For linear partial differential equation 〔 2t 2-a 2P( x)〕 m u=f(x,t), where m1,X∈R n,t∈R 1, the author gives the analytic solution of the initial value problem using the operators sh(tP( x) 1/2 )...For linear partial differential equation 〔 2t 2-a 2P( x)〕 m u=f(x,t), where m1,X∈R n,t∈R 1, the author gives the analytic solution of the initial value problem using the operators sh(tP( x) 1/2 )P( x) 1/2 . By representing the operators with integrals, explicit solutions are obtained with an integral form of a given function.展开更多
A high-order leap-frog based non-dissipative discontinuous Galerkin time- domain method for solving Maxwell＇s equations is introduced and analyzed. The pro- posed method combines a centered approximation for the eval...A high-order leap-frog based non-dissipative discontinuous Galerkin time- domain method for solving Maxwell＇s equations is introduced and analyzed. The pro- posed method combines a centered approximation for the evaluation of fluxes at the in- terface between neighboring elements, with a Nth-order leap-frog time scheme. More- over, the interpolation degree is defined at the element level and the mesh is refined locally in a non-conforming way resulting in arbitrary level hanging nodes. The method is proved to be stable under some CFL-like condition on the time step. The convergence of the semi-discrete approximation to Maxwelrs equations is established rigorously and bounds on the global divergence error are provided. Numerical experiments with high- order elements show the potential of the method.展开更多
In this paper, we establish the first variational formula and its Euler-Lagrange equation for the total 2p-th mean curvature functional .M2p of a submanifold Mn in a general Riemannian manifold gn^n＋m for p = 0, 1,.....In this paper, we establish the first variational formula and its Euler-Lagrange equation for the total 2p-th mean curvature functional .M2p of a submanifold Mn in a general Riemannian manifold gn^n＋m for p = 0, 1,..., [n/2]. As an example, we prove that closed complex submanifolds in complex projective spaces are critical points of the functional M2p, called relatively 2p-minimal submanifolds, for all p. At last, we discuss the relations between relatively 2p-minimal submanifoIds and austere submanifolds in real space forms, as well as a special variational problem.展开更多
文摘For linear partial differential equation 〔 2t 2-a 2P( x)〕 m u=f(x,t), where m1,X∈R n,t∈R 1, the author gives the analytic solution of the initial value problem using the operators sh(tP( x) 1/2 )P( x) 1/2 . By representing the operators with integrals, explicit solutions are obtained with an integral form of a given function.
基金supported by a grant from the French National Ministry of Education and Research(MENSR,19755-2005)
文摘A high-order leap-frog based non-dissipative discontinuous Galerkin time- domain method for solving Maxwell＇s equations is introduced and analyzed. The pro- posed method combines a centered approximation for the evaluation of fluxes at the in- terface between neighboring elements, with a Nth-order leap-frog time scheme. More- over, the interpolation degree is defined at the element level and the mesh is refined locally in a non-conforming way resulting in arbitrary level hanging nodes. The method is proved to be stable under some CFL-like condition on the time step. The convergence of the semi-discrete approximation to Maxwelrs equations is established rigorously and bounds on the global divergence error are provided. Numerical experiments with high- order elements show the potential of the method.
基金supported by National Natural Science Foundation of China(Grant No.11001016)the Specialized Research Fund for the Doctoral Program of Higher Education(Grant No. 20100003120003)the Program for Changjiang Scholars and Innovative Research Team in University
文摘In this paper, we establish the first variational formula and its Euler-Lagrange equation for the total 2p-th mean curvature functional .M2p of a submanifold Mn in a general Riemannian manifold gn^n＋m for p = 0, 1,..., [n/2]. As an example, we prove that closed complex submanifolds in complex projective spaces are critical points of the functional M2p, called relatively 2p-minimal submanifolds, for all p. At last, we discuss the relations between relatively 2p-minimal submanifoIds and austere submanifolds in real space forms, as well as a special variational problem.
