Variational Approach for the Adapted Solution of Backw ard Stochastic Differential Equations with Locally Lipschitz Diffusion Coefficients

One existence integral condition was obtained for the adapted solution of the general backward stochastic differential equations(BSDEs). Then by solving the integral constraint condition, and using a limit procedure, a new approach method is proposed and the existence of the solution was proved for the BSDEs if the diffusion coefficients satisfy the locally Lipschitz condition. In the special case the solution was a Brownian bridge. The uniqueness is also considered in the meaning of "F0-integrable equivalent class" . The new approach method would give us an efficient way to control the main object instead of the "noise".

Applications of F-expansion to Periodic Wave Solutions for Variant Boussinesq Equations

We present an F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the variant Boussinesq equations. When the modulus m approaches 1 and 0, the hyperbolic function solutions (including the solitary wave solutions) and trigonometric solutions are also given respectively.

Extended F-Expansion Method and Periodic Wave Solutions for Klein-Gordon-SchrSdinger Equations

我们在场为在数学物理发现非线性的进化方程的周期的波浪答案的一个扩大 F 扩大方法。由使用扩大 F 扩大方法，各种各样的 Jacobi 椭圆形的功能为 theKlein-Gordon-Schroedinger 方程表示的许多周期的波浪答案被获得。在限制盒子中，方程的独居的波浪解决方案和三角法的函数解决方案也被获得。

A Generalized F-expansion Method and Its Application in High-Dimensional Nonlinear Evolution Equation

A generalized F-expansion method is introduced and applied to (3+ 1)-dimensional Kadomstev-Petviashvili(KP) equation. As a result, some new Jacobi elliptic function solutions of the equation are found, from which the trigonometric function solutions and the solitary wave solutions can be obtained. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.

Self-Similar Solutions of Variable-Coefficient Cubic-Quintic Nonlinear Schrdinger Equation with an External Potential

An improved homogeneous balance principle and an F-expansion technique are used to construct exactself-similar solutions to the cubic-quintic nonlinear Schrodinger equation.Such solutions exist under certain conditions,and impose constraints on the functions describing dispersion,nonlinearity,and the external potential.Some simpleself-similar waves are presented.

Exact solutions for the coupled Klein-Gordon-Schrǒdinger equations using the extended F-expansion method

The extended F-expansion method or mapping method is used to construct exact solutions for the coupled KleinGordon Schr/Sdinger equations （K-G-S equations） by the aid of the symbolic computation system Mathematica. More solutions in the Jacobi elliptic function form are obtained, including the single Jacobi elliptic function solutions, combined Jacobi elliptic function solutions, rational solutions, triangular solutions, soliton solutions and combined soliton solutions.

Quintessence anisotropic stellar models in quadratic and Born-Infeld modified teleparallel Rastall gravity

This study aims to discuss anisotropic solutions that are spherically symmetric in the quintessence field,which describe compact stellar objects in the modified Rastall teleparallel theory of gravity.To achieve this goal,the Krori and Barua arrangement for spherically symmetric components of the line element is incorporated.We explore the field equations by selecting appropriate off-diagonal tetrad fields.Born-Infeld function of torsion f(T)=β√λT+1-1 and power law form h(T)=δTn are used.The Born-Infeld gravity was the first modified teleparallel gravity to discuss inflation.We use the linear equation of state pr=ξρto separate the quintessence density.After obtaining the field equations,we investigate different physical parameters that demonstrate the stability and physical acceptability of the stellar models.We use observational data,such as the mass and radius of the compact star candidates PSRJ 1416-2230,Cen X-3,&4U 1820-30,to ensure the physical plausibility of our findings.

Periodic Wave Solutions for Konopelchenko-Dubrovsky Equation

By using F-expansion method proposed recently, we derive the periodic wave solution expressed by Jacobi elliptic functions for Konopelchenko-Dubrovsky equation. In the limit case, the solitary wave solution and other type of the traveling wave solutions are derived.

