A new mixed scheme which combines the variation of constants and the H1-Galerkin mixed finite element method is constructed for nonlinear Sobolev equation with nonlinear con- vection term. Optimal error estimates are ...A new mixed scheme which combines the variation of constants and the H1-Galerkin mixed finite element method is constructed for nonlinear Sobolev equation with nonlinear con- vection term. Optimal error estimates are derived for both semidiscrete and fully discrete schemes. Finally, some numerical results are given to confirm the theoretical analysis of the proposed method.展开更多
Recently, a new（2＋1）-dimensional displacement shallow water wave equation（2DDSWWE） was constructed by applying the variational principle of analytic mechanics in the Lagrange coordinates. However, the simplificat...Recently, a new（2＋1）-dimensional displacement shallow water wave equation（2DDSWWE） was constructed by applying the variational principle of analytic mechanics in the Lagrange coordinates. However, the simplification of the nonlinear term related to the incompressibility of the shallow water in the 2DDSWWE is a disadvantage of this approach.Applying the theory of nonlinear continuum mechanics, we add some new nonlinear terms to the 2DDSWWE and construct a new fully nonlinear（2＋1）-dimensional displacement shallow water wave equation（FN2DDSWWE）. The presented FN2DDSWWE contains all nonlinear terms related to the incompressibility of shallow water. The exact travelling-wave solution of the proposed FN2DDSWWE is also obtained, and the solitary-wave solution can be deduced from the presented travelling-wave solution under a special selection of integral constants.展开更多
The de Sitter invariant Special Relativity （dS-SR） is SR with constant curvature, and a natural extension of usual Einstein SR （E-SR）. In this paper, we solve the dS-SR Dirac equation of Hydrogen by means of the a...The de Sitter invariant Special Relativity （dS-SR） is SR with constant curvature, and a natural extension of usual Einstein SR （E-SR）. In this paper, we solve the dS-SR Dirac equation of Hydrogen by means of the adiabatic approach and the quasi-stationary perturbation calculations of QM. Hydrogen atom is located in the light cone of the Universe. FRW metric and ACDM cosmological model are used to discuss this issue. To the atom, effects of de Sitter space-time geometry described by Beltrami metric are taken into account. The dS-SR Dirac equation turns out to be a time dependent quantum Hamiltonian system. We reveal that： （i） The fundamental physics constants me, h, e variate adiabatically along with cosmologic time in dS-SR QM framework. But the fine-structure constant α≡ - e^2/（hc） keeps to be invariant; （ii） （2s^1/2 - 2p^1/2）-splitting due to dS-SR QM effects： By means of perturbation theory, that splitting △E（z） are calculated analytically, which belongs to O（1/R^2）-physics of dS-SR QM. Numerically, we find that when ｜R｜ = {103 Gly, 104 Gly, 105 Gly}, and z = {1, or 2}, the AE（z） 〉〉 1 （Lamb shift）. This indicates that for these cases the hyperfine structure effects due to QED could be ignored, and the dS-SR fine structure effects are dominant. This effect could be used to determine the universal constant R in dS-SR, and be thought as a new physics beyond E-SR.展开更多
基金Supported by National Natural Science Fund of China (11061021)Key Project of Chinese Ministry of Education (12024)+2 种基金Natural Science Fund of Inner Mongolia Autonomous Region (2012MS0108,2012MS0106,2011BS0102)Scientific Research Projection of Higher Schools of Inner Mongolia (NJZZ12011,NJZY13199)Program of Higher-level talents of Inner Mongolia University (125119,Z200901004,30105-125132)
文摘A new mixed scheme which combines the variation of constants and the H1-Galerkin mixed finite element method is constructed for nonlinear Sobolev equation with nonlinear con- vection term. Optimal error estimates are derived for both semidiscrete and fully discrete schemes. Finally, some numerical results are given to confirm the theoretical analysis of the proposed method.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11272076 and 51609034)China Postdoctoral Science Foundation(Grant No.2016M590219)
文摘Recently, a new（2＋1）-dimensional displacement shallow water wave equation（2DDSWWE） was constructed by applying the variational principle of analytic mechanics in the Lagrange coordinates. However, the simplification of the nonlinear term related to the incompressibility of the shallow water in the 2DDSWWE is a disadvantage of this approach.Applying the theory of nonlinear continuum mechanics, we add some new nonlinear terms to the 2DDSWWE and construct a new fully nonlinear（2＋1）-dimensional displacement shallow water wave equation（FN2DDSWWE）. The presented FN2DDSWWE contains all nonlinear terms related to the incompressibility of shallow water. The exact travelling-wave solution of the proposed FN2DDSWWE is also obtained, and the solitary-wave solution can be deduced from the presented travelling-wave solution under a special selection of integral constants.
基金Supported in part by National Natural Science Foundation of China under Grant No. 10975128by the Chinese Science Academy Foundation under Grant No. KJCX-YW-N29
文摘The de Sitter invariant Special Relativity （dS-SR） is SR with constant curvature, and a natural extension of usual Einstein SR （E-SR）. In this paper, we solve the dS-SR Dirac equation of Hydrogen by means of the adiabatic approach and the quasi-stationary perturbation calculations of QM. Hydrogen atom is located in the light cone of the Universe. FRW metric and ACDM cosmological model are used to discuss this issue. To the atom, effects of de Sitter space-time geometry described by Beltrami metric are taken into account. The dS-SR Dirac equation turns out to be a time dependent quantum Hamiltonian system. We reveal that： （i） The fundamental physics constants me, h, e variate adiabatically along with cosmologic time in dS-SR QM framework. But the fine-structure constant α≡ - e^2/（hc） keeps to be invariant; （ii） （2s^1/2 - 2p^1/2）-splitting due to dS-SR QM effects： By means of perturbation theory, that splitting △E（z） are calculated analytically, which belongs to O（1/R^2）-physics of dS-SR QM. Numerically, we find that when ｜R｜ = {103 Gly, 104 Gly, 105 Gly}, and z = {1, or 2}, the AE（z） 〉〉 1 （Lamb shift）. This indicates that for these cases the hyperfine structure effects due to QED could be ignored, and the dS-SR fine structure effects are dominant. This effect could be used to determine the universal constant R in dS-SR, and be thought as a new physics beyond E-SR.
