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.

A Radial Basis Function Method with Improved Accuracy for Fourth Order Boundary Value Problems

Accurately approximating higher order derivatives is an inherently difficult problem. It is shown that a random variable shape parameter strategy can improve the accuracy of approximating higher order derivatives with Radial Basis Function methods. The method is used to solve fourth order boundary value problems. The use and location of ghost points are examined in order to enforce the extra boundary conditions that are necessary to make a fourth-order problem well posed. The use of ghost points versus solving an overdetermined linear system via least squares is studied. For a general fourth-order boundary value problem, the recommended approach is to either use one of two novel sets of ghost centers introduced here or else to use a least squares approach. When using either ghost centers or least squares, the random variable shape parameter strategy results in significantly better accuracy than when a constant shape parameter is used.

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.

A Class of Nonlocal Boundary Value Problems for Elliptic Systems in Unbounded Domains

A class of nonlocal boundary value probl em s for elliptic systems in the unbounded domains are considered. Under suitable c onditions, the existence of solution and the comparison theorem for the boundary value problems are studied.

A CLASS OF NONLOCAL BOUNDARY VALUE PROBLEMS OF NONLINEAR ELLIPTIC SYSTEMS IN UNBOUNDED DOMAINS

The nonlocal boundary value problems for nonlinear elliptic systems in the unbounded domain are considered. Under suitable conditions the existence of solution and comparison theorem for the boundary value problems are studied.

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.

Complex variable solution for boundary value problem with X-shaped cavity in plane elasticity and its application*

A new type of displacement pile, the X-section cast-in-place concrete （XCC） pile, has recently been developed in China. Extensive field tests and laboratory experi- ments are undertaken to evaluate its performance and quantify the non-uniform deforma- tion effect （NUDE） of the X-shaped cross section during installation. This paper develops a simplified theoretical model that attempts to capture the NUDE. Based on the theory of complex variable plane elasticity, closed-form solutions of the stress and displacement for the X-shaped cavity boundary value problem are given. Subsequently, the analytical solution is used to evaluate the NUDE, the concrete filling index （CFI）, and the perimeter reduction coefficient of the XCC pile cross section. The computed results are compared with field test results, showing reasonable agreement. The present simplified theoretical model reveals the deformation mechanism of the X-shaped cavity and facilitates applica- tion of the newly developed XCC pile technique in geotechnical engineering.

SUPERCONVERGENCE OF GENERALIZED DIFFERENCE METHOD FOR ELLIPTIC BOUNDARY VALUE PROBLEM

Some superconvergence results of generalized difference solution for elliptic boundary value problem are given. It is shown that optimal points of the stresses for generalized difference method are the same as that for finite element method.

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.

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.

A CLASS OF NONLINEAR BOUNDARY VALUE PROBLEMS FOR THE SECOND-ORDER E_2 CLASS ELLIPTIC SYSTEMS IN GENERAL FORM

A class of nonlinear boundary value problems(BVP) for the second_order E 2 class elliptic systems in general form is discussed. By introducing a kind of transformation,this kind of BVP is reduced to a class of generalized nonlinear Riemann_Hilbert BVP. And then some singular integral operators are introduced to establish the equivalent nonlinear singular integral equations. The solvability is proved under some suitable hypotheses by means of the properties of singular integral operators and the function theoretic methods.

MIXED BOUNDARY VALUE PROBLEM FOR SECOND－ORDER SYSTEM OF DIFFERENTIAL EQUATIONS OF THE ELLIPTIC TYPE

In this paper.the following ined boundary value problem for second-order system of differential equations of the elliptic type will be discussed to find the function u and v such that they satisfy:The solution of this problem is found by means of the theory of generalized analutic function and the integral equation method for solving boundary value problems.

A SPECIAL METHOD OF FOURIER SERIES WHICH IS EQUAL TO THE METHOD OF SEPARATION OF VARIABLES ON BOUNDARY VALUE PROBLEM

By the separation of singularity, a special Fourier series solution of the boundary value problem for plane is obtained, which can satisfy all boundary conditions and converges rapidly. II is proved that the solution is equal to the result of separation of variables. As a result, the non-linear characteristic equations resulting from the method of separation of variables are transformed into polynomial equations that can provide a foundation for approximate computation and asymptotic analysis.

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.

Dirichlet boundary value problem with variable growth

In this paper, we study higher order elliptic partial differential equations with variable growth, and obtain the existence of solutions in the setting of Wm,p（x） spaces by means of an abstract result for variational inequalities obtained by Gossez and Mustonen. Our result generalizes the corresponding one of Kováik and Rákosník.

Jacobi-Sobolev Orthogonal Polynomialsand Spectral Methods for Elliptic Boundary Value Problems

Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered,and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems,the Jacobi-Sobolev orthogonal basis functions are constructed,which allow the exact solutions and the approximate solutions to be represented in the forms of infinite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the effectiveness and the spectral accuracy.

Modified Tikhonov Method for Cauchy Problem of Elliptic Equation with Variable Coefficients

A Cauchy problem for the elliptic equation with variable coefficients is considered. This problem is severely ill-posed. Then, we need use the regularization techniques to overcome its ill-posedness and get a stable numerical solution. In this paper, we use a modified Tikhonov regularization method to treat it. Under the a-priori bound assumptions for the exact solution, the convergence estimates of this method are established. Numerical results show that our method works well.

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.

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.