In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal...In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal asymptotic solution by Lyuternik- Vishik 's method. Finally, by means of the energy estimates we obtain the bound of the remainder of the asymptotic solution.展开更多
In this paper, using nonuniform mesh and exponentially fitted difference method, a uniformly convergent difference scheme for an initial-boundary value problem of linear parabolic differential equation with the nonsmo...In this paper, using nonuniform mesh and exponentially fitted difference method, a uniformly convergent difference scheme for an initial-boundary value problem of linear parabolic differential equation with the nonsmooth boundary layer function with respect to small parameter e is given, and error estimate and numerical result are also given.展开更多
In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the a...In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the asymptotic solution are given. Then an exponentially fitted difference scheme is developed in an equidistant mesh. Finally, uniform convergence in small parameter is proved in the sense of discrete energy norm.展开更多
The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the sp...The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the special non-uniform grids. The uniform con vergence of this scheme is proved and some numerical examples are given.展开更多
In this paper we construct the finite-difference scheme for the singularly perturbed boundary value problem for the fourth-order elliptic differential equation on the basis of paper [1], and prove the uniform converge...In this paper we construct the finite-difference scheme for the singularly perturbed boundary value problem for the fourth-order elliptic differential equation on the basis of paper [1], and prove the uniform convergence of this scheme with respect to the small parameter Ε in the discrete energy norm. Finally, we give a numerical example.展开更多
In this paper, we first consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the priori estimation of the solution of the continuous problem. Then, we...In this paper, we first consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the priori estimation of the solution of the continuous problem. Then, we present an exponential fitted difference scheme and discuss the solution properties of the difference equations. Finally, the uniform convergence of this scheme with respect to the small parameter in the discrete energy norm, is proved.展开更多
This paper is taken up for the following difference equation problem (Pe);where e is a small parameter, c1, c2,constants and functions of k and e . Firstly, the case with constant coefficients is considered. Secondly,...This paper is taken up for the following difference equation problem (Pe);where e is a small parameter, c1, c2,constants and functions of k and e . Firstly, the case with constant coefficients is considered. Secondly, a general method based on extended transformation is given to handle (P.) where the coefficients may be variable and uniform asymptotic expansions are obtained. Finally, a numerical example is provided to illustrate the proposed method.展开更多
In this paper we consider the initial-boundary value problem for a second order hyperbolic equation with initial jump. The bounds on the derivatives of the exact solution are given. Then a difference scheme is constru...In this paper we consider the initial-boundary value problem for a second order hyperbolic equation with initial jump. The bounds on the derivatives of the exact solution are given. Then a difference scheme is constructed on a non-uniform grid. Finally, uniform convergence of the difference solution is proved in the sense of the discrete energy norm.展开更多
In this paper we discuss, an initial-boundary value problem of hyperbolic type with first derivative with respect to x. The asymptotic solution is constructed and its uniform validity is proved under weader compatibil...In this paper we discuss, an initial-boundary value problem of hyperbolic type with first derivative with respect to x. The asymptotic solution is constructed and its uniform validity is proved under weader compatibility conditions. Then we develop an exponentially fitted difference scheme and establish discrete energy inequality. Finally, we prove that the solution of difference problem uniformly converges to the solution of the original problem.展开更多
文摘In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal asymptotic solution by Lyuternik- Vishik 's method. Finally, by means of the energy estimates we obtain the bound of the remainder of the asymptotic solution.
文摘In this paper, using nonuniform mesh and exponentially fitted difference method, a uniformly convergent difference scheme for an initial-boundary value problem of linear parabolic differential equation with the nonsmooth boundary layer function with respect to small parameter e is given, and error estimate and numerical result are also given.
文摘In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the asymptotic solution are given. Then an exponentially fitted difference scheme is developed in an equidistant mesh. Finally, uniform convergence in small parameter is proved in the sense of discrete energy norm.
文摘The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the special non-uniform grids. The uniform con vergence of this scheme is proved and some numerical examples are given.
文摘In this paper we construct the finite-difference scheme for the singularly perturbed boundary value problem for the fourth-order elliptic differential equation on the basis of paper [1], and prove the uniform convergence of this scheme with respect to the small parameter Ε in the discrete energy norm. Finally, we give a numerical example.
文摘In this paper, we first consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the priori estimation of the solution of the continuous problem. Then, we present an exponential fitted difference scheme and discuss the solution properties of the difference equations. Finally, the uniform convergence of this scheme with respect to the small parameter in the discrete energy norm, is proved.
文摘This paper is taken up for the following difference equation problem (Pe);where e is a small parameter, c1, c2,constants and functions of k and e . Firstly, the case with constant coefficients is considered. Secondly, a general method based on extended transformation is given to handle (P.) where the coefficients may be variable and uniform asymptotic expansions are obtained. Finally, a numerical example is provided to illustrate the proposed method.
文摘In this paper we consider the initial-boundary value problem for a second order hyperbolic equation with initial jump. The bounds on the derivatives of the exact solution are given. Then a difference scheme is constructed on a non-uniform grid. Finally, uniform convergence of the difference solution is proved in the sense of the discrete energy norm.
文摘In this paper we discuss, an initial-boundary value problem of hyperbolic type with first derivative with respect to x. The asymptotic solution is constructed and its uniform validity is proved under weader compatibility conditions. Then we develop an exponentially fitted difference scheme and establish discrete energy inequality. Finally, we prove that the solution of difference problem uniformly converges to the solution of the original problem.
