In this paper we present and test a numerical method for computing eigenvalues of 4th order Sturm-Liouville (SL) differential operators on finite intervals with regular boundary conditions. This method is a 4th order ...In this paper we present and test a numerical method for computing eigenvalues of 4th order Sturm-Liouville (SL) differential operators on finite intervals with regular boundary conditions. This method is a 4th order shooting method based on Magnus expansions (MG4) which use MG4 shooting as the integrator. This method is similar to the SLEUTH (Sturm-Liouville Eigenvalues Using Theta Matrices) method of Greenberg and Marletta which uses the 2nd order Pruess method (also known as the MG2 shooting method) for the integrator. This method often achieves near machine precision accuracies, and some comparisons of its performance against the well-known SLEUTH software package are presented.展开更多
Rapidly solidified（RS） Mg-6Zn-1Y-1Ce ribbons were prepared by single roller melt-spinning technique.Transmission electron microscopy and energy dispersive X-ray spectroscopy were employed to characterize the microst...Rapidly solidified（RS） Mg-6Zn-1Y-1Ce ribbons were prepared by single roller melt-spinning technique.Transmission electron microscopy and energy dispersive X-ray spectroscopy were employed to characterize the microstructure of RS ribbons.The results show that there is high density of particles distributed within grains and at grain boundaries in the region near wheel side.The particle density is decreased in the middle region and free surface region.The alloy is predominantly composed of supersaturated--Mg solid solution,T phase and W phase;meanwhile,a few icosahedral quasicrystalline and Mg4Zn7 particles are also observed.The T phase is confirmed having a body-centered orthorhombic structure that is transformed from the body-centered tetragonal structure Mg12Ce phase due to the partial substitution of Mg atoms by Zn.展开更多
文摘In this paper we present and test a numerical method for computing eigenvalues of 4th order Sturm-Liouville (SL) differential operators on finite intervals with regular boundary conditions. This method is a 4th order shooting method based on Magnus expansions (MG4) which use MG4 shooting as the integrator. This method is similar to the SLEUTH (Sturm-Liouville Eigenvalues Using Theta Matrices) method of Greenberg and Marletta which uses the 2nd order Pruess method (also known as the MG2 shooting method) for the integrator. This method often achieves near machine precision accuracies, and some comparisons of its performance against the well-known SLEUTH software package are presented.
基金Project (50271054) supported by the National Natural Science Foundation of ChinaProject (20070700003) supported by the Doctorate Programs Foundation of Ministry of Education of China+1 种基金Project (102102210031) supported by the Science and Technologies Foundation of Henan Province,ChinaProject (2010A430008) supported by the Natural Science Foundation of Henan Educational Committee of China
文摘Rapidly solidified（RS） Mg-6Zn-1Y-1Ce ribbons were prepared by single roller melt-spinning technique.Transmission electron microscopy and energy dispersive X-ray spectroscopy were employed to characterize the microstructure of RS ribbons.The results show that there is high density of particles distributed within grains and at grain boundaries in the region near wheel side.The particle density is decreased in the middle region and free surface region.The alloy is predominantly composed of supersaturated--Mg solid solution,T phase and W phase;meanwhile,a few icosahedral quasicrystalline and Mg4Zn7 particles are also observed.The T phase is confirmed having a body-centered orthorhombic structure that is transformed from the body-centered tetragonal structure Mg12Ce phase due to the partial substitution of Mg atoms by Zn.
