• Journal of applied mathematics & informatics
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• v.31 no.1_2
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• pp.229-239
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• 2013

This paper studies the generalized fifth-order KdV equation with variable coefficients using Lie symmetry methods.Lie group classification with respect to the time dependent coefficients is performed. Then we get the similarity reductions using the symmetry and give some exact solutions.

• Journal of applied mathematics & informatics
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• v.29 no.3_4
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• pp.563-572
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• 2011

In this paper, the Lie symmetry analysis is performed on the three mixed second-order PDEs, which arise in fluid dynamics, nonlinear wave theory and plasma physics, etc. The symmetries and similarity reductions of the equations are obtained, and the exact solutions to the equations are investigated by the dynamical system and power series methods. Then, the exact solutions to the general types of PDEs are considered through a variable transformation. At last, the symmetry and integration method is employed for reducing the nonlinear ODEs.

• Bulletin of the Korean Chemical Society
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• v.10 no.3
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• pp.234-238
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• 1989

The crystal structure of partially $Ag^+$-exchanged zeolite A, $Ag_{3.2}Na_{8.8}$-A, vacuum dehydrated at $360^{\circ}C$ and then exposed to 0.1 torr of cesium vapor for 12 hours at $250^{\circ}C$ has been determined by single-crystal X-ray diffraction techniques in the cubic space group Pm3m (a = 12.262(2)${\AA})\;at\;21(1)^{\circ}C$. The structure was refined to the final error indexes $R_1=0.068\;and\;R_2=0.072$ by using 338 reflections for which $I_o\;>\;3{\sigma}(I_o)$ and the composition of unit cell is $Ag_{3.2}Cs_{8.8}-A.\;3\;Cs^+$ ions lie on the centers of the 8-rings at sites of D4h symmetry. Two crystallographycally different 6-ring $Cs^+$ ions were found: 1.5 $Cs^+$ ions at Cs(2) are located inside of sodalite cavity and 4.3 $Cs^+$ ions at Cs(3) are located in the large cavity. The fractional occupancies observed at Cs(2) and Cs(3) indicate that the existence of at least three types of unit cells with regard to the 6-ring $Cs^+$ ions. For example, 50% of unit cells may have two $Cs^+$ ions at Cs(2) and 4 $Cs^+$ ions at Cs(3). 30% of unit cells may have one Cs+ ion at Cs(2) and 5 $Cs^+$ ions at Cs(3). The remaining 20% would have one $Cs^+$ ion at Cs(2) and 4 $Cs^+$ ions at Cs(3). On threefold axes of the unit cell two non-equivalent Ag atom positions are found in the large cavity, each containing 0.64 and 1.92 Ag atoms, respectively. A crystallographic analysis may be interpreted to indicate that 0.64 $(Ag_5)^+$ clusters are present in each large cavity. This cluster may be viewed as a tetrasilver molecule $(Ag_4)^0$(bond length, 2.84${\AA}$) stabilized by the coordination of one $Ag^+$ ion.