Dispersive propagation of optical solitions and solitary wave solutions of Kundu-Eckhaus dynamical equation via modified mathematical method

In this research work,we constructed the optical soliton solutions of nonlinear complex Kundu-Eckhaus(KE)equation with the help of modified mathematical method.We obtained the solutions in the form of dark solitons,bright solitons and combined dark-bright solitons,travelling wave and periodic wave solutions with general coefficients.In our knowledge earlier reported results of the KE equation with specific coefficients.These obtained solutions are more useful in the development of optical fibers,dynamics of solitons,dynamics of adiabatic parameters,dynamics of fluid,problems of biomedical,industrial phenomena and many other branches.All calculations show that this technique is more powerful,effective,straightforward,and fruitfulness to study analytically other higher-order nonlinear complex PDEs involves in mathematical physics,quantum physics,Geo physics,fluid mechanics,hydrodynamics,mathematical biology,field of engineering and many other physical sciences.

RHOMBUS THINKING METHOD AND ITS APPLICATION IN SCHEME DESIGN

Rhombus thinking, a new creative thinking method,is the combination of divergent thinking process and convergent thinking process,in which qualitative analysis is carried out before quantitative analysis This method tries to solve the bottle neck problem in intelligent CAD based on the extension theory The rhombus thinking method to the scheme design of new products is applied In this process, firstly, the matter element expression for the know information are set up, and then a set of matter elements are opened up by matter elements extension method; Finally,the useful information are got by appraisal method of dependent degree It has been successfully applied to the scheme design for the cutter store of machining center Theoretical and experimental results demonstrated fhat the method is much more accurate,objective and efficient than the traditional one

Geological Environmental Adaptability of Qingdao Rural Urbanization Based on Fuzzy Mathematical Method

Fuzzy mathematics comprehensive evaluation method is used to evaluate the geological environment suitability of rural urbanization in Qingdao City,China.A total of 5 first-level evaluation factors are selected,including environmental geological condition,geological resources,engineering geological condition,geological disaster and environmental geological problem,and human engineering activity.And there are 27 second-level evaluation factors,such as topography,land type and vegetation,nature reserve,water source protection area,groundwater quality division,and major engineering project.Qingdao City is divided into four districts of suitable area,relatively suitable area,moderately suitable area and relatively unsuitable area of ecological environment.And their characteristics are introduced.Suggestions for the developing direction of urban construction are put forward.Region of Laoshan District lying to the west of the Shilaoren is suitable to set up high-rise building;west Hongshiya may establish a waste landfill site;Jiaozhou Bay,the downstream of Dagu River,and Jihongtan Reservoir should be built as the key geological environment protection area and water source protection area.And the north Hongdao should be strictly monitored in order to control the expansion of urban construction to Jihongtan Reservoir.Mocheng District and the area north of it,Jiaozhou District and the area east of it are the ideal urban construction development areas in Qingdao City in the future.

Consistency and Validity of the Mathematical Models and the Solution Methods for BVPs and IVPs Based on Energy Methods and Principle of Virtual Work for Homogeneous Isotropic and Non-Homogeneous Non-Isotropic Solid Continua

Energy methods and the principle of virtual work are commonly used for obtaining solutions of boundary value problems (BVPs) and initial value problems (IVPs) associated with homogeneous, isotropic and non-homogeneous, non-isotropic matter without using (or in the absence of) the mathematical models of the BVPs and the IVPs. These methods are also used for deriving mathematical models for BVPs and IVPs associated with isotropic, homogeneous as well as non-homogeneous, non-isotropic continuous matter. In energy methods when applied to IVPs, one constructs energy functional (I) consisting of kinetic energy, strain energy and the potential energy of loads. The first variation of this energy functional (δI) set to zero is a necessary condition for an extremum of I. In this approach one could use δI = 0 directly in constructing computational processes such as the finite element method or could derive Euler’s equations (differential or partial differential equations) from δI = 0, which is also satisfied by a solution obtained from δI = 0. The Euler’s equations obtained from δI = 0 indeed are the mathematical model associated with the energy functional I. In case of BVPs we follow the same approach except in this case, the energy functional I consists of strain energy and the potential energy of loads. In using the principle of virtual work for BVPs and the IVPs, we can also accomplish the same as described above using energy methods. In this paper we investigate consistency and validity of the mathematical models for isotropic, homogeneous and non-isotropic, non-homogeneous continuous matter for BVPs that are derived using energy functional consisting of strain energy and the potential energy of loads. Similar investigation is also presented for IVPs using energy functional consisting of kinetic energy, strain energy and the potential energy of loads. The computational approaches for BVPs and the IVPs designed using energy functional and principle of virtual work, their consistency and validity are also investigated. Classical continuum mechanics (CCM) principles i.e. conservation and balance laws of CCM with consistent constitutive theories and the elements of calculus of variations are employed in the investigations presented in this paper.

