A Modified Transitional Korteweg-De Vries Equation: Posed in the Quarter Plane

This paper is concerned with a modified transitional Korteweg-de Vries equation ut+f(t)u2ux+uxxx=0, (x,t)∈R+×R+with initial value u(x,0)=g(x)∈H4(R+)and inhomogeneous boundary value u(0,t)=Q(t)∈C2([ 0,∞ )). Under the conditions either 1) f(t)≤0, f′(t)≥0or 2) f(t)≤−αwhere α>0, we prove the existence of a unique global classical solution.

The Existence of Meromorphic Solutions to Non-Linear Delay Differential Equations

In this paper, we study the existence of the transcendental meromorphic solution of the delay differential equations , where a(z) is a rational function, and are polynomials in w(z) with rational coefficients, k is a positive integer. Under the assumption when above equations own transcendental meromorphic solutions with minimal hyper-type, we derive the concrete conditions on the degree of the right side of them. Specially, when w(z)=0 is a root of , its multiplicity is at most k. Some examples are given here to illustrate that our results are accurate.

A Kind of Diophantine Equations in Finite Simple Groups

In this paper, we prove that if p, q are distinct primes, (p,q)≡(1,7) (mod 12) and Legendres symbol pq=1 , then the equation 1+p a=2 bq c+2 dp eq f has only solutions of the form (a,b,c,d,e,f)=(t,0,0,0,t,0), where t is a non negative integer. We also give all solutions of a kind of generalized Ramanujan Nagell equations by using the theories of imaginary quadratic field and Pells equation.

Symmetry and general symmetry groups of the coupled Kadomtsev-Petviashvili equation

In this paper, the Lie symmetry algebra of the coupled Kadomtsev-Petviashvili （cKP） equation is obtained by the classical Lie group method and this algebra is shown to have a Kac-Moody-Virasoro loop algebra structure. Then the general symmetry groups of the cKP equation is also obtained by the symmetry group direct method which is proposed by Lou et alo From the general symmetry groups, the Lie symmetry group can be recovered and a group of discrete transformations can be derived simultaneously. Lastly, from a known simple solution of the cKP equation, we can easily obtain two new solutions by the general symmetry groups.

Conditional Symmetry Groups of Nonlinear Diffusion Equations with x-Dependent Convection and Absorption

The generalized conditional symmetry and sign-invarinat approaches are developed to study the nonlinear diffusion equations with x-dependent convection and source terms.We obtain conditions under which the equations admit the second-order generalized conditional symmetries and the first-order sign-invariants on the solutions.Several types of different generalized conditional symmetries and first-order sign-invariants for the equations with diffusion of power law are obtained.Exact solutions to the resulting equations are constructed.

Finite Symmetry Transformation Groups and Exact Solutions of Konopelchenko-Dubrovsky Equation

Based on the general direct method developed by Lou et al.[J.Phys.A:Math.Gen.38 (2005) L129],the symmetry group theorem is obtained,from that both the Lie point groups and the non-Lie symmetry groups of theKonopelchenko-Dubrovsky (KD) equation are obtained.From the theorem,some exact solutions of KD equation arederived from a simple travelling wave solution and a multi-soliton solution.

Symmetry groups and Gauss kernels of Schrdinger equations

The relationship between symmetries and Gauss kernels for the SchrSdinger equation iut = uxx ＋ f（x）u is established. It is shown that if the Lie point symmetries of the equation are nontrivial, a classical integral transformations of the Gauss kernels can be obtained. Then the Gauss kernels of Schroedinger equations are derived by inverting the integral transformations. Furthermore, the relationship between Gauss kernels for two equations related by an equivalence transformation is identified.

Lie Symmetry Groups of(2+1)-Dimensional BKP Equation and Its New Solutions

A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physicssystems.Applying the modified direct method to the well-known (2+1)-dimensional BKP equation we get its symmetry.Furthermore,the exact solutions of (2+1)-dimensional BKP equation are obtained through symmetry analysis.

Some New Lie Symmetry Groups of Differential-Difference Equations Obtained from a Simple Direct Method

In this paper,based on the symbolic computing system Maple,the direct method for Lie symmetry groupspresented by Sen-Yue Lou [J.Phys.A:Math.Gen.38 (2005) L129] is extended from the continuous differential equationsto the differential-difference equations.With the extended method,we study the well-known differential-difference KPequation,KZ equation and (2+1)-dimensional ANNV system,and both the Lie point symmetry groups and the non-Liesymmetry groups are obtained.

Diophantine Equations and the Freeness of Mobius Groups

Let p and q be two fixed non zero integers verifying the condition gcd(p,q) = 1. We check solutions in non zero integers a1,b1,a2,b2 and a3 for the following Diophantine equations: (B1) (B2) . The equations (B1) and (B2) were considered by R.C. Lyndon and J.L. Ullman in [1] and A.F. Beardon in [2] in connection with the freeness of the M?bius group generated by two matrices of namely and where .?They proved that if one of the equations (B1) or (B2) has solutions in non zero integers then the group is not free. We give algorithms to decide if these equations admit solutions. We obtain an arithmetical criteria on p and q for which (B1) admits solutions. We show that for all p and q the equations (B1) and (B2) have only a finite number of solutions.