BCKLUND TRANSFORMATIONS FOR THE EQUATION~2u/x^1x^1+~2u/x^2x^2=f(u)

Backlund transformations for the equation is an arbitrary function) is studied in this paper, using the procedure of Wahlquist and Estabrook (WEP). We conclude that the condition is sufficient for the existence of Backlund transformations for the equation of our interest. A special case of our results leads to the conclusion of Leibbrandt[1,2]

MAURER-CARTAN EQUATION OF A_∞-ALGEBRAS

The axioms of A ∞ algebras can be written as Maurer Cartan equation.Infi nitesimal multiplication of a graded associative algebra is defined and the integrability of infinitesimal multiplication is discussed through the Massey F product.

THE INERTIAL SETS OF EQUATION FOR THE FLOW AND MAGNETIC FIELD WITHIN THE EARTH

In this paper we consider a class of equations for the flow and magnetic field within the earth, for initial-boundary value problem, we prove existence of inertial sets and it's fractal dimension has been given, the squeezing rate to inertial sets from trajectory of absorbing set has been estimated.

岩体基本质量指标BQ与岩石坚固性系数f之间的关系研究

为了建立岩石坚固性系数f与岩体基本质量指标BQ、岩体各力学参数指标之间的关系,采用下列方法进行研究:(1)已知岩石单轴抗压强度R_C和BQ、f都存在相关关系,以R_C作为中间量,建立BQ与f的关系式;(2)已知岩体体积节理数J_V和岩石质量指标D_(RQD)、完整性指数K_V都存在相关关系,以J_V作为中间量,建立D_(RQD)与K_V的关系式,再经推导可得出BQ、f、D_(RQD)之间的关系式;(3)根据BQ、f、D_(RQD)之间的关系式,建立岩体各力学参数与f之间的关系方程;(4)将得到的研究结果用于吉尔吉斯斯坦南北路隧道工程的岩体分级及岩体各力学参数估计,据此确定施工方案,沉降变形监测结果表明,该工程施工效果良好。

Some New Nonlinear Wave Solutions for a Higher-Dimensional Shallow Water Wave Equation

In this manuscript, we first perform a complete Lie symmetry classification for a higher-dimensional shallow water wave equation and then construct the corresponding reduced equations with the obtained Lie symmetries. Moreover, with the extended F-expansion method, we obtain several new nonlinear wave solutions involving differentiable arbitrary functions, expressed by Jacobi elliptic function, Weierstrass elliptic function, hyperbolic function and trigonometric function.

变系数F展开法求解非线性Schrodinger方程的精确解

本文基于变系数F展开法,并借助Mathematica数学软件,求解了水平科氏力作用下Rossby波振幅满足的非线性Schrodinger方程,得到一系列Jacobi椭圆函数解,以及当模数m→1和m→0时由其退化的双曲函数解和三角函数解,并绘制它们的三维图形.扩展了变系数F展开法求解非线性偏微分方程的应用范畴.同时也为非线性Schr?dinger方程得到更多形式的精确解.

New Exact Solutions to Dispersive Long-Wave Equations in （2＋1）-Dimensional Space

Jacobi 椭圆形的函数表示的新准确解决方案被获得到(2+1 ) 由使用修改 F 扩大方法的维的散 Jong 波浪方程。Inthe 限制大小写，新独居的波浪答案和三角形的周期的波浪答案也被获得。

The exact solutions to (2+1)-dimensional nonlinear Schrdinger equation

By using the extended F-expansion method, the exact solutions,including periodic wave solutions expressed by Jacobi elliptic functions, for (2+1)-dimensional nonlinear Schrdinger equation are derived. In the limit cases, the solitary wave solutions and the other type of traveling wave solutions for the system are obtained.

利用通用F-展开法求解ZK-BBM方程

利用通用F-展开法对ZK-BBM方程进行求解,作行波变换,将偏微分方程转化为常微分方程.假设方程具有洛朗级数形式的解,其中φ满足一般椭圆方程.通过齐次平衡法,确定参数n.通过比较同次幂项的系数,将非线性方程的求解问题转化为代数方程组求解.运用扩展的椭圆方程方法,求解ZK-BBM方程的孤子解、三角函数解、有理解和Jacobi椭圆函数解.