Criteria for System of Three Second-Order Ordinary Differential Equations to Be Reduced to a Linear System via Restricted Class of Point Transformation

This paper is devoted to the study of the linearization problem of system of three second-order ordinary differential equations and . The necessary conditions for linearization by general point transformation and are found. The sufficient conditions for linearization by restricted class of point transformation and are obtained. Moreover, the procedure for obtaining the linearizing transformation is provided in explicit forms. Examples demonstrating the procedure of using the linearization theorems are presented.

On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods

Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects.Using the Laplace transform for solving differential equations,however,sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analyticalmeans.Thus,we need numerical inversionmethods to convert the obtained solution fromLaplace domain to a real domain.In this paper,we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with orderα,β.Our proposed numerical scheme is based on three main steps.First,we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense,and then into Caputo sense.Secondly,we transformthe fractional differential equation in Caputo sense to an equivalent equation in Laplace space.Then the solution of the transformed equation is obtained in Laplace domain.Finally,the solution is converted into the real domain using numerical inversion of Laplace transform.Three inversion methods are evaluated in this paper,and their convergence is also discussed.Three test problems are used to validate the inversion methods.We demonstrate our results with the help of tables and figures.The obtained results show that Euler’s and Talbot’s methods performed better than Stehfest’s method.

Combining Methods of Lyapunov for Exponential Stability of Linear Dynamic Systems on Time Scales

Consider the linear dynamic equation on time scales (1) where , ,?is a rd-continuous function, T is a time scales. In this paper, we shall investigate some results for the exponential stability of the dynamic Equation (1) by combinating the first approximate method and the second method of Lyapunov.

Some New Applications of Elzaki Transform for Solution of Linear Volterra Type Integral Equations

Volterra type integral equations have diverse applications in scientific and other fields. Modelling physical phenomena by employing integral equations is not a new concept. Similarly, extensive research is underway to find accurate and efficient solution methods for integral equations. Some of noteworthy methods include Adomian Decomposition Method (ADM), Variational Iteration Method (VIM), Method of Successive Approximation (MSA), Galerkin method, Laplace transform method, etc. This research is focused on demonstrating Elzaki transform application for solution of linear Volterra integral equations which include convolution type equations as well as one system of equations. The selected problems are available in literature and have been solved using various analytical, semi-analytical and numerical techniques. Results obtained after application of Elzaki transform have been compared with solutions obtained through other prominent semi-analytic methods i.e. ADM and MSA (limited to first four iterations). The results substantiate that Elzaki transform method is not only a compatible alternate approach to other analytic methods like Laplace transform method but also simple in application once compared with methods ADM and MSA.

Linearization of Emden Differential Equation via the Generalized Sundman Transformations

The Emden differential equation is one of the most widely studied and challenging nonlinear dynamics equations in literature. It finds applications in various areas of study such as celestial mechanics, fluid mechanics, Steller structure, isothermal gas spheres, thermionic currents and so on. Because of the importance of the equation, the method of generalized Sundman transformation (GST) as proposed by Nakpim and Meleshko is used for linearizing the Emden differential equation. The Emden differential equation considered here is a modification of the equation given by Berkovic. The results obtained in this paper imply that the Emden equation cannot be linearized by a point transformation. The general solution of the modified Emden equation is also obtained.

Breakdown Voltage Prediction by Utilizing the Behavior of Natural Ester for Transformer Applications

This research investigates the dielectric performance of Natural Ester(NE)using the Partial Differential Equation(PDE)tool and analyzes dielectric performance using fuzzy logic.NE nowadays is found to replace Mineral Oil(MO)due to its extensive dielectric properties.Here,the heat-tolerant Natural Esters Olive oil(NE1),Sunflower oil(NE2),and Ricebran oil(NE3)are subjected to High Voltage AC(HVAC)under different electrodes configurations.The break-down voltage and leakage current of NE1,NE2,and NE3 under Point-Point(P-P),Sphere-Sphere(S-S),Plane-Plane(PL-PL),and Rod-Rod(R-R)are measured,and survival probability is presented.The electricfield distribution is analyzed using the Partial Differential Equation(PDE)tool.NE shows better HVAC with stand capacity under all the electrodes configuration,especially in the S-S shape geometry.The exponential function is developed for the oils under different elec-trode geometry;NE shows a higher survival probability.Likewise,the most influ-ential dielectric properties such as breakdown voltage,kinematic viscosity,and water content are used to develop a Mamdani-based control system model that combines two variables to produce the surface model.The surface model indi-cates that the NE subjected for investigation is less susceptible to moisture effect and higher kinematic viscosity.Based on the surface models of PDE and fuzzy,it is concluded that NE possesses a high survival rate since its breakdown voltage does not react to changes in water content.Hence the application of NE in the transformer application is highly safe and possesses extended life.

