In this paper,we propose a new numerical method which is a least squares approximaton based on pseudospectral method for the Forward-Backward heat equation. The existence and uniqueness of the solution of the least sq...In this paper,we propose a new numerical method which is a least squares approximaton based on pseudospectral method for the Forward-Backward heat equation. The existence and uniqueness of the solution of the least squares approximation are proved. Error estimates for this approximation are given,which show that tile order of convergence depends only on the regularity of tile solution and the right hand of the Forward-Backward heat equation.展开更多
The backward heat equation is a typical ill-posed problem. In this paper, we shall apply a dual least squares method connecting Shannon wavelet to the following equation Motivated by Reginska＇s work, we shall give ...The backward heat equation is a typical ill-posed problem. In this paper, we shall apply a dual least squares method connecting Shannon wavelet to the following equation Motivated by Reginska＇s work, we shall give two nonlinear approximate methods to regularize the approximate solutions for high-dimensional backward heat equation, and prove that our methods are convergent.展开更多
The regularization of ill-posed problems has become a useful tool in studying initial value problems that do not adhere to certain desired properties such as continuous dependence of solutions on initial data.Because ...The regularization of ill-posed problems has become a useful tool in studying initial value problems that do not adhere to certain desired properties such as continuous dependence of solutions on initial data.Because direct computation of the solution becomes difficult in this situation,many authors have alternatively approximated the solution by the solution of a closely defined well posed problem.In this paper,we demonstrate this process of regularization for nonautonomous ill-posed problems including the backward heat equation with a time-dependent diffusion coefficient.In the process,we provide two different approximate well posed models and numerically compare convergence rates of their solutions to a known solution of the original ill-posed problem.展开更多
Abstract The authors establish the null controllability for some systems coupled by two backward stochastic heat equations. The desired controllability result is obtained by means of proving a suitable observability e...Abstract The authors establish the null controllability for some systems coupled by two backward stochastic heat equations. The desired controllability result is obtained by means of proving a suitable observability estimate for the dual system of the controlled system.展开更多
This paper deals with the inverse time problem for an axisymmetric heat equation. The problem is ill-posed. A modified Tikhonov regularization method is applied to formulate regularized solution which is stably conver...This paper deals with the inverse time problem for an axisymmetric heat equation. The problem is ill-posed. A modified Tikhonov regularization method is applied to formulate regularized solution which is stably convergent to the exact one. estimate between the approximate solution and exact technical inequality and improving a priori smoothness Meanwhile, a logarithmic-HSlder type error solution is obtained by introducing a rather assumption.展开更多
In this paper we propose the finite difference method for the forward-backward heat equation. We use a coarse-mesh second-order central difference scheme at the middle line mesh points and derive the error estimate. T...In this paper we propose the finite difference method for the forward-backward heat equation. We use a coarse-mesh second-order central difference scheme at the middle line mesh points and derive the error estimate. Then we discuss the iterative method based on the domain decomposition for our scheme and derive the bounds for the rates of convergence. Finally we present some numerical experiments to support our analysis.展开更多
A finite difference method is introduced to solve the forward-backward heat equation in two space dimensions. In this procedure, the backward and forward difference scheme in two subdomains and a coarse-mesh second-or...A finite difference method is introduced to solve the forward-backward heat equation in two space dimensions. In this procedure, the backward and forward difference scheme in two subdomains and a coarse-mesh second-order central difference scheme at the middle interface are used. Maximum norm error estimate for the procedure is derived. Then an iterative method based on domain decomposition is presented for the numerical scheme and the convergence of the given method is established. Then numerical experiments are presented to support the theoretical analysis.展开更多
The forward-backward heat equation arises in a remarkable variety of physical applications. A non-overlaping domain decomposition method was constructed to obtain numerical solutions of the forward-backward heat equa...The forward-backward heat equation arises in a remarkable variety of physical applications. A non-overlaping domain decomposition method was constructed to obtain numerical solutions of the forward-backward heat equation. The primary advantage is that the method reduces the computation time tremendously. The convergence of the given method is established. The numerical performance shows that the domain decomposition method is effective.展开更多
文摘In this paper,we propose a new numerical method which is a least squares approximaton based on pseudospectral method for the Forward-Backward heat equation. The existence and uniqueness of the solution of the least squares approximation are proved. Error estimates for this approximation are given,which show that tile order of convergence depends only on the regularity of tile solution and the right hand of the Forward-Backward heat equation.
