Fractional differential equations of motion in terms of combined Riemann-Liouville derivatives

In this paper, we focus on studying the fractional variational principle and the differential equations of motion for a fractional mechanical system. A combined Riemann-Liouville fractional derivative operator is defined, and a fractional Hamilton principle under this definition is established. The fractional Lagrange equations and the fractional Hamilton canonical equations are derived from the fractional Hamilton principle. A number of special cases are given, showing the universality of our conclusions. At the end of the paper, an example is given to illustrate the application of the results.

Bifurcation and Solitary Waves of the Combined KdV and KdV Equation

Bifurcation, bistability and solitary waves of the combined KdV and mKdV equation are investigatedsystematically. At first, bifurcation and bistability are analyzed by selecting an integral constant as the bifurcationparameter. Then, different conditions expressed in terms of the bifurcation parameter are obtained for the existence ofbreather-like, algebraic, pulse-like solitary waves, and shock waves. All types of the solitary wave and shock wave solutionsare given by direct integration. Finally, an approximate analytic method by employing the interpolation polynomials iscomplete and the theoretical methods are the simplest hitherto.

Fractional cyclic integrals and Routh equations of fractional Lagrange system with combined Caputo derivatives

In this paper, we develop a fractional cyclic integral and a Routh equation for fractional Lagrange system defined in terms of fractional Caputo derivatives. The fractional Hamilton principle and the fractional Lagrange equations of the system are obtained under a combined Caputo derivative. Furthermore, the fractional cyclic integrals based on the Lagrange equations are studied and the associated Routh equations of the system are presented. Finally, two examples are given to show the applications of the results.

Hierarchy of Combined TL-RTL Equations and an Associated （2＋l）-Dimensional Lattice Equation

A new （29-1）-dimensional lattice equation is presented based upon the first two members in the hierarchy of the combined Toda lattice and relativistic Toda lattice （TL-RTL） equations in （19991） dimensions. A Darboux transformation for the hierarchy of the combined TL-RTL equations is constructed. Solutions of the first two members in the hierarchy of the combined TL-RTL equations, as well as the new （29-1）-dimensional lattice equation are explicitly obtained by the Darboux transformation.

Exact Travelling Wave Solutions for a Combined KdV and Schwarzian KdV Equation

In this paper the ( G'/G )-expansion method is used to find exact travelling wave solutions for a combined KdV and Schwarzian KdV equation. As a result, multiple travelling wave solutions with arbitrary parameters are obtained, which are expressed by hyperbolic functions, trigonometric functions and rational functions. When the parameters are taken as special values, the solitary waves are derived from the travelling waves. The (G'/G)-expansion method presents a wider applicability for handling nonlinear wave equations.

Analytical Treatment of the Evolutionary (1 + 1)-Dimensional Combined KdV-mKdV Equation via the Novel (G'/G)-Expansion Method

The novel (G'/G)-expansion method is a powerful and simple technique for finding exact traveling wave solutions to nonlinear evolution equations (NLEEs). In this article, we study explicit exact traveling wave solutions for the (1 + 1)-dimensional combined KdV-mKdV equation by using the novel (G'/G)-expansion method. Consequently, various traveling wave solutions patterns including solitary wave solutions, periodic solutions, and kinks are detected and exhibited.

EXPLICIT EXACT SOLUTIONS FOR THE GENERALIZED COMBINED KdV AND MKdV EQUATION

With the aid of Mathematica and Wu elimination method,via using a new generalized ansatz and well known Riccati equation,thirty two families of explicit and exact solutions for the generalized combined KdV and mKdV equation are obtained,which contain new solitary wave solutions and periodic wave solutions.This approach can also be applied to other nonlinear evolution equations.

An Exact Solution to the Combined KS and EdV Equation

The exact solution for the combined KS and KdV equation is obtained via introducing a simple and effective nonlinear transformations.This method is very concise and primary and can be applied to other unintegrable nonlinear evolution equations.It is common knowledge that the Korteweg de Vries(KdV) equation [1] (1)has been proposed as model equation for the weakly nonlinear long waves which occur in many different physical systems; the Kuramoto-Sivashinsky (KS) equationis one of the simplest nonliaear partial differential equations that exhibit Chaotic behavior frequently encounted in the study of continous media [2-4] . Many interesting mathematical and physical properties of eqs. (1) and (2) have been studied widely. But, in several problems where a lonq wavelength oscilatory instability is found, the noulineai evolution of the perturbations near rriticality is governed by the dispersion modified Kuramoto-Sivashi nsky equation(3)ft is clear that this equation is a combination of the KdV and