Exponential Dichotomies and Fredholm Operators of Dynamic Equations on Time Scales

For time-varying non-regressive linear dynamic equations on a time scale with bounded graininess, we introduce the concept of the associative operator with linear systems on time scales. The purpose of this research is the characterizations of the exponential dichotomy obtained in terms of Fredholm property of that associative operator. Particularly, we use Perron’s method, which was generalized on time scales by J. Zhang, M. Fan, H. Zhu in?[1], to show that if the associative operator is semi-Fredholm then the corresponding linear nonautonomous equation has an exponential dichotomy on both?T?+?and?T-.??Moreover, we also give the converse result that the linear systems have?an exponential dichotomy on both?T?+?andT-??then the associative operator is Fredholm on?T.

Double Elzaki Transform Decomposition Method for Solving Non-Linear Partial Differential Equations

In this paper, we discuss a new method employed to tackle non-linear partial differential equations, namely Double Elzaki Transform Decomposition Method (DETDM). This method is a combination of the Double ELzaki Transform and Adomian Decomposition Method. This technique is hereafter provided and supported with necessary illustrations, together with some attached examples. The results reveal that the new method is very efficient, simple and can be applied to other non-linear problems.

STABILITY OF GLOBAL MAXWELLIAN FOR NON-LINEAR VLASOV-POISSON-FOKKER-PLANCK EQUATIONS

In this article, we establish the exponential time decay of smooth solutions around a global Maxwellian to the non-linear Vlasov–Poisson–Fokker–Planck equations in the whole space by uniform-in-time energy estimates. The non-linear coupling of macroscopic part and Fokker–Planck operator in the model brings new difficulties for the energy estimates, which is resolved by adding tailored weighted-in-v energy estimates suitable for the Fokker–Planck operator.

NUMERICAL ANALYSIS FOR VOLTERRA INTEGRAL EQUATION WITH TWO KINDS OF DELAY

In this article, we study the Volterra integral equations with two kinds of delay that are proportional delay and nonproportional delay. We mainly use Chebyshev spectral collocation method to analyze them. First, we use variable transformation to transform the equation into an new equation which is defined in [-1,1]. Then, with the help of Gronwall inequality and some other lemmas, we provide a rigorous error analysis for the proposed method, which shows that the numerical error decay exponentially in L~∞ and L_(ω~c)~2-norm. In the end, we give numerical test to confirm the conclusion.

Applications of Lodrigues Matrix in 3D Coordinate Transformation

Three transformation models （Bursa-Wolf, Molodensky, and WTUSM） are generally used between two data systems transformation. The linear models are used when the rotation angles are small; however, when the rotation angles get bigger, model errors will be produced. In this paper, we present a method with three main terms：① the traditional rotation angles θ,φ,ψ are substituted with a,b,c which are three respective values in the anti-symmetrical or Lodrigues matrix; ② directly and accurately calculating the formula of seven parameters in any value of rotation angles; and ③ a corresponding adjustment model is established. This method does not use the triangle function. Instead it uses addition, subtraction, multiplication and division, and the complexity of the equation is reduced, making the calculation easy and quick.

Linear Complementary Method for Moving Boundary Problems with Phase Transformation

In this paper, the linear complementary method for moving boundary problems with phase transformation is presented, in which a pair of unknown vectors of heat source with phase transforming and the temperature field can be solved exactly, and a large amount of iterative calculations can be avoided.

Simple Procedure for Entanglement Transformation of Bipartite Pure States

An explicit procedure for transforming one bipartite entangled state into another via local operations and classical communication （LOCC） is presented. Our procedure is much simper than the previous ones in the sense that, it only involves two steps and the explicit expression of local general measurement used in the procedure can be obtained by solving a set of linear equations. Furthermore, this procedure is still applicable in high dimensional case.

Continuity Equation in Presence of a Non-Local Potential in Non-Commutative Phase-Space

We studied the continuity equation in presence of a local potential, and a non-local potential arising from electron-electron interaction in both commutative and non-commutative phase-space. Furthermore, we examined the influence of the phase-space non-commutativity on both the locality and the non-locality, where the definition of current density in commutative phase-space cannot satisfy the condition of current conservation, but with the steady state, in order to solve this problem, we give a new definition of the current density including the contribution due to the non-local potential. We showed that the calculated current based on the new definition of current density maintains the current. As well for the case when the non- commutativity in phase-space considered, we found that the conservation of the current density completely violated;and the non-commutativity is not suitable for describing the current density in presence of non-local and local potentials. Nevertheless, under some conditions, we modified the current density to solve this problem. Subsequently, as an application we studied the Frahn-Lemmer non-local potential, taking into account that the employed methods concerning the phase-space non-commutativity are both of Bopp-shift linear transformation through the Heisenberg-like commutation relations, and the Moyal-Weyl product.