Dynamic Factor Method of Computing Dynamic Mathematical Model for System Simulation

The computational methods of a typical dynamic mathematical model that can describe the differential element and the inertial element for the system simulation are researched. The stability of numerical solutions of the dynamic mathematical model is researched. By means of theoretical analysis, the error formulas, the error sign criteria and the error relationship criterion of the implicit Euler method and the trapezoidal method are given, the dynamic factor affecting the computational accuracy has been found, the formula and the methods of computing the dynamic factor are given. The computational accuracy of the dynamic mathematical model like this can be improved by use of the dynamic factor.

An Effective Numerical Calculation Method for Multi-Time-Scale Mathematical Models in Systems Biology

The improvements of high-throughput experimental devices such as microarray and mass spectrometry have allowed an effective acquisition of biological comprehensive data which include genome, transcriptome, proteome, and metabolome (multi-layered omics data). In Systems Biology, we try to elucidate various dynamical characteristics of biological functions with applying the omics data to detailed mathematical model based on the central dogma. However, such mathematical models possess multi-time-scale properties which are often accompanied by time-scale differences seen among biological layers. The differences cause time stiff problem, and have a grave influence on numerical calculation stability. In the present conventional method, the time stiff problem remained because the calculation of all layers was implemented by adaptive time step sizes of the smallest time-scale layer to ensure stability and maintain calculation accuracy. In this paper, we designed and developed an effective numerical calculation method to improve the time stiff problem. This method consisted of ahead, backward, and cumulative algorithms. Both ahead and cumulative algorithms enhanced calculation efficiency of numerical calculations via adjustments of step sizes of each layer, and reduced the number of numerical calculations required for multi-time-scale models with the time stiff problem. Backward algorithm ensured calculation accuracy in the multi-time-scale models. In case studies which were focused on three layers system with 60 times difference in time-scale order in between layers, a proposed method had almost the same calculation accuracy compared with the conventional method in spite of a reduction of the total amount of the number of numerical calculations. Accordingly, the proposed method is useful in a numerical analysis of multi-time-scale models with time stiff problem.

SIMULTANEOUS DETERMINATION OF MULTICOMPONENT IN MIXED ACIDS BY MATHEMATICAL METHOD

A new method for the determination of components in mixed acids has been developed.The mathematical model is obtained from samples of known composition and is then used to predict the concentrations of components in unknown sample.The practical utility of this method is demonstrated for simultaneous determination of two systems of ternary mixed acids and the results are satisfactory.

A Numerical Method for Rigid-plastic FEM Analysis Basing on Mathematical Programming

The rigid-plastic analysis of mental forming simulation is formulated as a discrete nonlinear mathematical programming problem with equality and inequality constraints by means of the finite element technique. An iteration algorithm is used to solve this formulation, which distinguishes the integration points of the rigid zones and the plastic zones and solves a series of the quadratic programming to overcome the difficulties caused by the nonsmoothness and the nonlinearity of the objective function. This method has been used to carry out the rigid-plastic FEM analysis. An example is given to demonstrate the effectiveness of this method.

Mathematical modelling of a biofilm: The Adomian decomposition method

A mathematical modelling by a biofilm under steady state conditions is discussed. The nonlinear differential Equations in biofilm reaction is solved using the Adomian decomposition method. Approximate analytical expressions for substrate concentration have been derived for all values of parameters δ and SL. These analytical results are compared with the available numerical results and are found to be in good agreement.

A two-parameter mathematical model for immobilizedenzymes and Homotopy analysis method

A two parameter mathematical model was developed to find the concentration for immobilized enzyme systems in porous spherical particles. This model contains a non-linear term related to reversible Michaelies-Menten kinetics. Analytical expression pertaining to the substrate concentration was reported for all possible values of Thiele module φ and α . In this work, we report the theoretically evaluated steady-state effectiveness factor for immobilized enzyme systems in porous spherical particles. These analytical results were found to be in good agreement with numerical results. Moreover, herein we employ new “Homotopy analysis method” (HAM) to solve non-linear reaction/diffusion equation.

Levenberg-Marquardt Method for Mathematical Programs with Linearly Complementarity Constraints

In this paper, a new method for solving a mathematical programming problem with linearly complementarity constraints (MPLCC) is introduced, which applies the Levenberg-Marquardt (L-M) method to solve the B-stationary condition of original problem. Under the MPEC-LICQ, the proposed method is proved convergent to B-stationary point of MPLCC.

Ignite the Spark of Wisdom——Thinking on the Cultivation of Elementary Students’ Mathematical Intuition Thinking Ability

In the learning process of primary mathematics,intuitive thinking remains an important section for students to analyze and solve mathematical problems,which has played an irreplaceable role in enlightening and developing the underlying intellectual and non-intellectual factors of students.By analyzing and comparing the relevant theories and research results regarding mathematical intuition thinking,as well as taking into account the learning characteristics of elementary students,the author has summarized four kinds of strategies suitable for training the mathematical intuition thinking ability of elementary students.

