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 suff...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.展开更多
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 ar...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.展开更多
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 expansio...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.展开更多
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 fo...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.展开更多
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...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.展开更多
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 o...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.展开更多
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 genera...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.展开更多
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) (Br...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.展开更多
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 ...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.展开更多
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 bo...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.展开更多
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...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.展开更多
In this paper,a class of singular perturbation of nonlocal boundary value problems for elliptic partial differential equations of higher order is considered by using the differential inequalities.The uniformly valid a...In this paper,a class of singular perturbation of nonlocal boundary value problems for elliptic partial differential equations of higher order is considered by using the differential inequalities.The uniformly valid asymptotic expansion of solution is obtained.展开更多
This paper investigates the boundary value problems for a class of singularly perturbed nonlinear elliptic equations. By means of the theory of partial differential in- equalities the author obtains the existence and ...This paper investigates the boundary value problems for a class of singularly perturbed nonlinear elliptic equations. By means of the theory of partial differential in- equalities the author obtains the existence and asymptotic estimation of the solutions, involving the boundary and interior layer behavior, of the problems as described.展开更多
We show a result of symmetry for a big class of problems with condition of Neumann on the boundary in the case one dimensional. We use the method of reflection of Alexandrov and we show one application of this method ...We show a result of symmetry for a big class of problems with condition of Neumann on the boundary in the case one dimensional. We use the method of reflection of Alexandrov and we show one application of this method and the maximum principle for elliptic operators in problems with conditions of Neumann. Some results of symmetry for elliptic problems with condition of Neumann on the boundary may be extended to elliptic operators more general than the Laplacian.展开更多
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 variationa...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.展开更多
In [1], I. N. Vekua propose the Poincaré problem for some second order elliptic equations, but it can not be solved. In [2], the authors discussed the boundary value problem for nonlinear elliptic equations of se...In [1], I. N. Vekua propose the Poincaré problem for some second order elliptic equations, but it can not be solved. In [2], the authors discussed the boundary value problem for nonlinear elliptic equations of second order in some bounded domains. In this article, the Poincaré boundary value problem for general nonlinear elliptic equations of second order in unbounded multiply connected domains have been completely investigated. We first provide the formulation of the above boundary value problem and corresponding modified well posed-ness. Next we obtain the representation theorem and a priori estimates of solutions for the modified problem. Finally by the above estimates of solutions and the Schauder fixed-point theorem, the solvability results of the above Poincaré problem for the nonlinear elliptic equations of second order can be obtained. The above problem possesses many applications in mechanics and physics and so on.展开更多
In this paper we present a numerical method for solving Riemann type problem for the fifth order improperly elliptic equation in complex plane .We reduce this problem to the boundary value problems for properly ellipt...In this paper we present a numerical method for solving Riemann type problem for the fifth order improperly elliptic equation in complex plane .We reduce this problem to the boundary value problems for properly elliptic equations, and then solve those problems by the grid methods.展开更多
The authors investigatc relations between multiplicity of solutions and sourceterms of the fourth order nonlinear elliptic boundary value problem under Dirichlet boundary condition △2u+c△u = bu++f inΩ, wherc Ω i...The authors investigatc relations between multiplicity of solutions and sourceterms of the fourth order nonlinear elliptic boundary value problem under Dirichlet boundary condition △2u+c△u = bu++f inΩ, wherc Ω is a bounded open set in Rn with smoothbonndary and the nonlinearity bu+ crosses eigenvalues of △2 +c△. They investigate therelatiolls when the source term is constant and when it is generated by two eigenfuntions.展开更多
文摘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.
基金The project supported by the National Natural Science Foundation of China
文摘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.
文摘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.
基金This work is supported by the Foundatiorl of Zhongshan University Advanced Research Centre
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
基金the National Natural Science Foundation of China (Nos.11571238,11601332,91130014,11471312 and 91430216).
文摘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.
文摘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.
文摘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.
文摘In this paper,a class of singular perturbation of nonlocal boundary value problems for elliptic partial differential equations of higher order is considered by using the differential inequalities.The uniformly valid asymptotic expansion of solution is obtained.
文摘This paper investigates the boundary value problems for a class of singularly perturbed nonlinear elliptic equations. By means of the theory of partial differential in- equalities the author obtains the existence and asymptotic estimation of the solutions, involving the boundary and interior layer behavior, of the problems as described.
文摘We show a result of symmetry for a big class of problems with condition of Neumann on the boundary in the case one dimensional. We use the method of reflection of Alexandrov and we show one application of this method and the maximum principle for elliptic operators in problems with conditions of Neumann. Some results of symmetry for elliptic problems with condition of Neumann on the boundary may be extended to elliptic operators more general than the Laplacian.
文摘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.
文摘In [1], I. N. Vekua propose the Poincaré problem for some second order elliptic equations, but it can not be solved. In [2], the authors discussed the boundary value problem for nonlinear elliptic equations of second order in some bounded domains. In this article, the Poincaré boundary value problem for general nonlinear elliptic equations of second order in unbounded multiply connected domains have been completely investigated. We first provide the formulation of the above boundary value problem and corresponding modified well posed-ness. Next we obtain the representation theorem and a priori estimates of solutions for the modified problem. Finally by the above estimates of solutions and the Schauder fixed-point theorem, the solvability results of the above Poincaré problem for the nonlinear elliptic equations of second order can be obtained. The above problem possesses many applications in mechanics and physics and so on.
文摘In this paper we present a numerical method for solving Riemann type problem for the fifth order improperly elliptic equation in complex plane .We reduce this problem to the boundary value problems for properly elliptic equations, and then solve those problems by the grid methods.
文摘The authors investigatc relations between multiplicity of solutions and sourceterms of the fourth order nonlinear elliptic boundary value problem under Dirichlet boundary condition △2u+c△u = bu++f inΩ, wherc Ω is a bounded open set in Rn with smoothbonndary and the nonlinearity bu+ crosses eigenvalues of △2 +c△. They investigate therelatiolls when the source term is constant and when it is generated by two eigenfuntions.