一个解特定二阶驻定方程的新方法——特征伴随方程法

本文对文献[1]中提出的一类二阶驻定方程G″/G=∑^(m)_(j=0)P_(j)(G′/G)^(j)的求法进行探讨,提出了全新的求解方法——特征伴随方程法,通过该求法得到这类方程当m取不同非负整数(本文仅讨论m=2)时的通解,并相应给出该方程作为求解非线性偏微分方程的辅助方程时所需要的G′/G的解析表达式;同时给出一个作为常微分方程中驻定方程的应用实例,以及该方程作为非线性偏微分方程的辅助方程时利用扩展G′/G展开法的求解例子,通过该实例给出方程的精确行波解。

线性正则正余弦加权卷积及其应用

针对积分方程的求解问题,本文提出了利用卷积运算及其卷积定理来讨论两类卷积类积分方程组的解。首先,在线性正则正弦变换与线性正则余弦变换的基础上,定义了线性正则正余弦卷积运算及其加权卷积运算;其次,推导了相应的卷积定理,研究了该卷积与傅里叶正余弦变换卷积运算的关系;最后,讨论了两类卷积类积分方程组的解,给出了该方程解的一般形式。

Interaction analysis between slenderness ratio and resin content on mechanical properties of particleboard

The interaction between particle size and resin content is one of the most important structural parameters that can influence the accuracy of predictions about wood-composite properties. We developed three kinds of equation （linear, quadratic, and exponential） for each mechanical property of particleboard based on slenderness ratio and resin content at a constant density （0.7g cm -3 ）. Results from SHAZAM software （version 9） suggested that the quadratic function was not significant, but the linear and exponential functions were significant. The interaction between particle size and resin content was analyzed by Maple 9 software. The results indicated that an exponential function can better describe the simultaneous effect of slenderness and resin content than a linear equation. Under constant resin content, particles with higher slenderness ratios increased more in modulus of rupture （MOR） and modulus of elasticity （MOE） than did particles with lower slenderness ratios. Edge withdrawal resistance （SWRe） values did not increase with increasing slenderness ratio.

Exponential Stability of Traveling Pulse Solutions of a Singularly Perturbed System of Integral Differential Equations Arising From Excitatory Neuronal Networks

We establish the exponential stability of fast traveling pulse solutions to nonlinear singularly per-turbed systems of integral differential equations arising from neuronal networks.It has been proved that expo-nential stability of these orbits is equivalent to linear stability.Let (?) be the linear differential operator obtainedby linearizing the nonlinear system about its fast pulse,and let σ((?)) be the spectrum of (?).The linearizedstability criterion says that if max{Reλ:λ∈σ((?)),λ≠0}(?)-D,for some positive constant D,and λ=0 is asimple eigenvalue of (?)(ε),then the stability follows immediately (see [13] and [37]).Therefore,to establish theexponential stability of the fast pulse,it suffices to investigate the spectrum of the operator (?).It is relativelyeasy to find the continuous spectrum,but it is very difficult to find the isolated spectrum.The real part ofthe continuous spectrum has a uniformly negative upper bound,hence it causes no threat to the stability.Itremains to see if the isolated spectrum is safe.Eigenvalue functions (see [14] and [35,36]) have been a powerful tool to study the isolated spectrum of the as-sociated linear differential operators because the zeros of the eigenvalue functions coincide with the eigenvaluesof the operators.There have been some known methods to define eigenvalue functions for nonlinear systems ofreaction diffusion equations and for nonlinear dispersive wave equations.But for integral differential equations,we have to use different ideas to construct eigenvalue functions.We will use the method of variation of param-eters to construct the eigenvalue functions in the complex plane C.By analyzing the eigenvalue functions,wefind that there are no nonzero eigenvalues of (?) in {λ∈C:Reλ(?)-D} for the fast traveling pulse.Moreoverλ=0 is simple.This implies that the exponential stability of the fast orbits is true.

Weak solutions to one-dimensional quantum drift-diffusion equations for semiconductors

The weak solutions to the stationary quantum drift-diffusion equations （QDD） for semiconductor devices are investigated in one space dimension. The proofs are based on a reformulation of the system as a fourth-order elliptic boundary value problem by using an exponential variable transformation. The techniques of a priori estimates and Leray-Schauder＇s fixed-point theorem are employed to prove the existence. Furthermore, the uniqueness of solutions and the semiclassical limit δ→0 from QDD to the classical drift-diffusion （DD） model are studied.

Non-Metric CCD Camera Calibration Algorithm in a Digital Photogrammetry System

Camera calibration is a critical process in photogrammetry and a necessary step to acquire 3D information from a 2D image. In this paper, a flexible approach for CCD camera calibration using 2D direct linear transformation （DLT） and bundle adjustment is proposed. The proposed approach assumes that the camera interior orientation elements are known, and addresses a new closed form solution in planar object space based on homogenous coordinate representation and matrix factorization. Homogeneous coordinate representation offers a direct matrix correspondence between the parameters of the 2D DLT and the collinearity equation. The matrix factorization starts by recovering the elements of the rotation matrix and then solving for the camera position with the collinearity equation. Camera calibration with high precision is addressed by bundle adjustment using the initial values of the camera orientation elements. The results show that the calibration precision of principal point and focal length is about 0.2 and 0.3 pixels respectivelv, which can meet the requirements of close-range photogrammetry with high accuracy.