A Class of Smoothing-regularization Methods to Mathematical Programs with Vanishing Constraints

this paper,we propose a class of smoothing-regularization methods for solving the mathematical programming with vanishing constraints.These methods include the smoothing-regularization method proposed by Kanzow et al.in[Comput.Optim.Appl.,2013,55(3):733-767]as a special case.Under the weaker conditions than the ones that have been used by Kanzow et al.in 2013,we prove that the Mangasarian-Fromovitz constraint qualification holds at the feasible points of smoothing-regularization problem.We also analyze that the convergence behavior of the proposed smoothing-regularization method under mild conditions,i.e.,any accumulation point of the stationary point sequence for the smoothing-regularization problem is a strong stationary point.Finally,numerical experiments are given to show the efficiency of the proposed methods.

An Empirical Study of the Effect of Inquiry Teaching on the Development of High School Students'Critical Thinking Under the New Curriculum Standards

The Ministry of Education of the People’s Republic of China has issued three guideline documents to encourage inquiry teaching to help students develop their ability to question,which is a defining feature of critical thinking.Two of the documents have clearly stated that English subject teaching should focus on building students’thinking capacity.Critical thinking is an essential feature of building thinking capacity and a key element of the core competencies the English subject seeks to cultivate.This research measures the performance of students of two high school classes in the exams conducted at the beginning,in the middle,and at the end of the experiment period during which inquiry teaching was applied to enhance students’critical thinking ability.The experiment results show that inquiry teaching can improve students’academic performance in English study and help them better develop their language skills.The implementation of the inquiry design in teaching listening,reading,and writing conforms to the law that governs the development of critical thinking,and the adoption of inquiry teaching can promote the development of critical thinking.

BLOCK PIVOT METHODS FOR SOLVING FRICTIONAL CONTACT PROBLEMS

Based on elementary group theory, the block pivot methods for solving two-dimensional elastic frictional contact problems are presented in this paper. It is proved that the algorithms converge within a finite number of steps when the friction coefficient is ''relative small''. Unlike most mathematical programming methods for contact problems, the block pivot methods permit multiple exchanges of basic and nonbasic variables.

Some Equivalence Transform Methods for Simulating Models in Control Systems

In this paper two classes of equivalence transform methods for solving ordinary differential equations are proposed. One class of method is the equivalence integral transform method for special differential algebraic problems. The advantage of this class of method is such that the amount of work calculating one integration with parameters becomes that of two interpolations, when the system of nonlinear equations is solved on the right hand side function. The other class of method is the equivalence substitution method for avoiding calculating derivative on the right hand side function. In order to avoid calculation derivatives, two equivalence substitution methods are proposed here. The application instances of some special effect of the equivalence substitution methods are given.

DAMAGE ANALYSIS BY BOUNDARY ELEMENT METHOD

The response of cracked bodies subjected to loading was investigated by the boundary element method in this paper. The two-law elastic-cohesive-softening model was used for crack propagation analysis. The interface conditions for uncracked, craze, open crack, adhesive crack and slid crack parts were discussed and the corresponding incremental iteration algorithm was given. A simplified damage propagation model was presented. The technique has been applied to some specific examples which give the evidence that the method is satisfactory and efficient.

Zonal method solution of radiative heat transfer in a one-dimensional long roller-hearth furnace in CSP

A radiative heat transfer mathematical model for a one-dimensional long furnace was set up in a through-type roller-hearth furnace （TTRHF） in compact strip production （CSP）. To accurately predict the heat exchange in the furnace, modeling of the complex gas energy-balance equation in volume zones was considered, and the heat transfer model of heating slabs and wall lines was coupled with the radiative heat transfer model to identify the surface zonal temperature. With numerical simulation, the temperature fields of gas, slabs, and wall lines in the furnace under one typical working condition were carefully accounted and analyzed. The fundamental theory for analyzing the thermal process in TI＇RI-IF was provided.

Mathematical modeling of elastic inverted pendulum control system

Inverted pendulums are important objects of theoretical investigation and experiment in the area of control theory and engineering. The researches concentrate on the rigid finite dimensional models which are described by ordinary differential equations (ODEs) .Complete rigidity is the approximation of practical models ; Elasticity should be introduced into mathematical models in the analysis of system dynamics and integration of highly precise controller. A new kind of inverted pendulum, elastic inverted pendulum was proposed, and elasticity was considered. Mathematical model was derived from Hamiltonian principle and variational methods, which were formulated by the coupling of partial differential equations (PDE) and ODE. Because of infinite dimensional, system analysis and control of elastic inverted pendulum is more sophisticated than the rigid one.