Symmetry Groups and New Exact Solutions of(2+1)-Dimensional Dispersive Long-Wave Equations

用修改 CK 的直接方法，我们导出一条对称组定理(2+1 ) 维的散长波浪的方程。在定理，谎言点对称组和新准确答案之上基于(2+1 ) 维的散长波浪的方程被获得。

关于“DIOPHANTINE EQUATIONS RELATED TO FINITE GROUPS”

本文利用推广的Ｐｅｌｌ方程法给出了文［１］中某些定理的简化证明。

ON THE HOLOMORPHIC SOLUTION OF NON-LINEAR TOTALLY CHARACTERISTIC EQUATIONS WITH SEVERAL SPACE VARIABLES

In this paper the authors study a class of non-linear singular partial differential equation in complex domain C-t x C-x(n). Under certain assumptions, they prove the existence and uniqueness of holomorphic solution near origin of C-t x C-x(n).

Developing allometric equations to estimate forest biomass for tree species categories based on phylogenetic relationships

The development of allometric biomass models is important process in biomass estimation because the reliability of forest biomass and carbon estimations largely depends on the accuracy and precision of such models.National Forest Inventories(NFI)are detailed assessments of forest resources at national and regional levels that provide valuable data for forest biomass estimation.However,the lack of biomass allometric equations for each tree species in the NFI currently hampers the estimation of national-scale forest biomass.The main objective of this study was to develop allometric biomass regression equations for each tree species in the NFI of China based on limited biomass observations.These equations optimally grouped NFI and biomass observation species according to their phylogenetic relationships.Significant phylogenetic signals demonstrated phylogenetic conservation of the crown-to-stem biomass ratio.Based on phylogenetic relationships,we grouped and matched NFI and biomass observation species into 22 categories.Allometric biomass regression models were developed for each of these 22 species categories,and the models performed successfully(R^(2)=0.97,root mean square error(RMSE)=12.9​t·ha^(–1),relative RMSE=11.5%).Furthermore,we found that phylogeny-based models performed more effectively than wood density-based models.The results suggest that grouping species based on their phylogenetic relationships is a reliable approach for the development and selection of accurate allometric equations.

On the Construction and Classification of the Common Invariant Solutions for Some P(1,4) -Invariant Partial Differential Equations

We consider the following (1 + 3)-dimensional P(1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inhomogeneous Monge-Ampère equation. The purpose of this paper is to construct and classify the common invariant solutions for those equations. For this aim, we have used the results concerning construction and classification of invariant solutions for the (1 + 3)-dimensional P(1,4)-invariant Eikonal equation, since this equation is the simplest among the equations under investigation. The direct checked allowed us to conclude that the majority of invariant solutions of the (1 + 3)-dimensional Eikonal equation, obtained on the base of low-dimensional (dimL ≤ 3) nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4), satisfy all the equations under investigation. In this paper, we present obtained common invariant solutions of the equations under study as well as the classification of those invariant solutions.

Analytical Solutions of System of Non-Linear Differential Equations in the Single-Enzyme, Single-Substrate Reaction with Non-Mechanism-Based Enzyme Inactivation

A closed form of an analytical expression of concentration in the single-enzyme, single-substrate system for the full range of enzyme activities has been derived. The time dependent analytical solution for substrate, enzyme-substrate complex and product concentrations are presented by solving system of non-linear differential equation. We employ He’s Homotopy perturbation method to solve the coupled non-linear differential equations containing a non-linear term related to basic enzymatic reaction. The time dependent simple analytical expressions for substrate, enzyme-substrate and free enzyme concentrations have been derived in terms of dimensionless reaction diffusion parameters ε, λ1, λ2 and λ3 using perturbation method. The numerical solution of the problem is also reported using SCILAB software program. The analytical results are compared with our numerical results. An excellent agreement with simulation data is noted. The obtained results are valid for the whole solution domain.

Computer Methodologies for the Comparison of Some Efficient Derivative FreeSimultaneous Iterative Methods for Finding Roots of Non-Linear Equations

In this article,we construct the most powerful family of simultaneous iterative method with global convergence behavior among all the existing methods in literature for finding all roots of non-linear equations.Convergence analysis proved that the order of convergence of the family of derivative free simultaneous iterative method is nine.Our main aim is to check out the most regularly used simultaneous iterative methods for finding all roots of non-linear equations by studying their dynamical planes,numerical experiments and CPU time-methodology.Dynamical planes of iterative methods are drawn by using MATLAB for the comparison of global convergence properties of simultaneous iterative methods.Convergence behavior of the higher order simultaneous iterative methods are also illustrated by residual graph obtained from some numerical test examples.Numerical test examples,dynamical behavior and computational efficiency are provided to present the performance and dominant efficiency of the newly constructed derivative free family of simultaneous iterative method over existing higher order simultaneous methods in literature.

Approximate Solution of Non-Linear Reaction Diffusion Equations in Homogeneous Processes Coupled to Electrode Reactions for CE Mechanism at a Spherical Electrode

A mathematical model of CE reaction schemes under first or pseudo-first order conditions with different diffusion coefficients at a spherical electrode under non-steady-state conditions is described. The model is based on non-stationary diffusion equation containing a non-linear reaction term. This paper presents the complex numerical method (Homotopy perturbation method) to solve the system of non-linear differential equation that describes the homogeneous processes coupled to electrode reaction. In this paper the approximate analytical expressions of the non-steady-state concentrations and current at spherical electrodes for homogeneous reactions mechanisms are derived for all values of the reaction diffusion parameters. These approximate results are compared with the available analytical results and are found to be in good agreement.

Extended Group Foliation Method and Functional Separation of Variables to Nonlinear Wave Equations

Generalized functional separation of variables to nonlinear evolution equations is studied in terms of the extended group foliation method, which is based on the Lie point symmetry method. The approach is applied to nonlinear wave equations with variable speed and external force. A complete classification for the wave equation which admits functional separable solutions is presented. Some known results can be recovered by this approach.