Preliminary Group Classification for One Class of Nonlinear Wave Equations

The group classification is carried out on the nonlinear wave equation utt = f（x,u, ux）uzz ＋ g（x,u,uz） by using the preliminary group classification approach. The generators of equivalence group are determined and the corresponding reduced forms are obtained. The result of the work is shown in table form.

Group classification for path equation describing minimum drag work and symmetry reductions

The path equation describing the minimum drag work first proposed by Pakdemirli is reconsidered （Pakdemirli, M. The drag work minimization path for a fly- ing object with altitude-dependent drag parameters. Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science 223（5）, 1113- 1116 （2009））. The Lie group theory is applied to the general equation. The group classi- fication with respect to an altitude-dependent arbitrary function is presented. Using the symmetries, the group-invariant solutions are determined, and the reduction of order is performed by the canonical coordinates.

Preliminary group classification of quasilinear third-order evolution equations

Group classification of quasilinear third-order evolution equations is given by using the classical infinitesimal Lie method, the technique of equivalence transformations, and the theory of classification of abstract low-dimensional Lie algebras. We show that there are three equations admitting simple Lie algebras of dimension three. All non-equivalent equations admitting simple Lie algebras are nothing but these three. Furthermore, we also show that there exist two, five, twenty-nine and twenty-six non- equivalent third-order nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively.

Lie group classification of the N-th-order nonlinear evolution equations

In this paper, Lie group classification to the N-th-order nonlinear evolution equation Ut ： UNx ＋ F（x, t, u, ux, . . . , U（N-1）x）is performed. It is shown that there are three, nine, forty-four and sixty-one inequivalent equations admitting one-, two-, three- and four-dimensionM solvable Lie algebras, respectively. We also prove that there are no semisimple Lie group 50（3） as the symmetry group of the equation, and only two realizations oral（2, R） are admitted by the equation. The resulting invariant equations contain both the well-known equations and a variety of new ones.

Group Classification of Differential-difference Equations： Low-dimensional Lie Algebras

Differential-difference equations of the form un = Fn（t, un-1,Un,Unn＋1,Un-1,un,Un＋1） are clas- sifted according to their intrinsic Lie point symmetries, equivalence group and some low-dimensional Lie algebras including the Abelian symmetry algebras, nilpotent nonAbelian symmetry algebras, solvable symmetry algebras with nonAbelian nilradicals, solvable symmetry algebras with Abelian nilradicals and nonsolvable symmetry al- gebras. Here Fn is a nonlinear function of its arguments and the dot over u denotes differentiation with respect to t.

Orders of Classical Groups over Finite Rings

It is observed that a classical group over a finite ring R with identity can be reduced to that over finite fields after the procedures of taking “modulo the radical”, “direct sum” and “tensor products”. Basing on that fact, we calculate the orders of classical groups over R and the number of k dimensional free submodules of an n dimensional free module over R .

Algae functional group characteristics in reservoirs and lakes with different trophic levels in northwestern semi-humid and semi-arid regions in China

In order to study the differences in algae species and their biomass in water bodies in a region, three reservoirs and two lakes at the center of Guanzhong Plain were chosen to identify algae functional groups, measure biomass, and assess water quality, from January2013 to December 2014. The water bodies represented different trophic levels： one oligotrophic, three mesotrophic, and one eutrophic. Based on the Reynolds’ functional groups, they had 10 groups in common—B, P, D, X1, M, MP, F, S1, J, and G, but the algae biomasses and proportions were different. In the oligotrophic reservoir, functional group B reached a peak biomass of 576 × 104 L-1, which accounted for 31.27%. In the eutrophic lake,functional group D reached a peak biomass of 3227 × 104 L-1, which accounted for only13.38%. When samples collected from other water bodies with similar trophic levels were compared, we found differences in the algae species functional groups. The potential reasons for the differences in algae functional group characteristics in the different water bodies in the region were water temperature and nutritional states.

ONEOptimal:A Maple Package for Generating One-Dimensional Optimal System of Finite Dimensional Lie Algebra

We present a Maple computer algebra package, ONEOptimal, which can calculate one-dimensional optimal system of finite dimensional Lie algebra for nonlinear equations automatically based on Olver's theory. The core of this theory is viewing the Killing form of the Lie algebra as an invariant for the adjoint representation. Some examples are given to demonstrate the validity and efficiency of the program.

Symmetries, Symmetry Reductions and Exact Solutions to the Generalized Nonlinear Fractional Wave Equations

In this paper, the Lie group classification method is performed on the fractional partial differential equation(FPDE), all of the point symmetries of the FPDEs are obtained. Then, the symmetry reductions and exact solutions to the fractional equations are presented, the compatibility of the symmetry analysis for the fractional and integer-order cases is verified. Especially, we reduce the FPDEs to the fractional ordinary differential equations(FODEs) in terms of the Erd′elyi-Kober(E-K) fractional operator method, and extend the power series method for investigating exact solutions to the FPDEs.