基金Supported by Beijing Natural Science Foundation(Grant No.1092003)Beijing Educational Committee Foundation(Grant No.PHR201008022) National Natural Science Foundation of China(Grant No.11271038)1)Corresponding author
文摘The backward heat equation is a typical ill-posed problem. In this paper, we shall apply a dual least squares method connecting Shannon wavelet to the following equation Motivated by Reginska＇s work, we shall give two nonlinear approximate methods to regularize the approximate solutions for high-dimensional backward heat equation, and prove that our methods are convergent.
文摘The regularization of ill-posed problems has become a useful tool in studying initial value problems that do not adhere to certain desired properties such as continuous dependence of solutions on initial data.Because direct computation of the solution becomes difficult in this situation,many authors have alternatively approximated the solution by the solution of a closely defined well posed problem.In this paper,we demonstrate this process of regularization for nonautonomous ill-posed problems including the backward heat equation with a time-dependent diffusion coefficient.In the process,we provide two different approximate well posed models and numerically compare convergence rates of their solutions to a known solution of the original ill-posed problem.
基金Project supported by the National Natural Science Foundation of China(No.11101070)the Grant MTM2011-29306-C02-00 of the MICINN,Spain+4 种基金the Project PI2010-04 of the Basque Governmentthe ERC Advanced Grant FP7-246775 NUMERIWAVESthe ESF Research Networking Programme OPT-PDEScientific Research Fund of Sichuan Provincial Education Department of China(No.10ZC110) the project Z1062 of Leshan Normal University of China
文摘Abstract The authors establish the null controllability for some systems coupled by two backward stochastic heat equations. The desired controllability result is obtained by means of proving a suitable observability estimate for the dual system of the controlled system.
基金Supported by National Natural Science Foundation of China (Grant No.10671085)Fundamental Research Fund for Natural Science of Education Department of He'nan Province of China (Grant No.2009Bl10007)Hight-level Personnel fund of He'nan University of Technology (Grant No.2007BS028)
文摘This paper deals with the inverse time problem for an axisymmetric heat equation. The problem is ill-posed. A modified Tikhonov regularization method is applied to formulate regularized solution which is stably convergent to the exact one. estimate between the approximate solution and exact technical inequality and improving a priori smoothness Meanwhile, a logarithmic-HSlder type error solution is obtained by introducing a rather assumption.
基金The project was supported by National Science Foundation （Grant No. 10471129）.
文摘In this paper we propose the finite difference method for the forward-backward heat equation. We use a coarse-mesh second-order central difference scheme at the middle line mesh points and derive the error estimate. Then we discuss the iterative method based on the domain decomposition for our scheme and derive the bounds for the rates of convergence. Finally we present some numerical experiments to support our analysis.
基金Supported by National Science Foundation of China(Grant 10871179)the National Basic Research Programme of China(Grant 2008CB717806)the Department of Education of Zhejiang Province(GrantY200803559).
文摘A finite difference method is introduced to solve the forward-backward heat equation in two space dimensions. In this procedure, the backward and forward difference scheme in two subdomains and a coarse-mesh second-order central difference scheme at the middle interface are used. Maximum norm error estimate for the procedure is derived. Then an iterative method based on domain decomposition is presented for the numerical scheme and the convergence of the given method is established. Then numerical experiments are presented to support the theoretical analysis.
基金Supported by the Special Funds for Major State BasicResearch Projects of China (No.G19990 32 80 2 )
文摘The forward-backward heat equation arises in a remarkable variety of physical applications. A non-overlaping domain decomposition method was constructed to obtain numerical solutions of the forward-backward heat equation. The primary advantage is that the method reduces the computation time tremendously. The convergence of the given method is established. The numerical performance shows that the domain decomposition method is effective.
