Examines the development of the composite legendre approximation in unbounded domains. Proof of the stability and convergence of a proposed scheme; Discussion of two-dimensional exterior problems; Error estimations.
In this paper we consider the numerical solution of the one-dimensional heat equation on unbounded domains. First an exact semi-discrete artificial boundary condition is derived by discretizing the time variable with ...In this paper we consider the numerical solution of the one-dimensional heat equation on unbounded domains. First an exact semi-discrete artificial boundary condition is derived by discretizing the time variable with the Crank-Nicolson method. The semi-discretized heat equation equipped with this boundary condition is then proved to be unconditionally stable, and its solution is shown to have second-order accuracy. In order to reduce the computational cost, we develop a new fast evaluation method for the convolution operation involved in the exact semi-discrete artificial boundary condition. A great advantage of this method is that the unconditional stability held by the semi-discretized heat equation is preserved. An error estimate is also given to show the dependence of numerical errors on the time step and the approximation accuracy of the convolution kernel. Finally, a simple numerical example is presented to validate the theoretical results.展开更多
Some specific non-isotropic Jacobi approximations in multiple-dimensions are investigated, which are used for numerical solutions of differential equations on various unbounded domains. The convergence of proposed sch...Some specific non-isotropic Jacobi approximations in multiple-dimensions are investigated, which are used for numerical solutions of differential equations on various unbounded domains. The convergence of proposed schemes are proved. Some efficient algorithms are provided. Numerical results are presented to illustrate the efficiency of this new approach.展开更多
Assuming that the external forces of the system are small enough, the reference temperature being a periodic function, we study the existence, the uniqueness and the regularity of time-periodic solutions for the Bouss...Assuming that the external forces of the system are small enough, the reference temperature being a periodic function, we study the existence, the uniqueness and the regularity of time-periodic solutions for the Boussinesq equations in several classes of unbounded domains of Rn. Our analysis is based on the framework of weak-Lp spaces.展开更多
The exact boundary condition on a spherical artificial boundary is derived for the three-dimensional exterior problem of linear elasticity in this paper. After this boundary condition is imposed on the artificial boun...The exact boundary condition on a spherical artificial boundary is derived for the three-dimensional exterior problem of linear elasticity in this paper. After this boundary condition is imposed on the artificial boundary, a reduced problem only defined in a bounded domain is obtained. A series of approximate problems with increasing accuracy can be derived if one truncates the series term in the variational formulation, which is equivalent to the reduced problem. An error estimate is presented to show how the error depends on the finite element discretization and the accuracy of the approximate problem. In the end, a numerical example is given to demonstrate the performance of the proposed method.展开更多
研究GBBM方程ut-aΔut-bΔu+ F(u)+γu=h(x),其中F(u)=(F1(u),…,Fn(u)), F / xiFi,Fi(0)=0,Fi是R1上二阶导数连续的函数,fi(s)=d/dsFi(s),fi满足fi(0)=0,｜fi(s)｜<c(1=∑ni=1+｜s｜m),i=1,…,n,其中当n 2时,0 m<∞;当n 3时,0 m ...研究GBBM方程ut-aΔut-bΔu+ F(u)+γu=h(x),其中F(u)=(F1(u),…,Fn(u)), F / xiFi,Fi(0)=0,Fi是R1上二阶导数连续的函数,fi(s)=d/dsFi(s),fi满足fi(0)=0,｜fi(s)｜<c(1=∑ni=1+｜s｜m),i=1,…,n,其中当n 2时,0 m<∞;当n 3时,0 m 2/(n-2).在空间Rn上整体解的存在唯一性用Galerkin逼近方法和作极限的方法获得.展开更多
In this paper,we study the TE,TM model by using the decompositions of the vector fields in 2-D bounded multiply connected domains and 2-D unbounded domains,respectively.We find that the TE,TM model and the Darwin mode...In this paper,we study the TE,TM model by using the decompositions of the vector fields in 2-D bounded multiply connected domains and 2-D unbounded domains,respectively.We find that the TE,TM model and the Darwin models are equivalent if we assume some regularity of the initial data.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
We consider the numerical solution by finite difference methods of the heat equation in one space dimension, with a nonlocal integral boundary condition, resulting from the truncation to a finite interval of the probl...We consider the numerical solution by finite difference methods of the heat equation in one space dimension, with a nonlocal integral boundary condition, resulting from the truncation to a finite interval of the problem on a semi-infinite interval. We first analyze the forward Euler method, and then the 0-method for 0 〈 θ ≤ 1, in both cases in maximum-norm, showing O（h2 ＋ k） error bounds, where h is the mesh-width and k the time step. We then give an alternative analysis for the case θ= 1/2, the Crank-Nicolson method, using energy arguments, yielding a O（h2 ＋ k3/2） error bound. Special attention is given the approximation of the boundary integral operator. Our results are illustrated by numerical examples.展开更多
Longtime behavior of degenerate equations with the nonlinearity of polynomial growth of arbitrary order on the whole space RN is considered. By using-tra jectories methods, we proved that weak solutions generated by d...Longtime behavior of degenerate equations with the nonlinearity of polynomial growth of arbitrary order on the whole space RN is considered. By using-tra jectories methods, we proved that weak solutions generated by degenerate equations possess an（LU^2（R^N）, Lloc^2（R^N））-global attractor.Moreover, the upper bounds of the Kolmogorov ε-entropy for such global attractor are also obtained.展开更多
文摘Examines the development of the composite legendre approximation in unbounded domains. Proof of the stability and convergence of a proposed scheme; Discussion of two-dimensional exterior problems; Error estimations.
