THE REPRESENTATION OF THE SOLUTION OF STURM-LIOUVILLE EQUATION WITH DISCONTINUITY CONDITIONS

The aim of this paper is to construct the integral representation of the solution of Sturm-Liouville equation with eigenparameter-dependent discontinuity conditions at an interior point of the finite interval. Moreover, we examine the properties of the kernel function of this integral representation and obtain the partial differential equation provided by this kernel function.

Existence and Multiplicities of Solutions for Asymptotically Linear Ordinary Differential Equations Satisfying Sturm-Liouville BVPs with Resonance

In this paper, we prove existence and multiplicities of solutions for asymptotically linear ordinary differential equations satisfying Sturm-Liouville boundary value conditions with resonance. Adding assumption H3 that is similar to (LL) in Theorem 1.1, by index theory and Morse theory, we obtain more nontrivial solutions.

On a Boundary Value Problem for a Polynomial Pencil of the Sturm-Liouville Equation with Spectral Parameter in Boundary Conditions

The boundary value problem with a spectral parameter in the boundary conditions for a polynomial pencil of the Sturm-Liouville operator is investigated. Using the properties of the transformation operators for such operators, the asymptotic formulas for eigenvalues of the boundary value problem are obtained.

EXISTENCE OF SOLUTIONS OF STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS FOR NONLINEAR SECOND ORDER IMPULSIVE DIFFERENTIAL EQUATIONS IN BANACH SPACES

Abstract In this paper, the fixed point theorem is applied to investigate the existence of solutions of Sturm Liouville boundary value problems for nonlinear second order impulsive differential equations in Banach spaces.

Higher Order Solitary Wave Solutions of the Standard KdV Equations

Considered under their standard form, the fifth-order KdV equations are a sort of reading table on which new prototypes of higher order solitary waves residing there, have been uncovered and revealed to broad daylight. The mathematical tool that made it possible to explore and analyze this equation is the Bogning-Djeumen Tchaho-Kofané method extended to the new implicit Bogning' functions. The analytical form of the solutions chosen in this manuscript is particular in the sense that it contains within its bosom, a package of solitary waves made up of three solitons, especially, the bright type soliton, the hybrid soliton and the dark type soliton which we estimate capable in their interactions of generating new hybrid or multi-form solitons. Existence conditions of the obtained solitons have been determined. It emerges that, these existence conditions of the chosen ansatz could open the way to other new varieties of fifth-order KdV equations including to which it will be one of the solutions. Some of the obtained solitons are exact solutions. Intense numerical simulations highlighted numerical stability and confirmed the hybrid character of the obtained solutions. These results will help to model new nonlinear wave phenomena, in plasma media and in fluid dynamics, especially, on the shallow water surface.

Analytical and Traveling Wave Solutions to the Fifth Order Standard Sawada-Kotera Equation via the Generalized exp(-Φ(ξ))-Expansion Method

In this article, we propose a generalized exp(-Φ(ξ))-expansion method and successfully implement it to find exact traveling wave solutions to the fifth order standard Sawada-Kotera (SK) equation. The exact traveling wave solutions are established in the form of trigonometric, hyperbolic, exponential and rational functions with some free parameters. It is shown that this method is standard, effective and easily applicable mathematical tool for solving nonlinear partial differential equations arises in the field of mathematical physics and engineering.

Boundary Value Problems for Burgers Equations, through Nonstandard Analysis

In this paper we study inviscid and viscid Burgers equations with initial conditions in the half plane . First we consider the Burgers equations with initial conditions admitting two and three shocks and use the HOPF-COLE transformation to linearize the problems and explicitly solve them. Next we study the Burgers equation and solve the initial value problem for it. We study the asymptotic behavior of solutions and we show that the exact solution of boundary value problem for viscid Burgers equation as viscosity parameter is sufficiently small approach the shock type solution of boundary value problem for inviscid Burgers equation. We discuss both confluence and interacting shocks. In this article a new approach has been developed to find the exact solutions. The results are formulated in classical mathematics and proved with infinitesimal technique of non standard analysis.

Applicability of standard forward column/row recurrence equations for ALFs

Fully normalized associated Legendre functions(fnALFs)are a set of orthogonal basis functions that are usually calculated by using the recurrence equation.This paper presented the applicability and universality of the standard forward column/row recurrence equation based on the isolated singular factor method and extended-range arithmetic.Isolating a singular factor is a special normalization method that can improve the universality of the standard forward row recurrence equation to a certain extent,its universality can up to degree hundreds.However,it is invalid for standard forward column recurrence equation.The extended-range arithmetic expands the double-precision number field to the quad-precision numberfield.The quad-precision numberfield can retain more significant digits in the operation process and express larger and smaller numbers.The extended-range arithmetic can significantly improve the applicability and universality of the standard forward column/row recurrence equations,its universality can up to degree several thousand.However,the quad-precision numberfield operation needs to occupy more storage space,which is why its operation speed is slow and undesirable in practical applications.In this paper,the X-number method is introduced in the standard forward row recurrence equation for thefirst time.With the use of the X-number method,fnALFs can be recursed to 4.2 billion degree by using standard forward column/row recurrence equations.