The Effect of Ecological Factors on the Indexes of Yield Property Equation and Quantitative Analysis

The mechanism of action of ecological factors affecting crop growth and development was complicated. In order to study the relationships between ecological factors and the indexes of yield property equation and determine the main ecological factors affecting yield, using 3-yr field experimental results for different yielding spring maize （Zea mays L.） populations and the relative meteorological observation data in Huadian of Jilin Province in China, and analyzing on the base of the yield property equation （MLAI × D × MNAR × HI = EN × GN × GW）, the main ecological factors were screened, and further mechanisms of action affecting yield were analyzed. Stepwise regression analysis showed that yield was affected mainly by effective accumulated temperature, daily mean minimum temperature, daily mean maximum temperature in July, the ratios of growth days, and the sunshine hour before and after silking. In yield property equation, four indexes of MLAI, growth days, ear number and grain number （total grain number） affected principally yield, the ecological factors affecting predominantly yield were effective accumulated temperature, daily mean temperature, daily mean minimum temperature, daily mean maximum temperature in July, the ratios of growth days, rainfall, accumulated temperature, and sunshine hours before and after silking. Combined with the two analytical methods, it could be deduced that the temperature and the allocated ratios before and after silking of ecological factors were the key factors to achieve high yield. Therefore, appropriate sowing data should be adjusted to achieve the suitable temperature indexes during the whole growth stage and the rational allocated ratios of ecological factors before and after silking.

New Travelling Wave Solutions to Compound KdV-Burgers Equation

The compound KdV-Burgers equation and combined KdV-mKdV equation are real physical models concerning many branches in physics.In this paper,applying the improved trigonometric function method to these equations,rich explicit and exact travelling wave solutions,which contain solitary-wave solutions,periodic solutions,and combined formal solitary-wave solutions,are obtained.

FINITE TIME BLOW UP OF THE SOLUTIONS TO BOUSSINESQ EQUATION WITH LINEAR RESTORING FORCE AND ARBITRARY POSITIVE ENERGY

Finite time blow up of the solutions to Boussinesq equation with linear restoring force and combined power nonlinearities is studied. Sufficient conditions on the initial data for nonexistence of global solutions are derived. The results are valid for initial data with arbitrary high positive energy. The proofs are based on the concave method and new sign preserving functionals.

Solution of nonlinear wave equation of elastic rod

The longitudinal oscillation of a nonlinear elastic rod with lateral inertia are studied. A nonlinear wave equation is derived. The equation is solved by the method of full approximation.

New explicit and exact solutions of the Benney-Kawahara-Lin equation

In this paper, we present a combination method of constructing the explicit and exact solutions of nonlinear partial differential equations. And as an illustrative example, we apply the method to the Benney-Kawahara-Lin equation and derive its many explicit and exact solutions which are all new solutions.

Application of Combined Prediction Model in Predicting Total Water Consumption in Ningxia

The gray GM( 1,1) prediction model and Logistic equation gray prediction model were established separately,and then the combined prediction model was established. Taking the water consumption in Ningxia Hui Autonomous Region from 2006 to 2012 as modeling data,the total water consumption of the whole region of Ningxia in 2018-2020 was analyzed and predicted. The results show that the accuracy of the three prediction models meets the accuracy requirements,but the gray GM( 1,1) and combined prediction models better conform to the actual situation and have better applicability.

Exact Solution of Dirac Equation with Charged Harmonic Oscillator in Electric Field: Bound States

In some quantum chemical applications, the potential models are linear combination of single exactly solvable potentials. This is the case equivalent of the Stark effect for a charged harmonic oscillator (HO) in a uniform electric field of specific strength (HO in an external dipole field). We obtain the exact s-wave solutions of the Dirac equation for some potential models which are linear combination of single exactly solvable potentials (ESPs). In the framework of the spin and pseudospin symmetric concept, we calculate analytical expressions for the energy spectrum and the corresponding two-component upper- and lower-spinors of the two Dirac particles by using the Nikiforov-Uvarov (NU) method, in closed form. The nonrelativistic limit of the solution is also studied and compared with the other works.

