theorem of at least one solution has been established. An example is given to illustrate our result.