基金Acknowledgments. This work is supported partially by the National Natural Science Foundation of China under Grant No. 10401020, the Alexander von Humboldt Foundation, and the Key Project of China High Performance Scientific Computation Research 2005CB321701.
文摘In this paper we consider the numerical solution of the one-dimensional heat equation on unbounded domains. First an exact semi-discrete artificial boundary condition is derived by discretizing the time variable with the Crank-Nicolson method. The semi-discretized heat equation equipped with this boundary condition is then proved to be unconditionally stable, and its solution is shown to have second-order accuracy. In order to reduce the computational cost, we develop a new fast evaluation method for the convolution operation involved in the exact semi-discrete artificial boundary condition. A great advantage of this method is that the unconditional stability held by the semi-discretized heat equation is preserved. An error estimate is also given to show the dependence of numerical errors on the time step and the approximation accuracy of the convolution kernel. Finally, a simple numerical example is presented to validate the theoretical results.
文摘Some specific non-isotropic Jacobi approximations in multiple-dimensions are investigated, which are used for numerical solutions of differential equations on various unbounded domains. The convergence of proposed schemes are proved. Some efficient algorithms are provided. Numerical results are presented to illustrate the efficiency of this new approach.
基金supported by M.E.C. (Spain), Project MTM 2006-07932supported by Junta de Andalucía, Project P06- FQM- 02373supported by Fondecyt-Chile (Grant No. 1080628)
文摘Assuming that the external forces of the system are small enough, the reference temperature being a periodic function, we study the existence, the uniqueness and the regularity of time-periodic solutions for the Boussinesq equations in several classes of unbounded domains of Rn. Our analysis is based on the framework of weak-Lp spaces.
文摘The exact boundary condition on a spherical artificial boundary is derived for the three-dimensional exterior problem of linear elasticity in this paper. After this boundary condition is imposed on the artificial boundary, a reduced problem only defined in a bounded domain is obtained. A series of approximate problems with increasing accuracy can be derived if one truncates the series term in the variational formulation, which is equivalent to the reduced problem. An error estimate is presented to show how the error depends on the finite element discretization and the accuracy of the approximate problem. In the end, a numerical example is given to demonstrate the performance of the proposed method.
文摘研究GBBM方程ut-aΔut-bΔu+ F(u)+γu=h(x),其中F(u)=(F1(u),…,Fn(u)), F / xiFi,Fi(0)=0,Fi是R1上二阶导数连续的函数,fi(s)=d/dsFi(s),fi满足fi(0)=0,｜fi(s)｜<c(1=∑ni=1+｜s｜m),i=1,…,n,其中当n 2时,0 m<∞;当n 3时,0 m 2/(n-2).在空间Rn上整体解的存在唯一性用Galerkin逼近方法和作极限的方法获得.
文摘In this paper,we study the TE,TM model by using the decompositions of the vector fields in 2-D bounded multiply connected domains and 2-D unbounded domains,respectively.We find that the TE,TM model and the Darwin models are equivalent if we assume some regularity of the initial data.
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We consider the numerical solution by finite difference methods of the heat equation in one space dimension, with a nonlocal integral boundary condition, resulting from the truncation to a finite interval of the problem on a semi-infinite interval. We first analyze the forward Euler method, and then the 0-method for 0 〈 θ ≤ 1, in both cases in maximum-norm, showing O（h2 ＋ k） error bounds, where h is the mesh-width and k the time step. We then give an alternative analysis for the case θ= 1/2, the Crank-Nicolson method, using energy arguments, yielding a O（h2 ＋ k3/2） error bound. Special attention is given the approximation of the boundary integral operator. Our results are illustrated by numerical examples.
基金supported by the Fundamental Research Funds for the Central Universities(Grant No.NS2014075)
文摘Longtime behavior of degenerate equations with the nonlinearity of polynomial growth of arbitrary order on the whole space RN is considered. By using-tra jectories methods, we proved that weak solutions generated by degenerate equations possess an（LU^2（R^N）, Lloc^2（R^N））-global attractor.Moreover, the upper bounds of the Kolmogorov ε-entropy for such global attractor are also obtained.