New Solutions for an Elliptic Equation Method and Its Applications in Nonlinear Evolution Equations

In this paper, we study an elliptic equation with four distinct real roots and obtain five new solutions to this type of elliptic equation. Using these obtained new elliptic function solutions we can construct a series of explicit exact solutions for many nonlinear evolution equations. As examples, we choose combined KdV-MKdV equation, a fourth-order integrable nonlinear Schrödinger equation and generalized Dullin-Gottwald-Holm equation to demonstrate the effectiveness of these new elliptic function solutions. These new elliptic function solutions can be applied to many other nonlinear evolution equations.

Soliton molecules for combined mKdV-type bilinear equation

Starting from the multi-soliton solutions obtained by the Hirota bilinear method,the soli ton molecule structures for the combined mKdV-type bilinear equation(Dt+∑n=1NαnDx2n+1)f*·f=0 are investigated using the velocity resonance mechanism.The two-soliton molecules of the mKdV-35 equation and the three-soliton molecules of the mKdV-357 equation are specifically demonstrated in this paper.With particular selections of the involved arbitrary parameters,especially the wave numbers,it is confirmed that,besides the usual multi-bright soliton molecules,the multi-dark soliton molecules and the mixed multibright-dark soliton molecules can also be obtained.In addition,we discuss the existence of the multi-soliton molecules for the combined mKdV-type bilinear equation with more higher order nonlinear terms and dispersions.The results demonstrate that when N≥4,the combined mKdVtype bilinear equation no longer admits soliton molecules comprising more than four solitons.

Multi-mode Multi-frequency GNSS-IR Combination System for Sea Level Retrieval

With the development of Global Navigation Satellite Systems(GNSS),geodetic GNSS receivers have been utilized to monitor sea levels using GNSS-Interferometry Reflectometry(GNSS-IR)technology.The multi-mode,multi-frequency signals of GPS,GLONASS,Galileo,and Beidou can be used for GNSS-IR sea level retrieval,but combining these retrievals remains problematic.To address this issue,a GNSS-IR sea level retrieval combination system has been developed,which begins by analyzing error sources in GNSS-IR sea level retrieval and establishing and solving the GNSS-IR retrieval equation.This paper focuses on two key points:time window selection and equation stability.The stability of the retrieval combination equations is determined by the condition number of the coefficient matrix within the time window.The impact of ill-conditioned coefficient matrices on the retrieval results is demonstrated using an extreme case of SNR data with only ascending or descending trajectories.After determining the time window and removing ill-conditioned equations,the multi-mode,multi-frequency GNSS-IR retrieval is performed.Results from three International GNSS Service(IGS)stations show that the combination method produces high-precision,high-resolution,and high-reliability sea level retrieval combination sequences.

职住学空间测度及儿童通学决策的影响机理——以昆明市双职工家庭为例

当代城市家庭选择居住地时往往将临近学校作为重要参考,在缩小学住距离的同时加剧了职住分离。选取昆明市561个有学龄儿童的双职工家庭为研究对象,分析家庭空间分异对儿童通学决策的影响。首先,基于时间地理学理论构建考虑职住学三个锚点的家庭活动空间及测度理论模型。其次,根据测度结果将家庭活动空间划分为职住学就近型和职住学分离型两类。最后,建立结构方程模型分别探究两种模式下父母工作时长以及父母儿童组合时间窗对儿童通学时空行为的影响。研究结果表明:相对于学住距离,父母的职住距离和工作时长对通学行为的影响更为显著,因此就近入学政策对于促进儿童步行上学的作用有限;对于职住学分离型家庭,父母下班与儿童放学组合时间窗的宽裕程度对儿童出行影响显著,组合时间窗越宽裕,儿童小汽车出行的可能越大,因此弹性放学时间的效果也有待商榷;儿童在通学行为中的性别差异主要在职住学分离型家庭中体现。

Cahn-Hilliard方程的一个超紧致有限差分格式

研究四阶Cahn-Hilliard方程的数值求解方法。给出组合型超紧致差分格式,将其用于四阶Cahn-Hilliard方程的空间导数离散,采用四阶Runge-Kutta格式离散时间导数,将二者结合得到四阶Cahn-Hilliard方程的离散格式,并给出了该格式的误差估计。通过编程计算得到其数值解,并与精确解进行对比,结果表明本文的数值方法误差小,验证了所提方法的有效性和可行性。