A New Two-Parameter Heteromorphic Elliptic Equation: Properties and Applications

The ellipse and the superellipse are both planar closed curves with a double axis of symmetry. Here we show the isoconcentration contour of the simplified two-dimensional advection-diffusion equation from a stable line source in the center of a wide river. A new two-parameter heteromorphic elliptic equation with a single axis of symmetry is defined. The values of heights, at the point of the maximum width and that of the centroid of the heteromorphic ellipse, are derived through mathematical analysis. Taking the compression coefficient θ = b/a = 1 as the criterion, the shape classification of H-type, Standard-type and W-type for heteromorphic ellipse have been given. The area formula, the perimeter theorem, and the radius of curvature of heteromorphic ellipses, and the geometric properties of the rotating body are subsequently proposed. An illustrative analysis shows that the inner contour curve of a heteromorphic elliptic tunnel has obvious advantages over the multiple- arc splicing cross section. This work demonstrates that the heteromorphic ellipses have extensive prospects of application in all categories of tunnels, liquid transport tanks, aircraft and submarines, bridges, buildings, furniture, and crafts.

Validation of general linear modeling for identifying factors associated with Quality of Life: A comparison with structural equation modeling

Purpose: General linear modeling (GLM) is usually applied to investigate factors associated with the domains of Quality of Life (QOL). A summation score in a specific sub-domain is regressed by a statistical model including factors that are associated with the sub-domain. However, using the summation score ignores the influence of individual questions. Structural equation modeling (SEM) can account for the influence of each question’s score by compositing a latent variable from each question of a sub-domain. The objective of this study is to determine whether a conventional approach such as GLM, with its use of the summation score, is valid from the standpoint of the SEM approach. Method: We used the Japanese version of the Maugeri Foundation Respiratory Failure Questionnaire, a QOL measure, on 94 patients with heart failure. The daily activity sub-domain of the questionnaire was selected together with its four accompanying factors, namely, living together, occupation, gender, and the New York Heart Association’s cardiac function scale (NYHA). The association level between individual factors and the daily activity sub-domain was estimated using SEM?and GLM, respectively. The standard partial regression coefficients of GLM and standardized path coefficients of SEM were compared. If?these coefficients were similar (absolute value of the difference -0.06 and -0.07 for the GLM and SEM. Likewise, the estimates of occupation, gender, and NYHA were -0.18 and -0.20, -0.08 and -0.08, 0.51 and 0.54, respectively. The absolute values of the difference for each factor were 0.01, 0.02, 0.00, and 0.03, respectively. All differences were less than 0.05. This means that these two approaches lead to similar conclusions. Conclusion: GLM is a valid method for exploring association factors with a domain in QOL.

Numerical solution to the Falkner-Skan equation:a novel numerical approach through the new rational α-polynomials

The new rationalα-polynomials are used to solve the Falkner-Skan equation.These polynomials are equipped with an auxiliary parameter.The approximated solution to the Falkner-Skan equation is obtained by the new rational a-polynomials with unknown coefficients.To find the unknown coefficients and the auxiliary parameter contained in the polynomials,the collocation method with Chebyshev-Gauss points is used.The numerical examples show the efficiency of this method.

Semiclassical Method to Schroedinger Equation with Position-Dependent Effective Mass

In this paper, two novel semiclassical methods including the standard and supersymmetric WKB quantization conditions are suggested to discuss the Schroedinger equation with position-dependent effective mass. From a proper coordinate transformation, the formalism of the Schroedinger equation with position-dependent effective mass is mapped into isospectral one with constant mass and therefore for a given mass distribution and physical potential function the bound state energy spectrum can be determined easily by above method associated with a simple integral formula. It is shown that our method can give the analytical results for some exactly-solvable quantum systems.

A Generalization of Ince’s Equation

We investigate the Hill differential equation ?where A(t), B(t), and D(t) are trigonometric polynomials. We are interested in solutions that are even or odd, and have period π or semi-period π. The above equation with one of the above conditions constitutes a regular Sturm-Liouville eigenvalue problem. We investigate the representation of the four Sturm-Liouville operators by infinite banded matrices.

NONCONFORMING FINITE ELEMENT PENALTY METHOD FOR STOKES EQUATION

a special penalty method is presented to improve the accuracy of the standard penalty method for solving Stokes equation with nonconforming finite element.It is shown that this method with a larger penalty parameter can achieve the same accuracy as the standard method with a smaller penalty parameter. The convergence rate of the standard method is just half order of this penalty method when using the same penalty parameter, while the extrapolation method proposed by Faik et al can not yield so high accuracy of convergence.At last, we also get the super convergence estimates for total flux. Received October 14, 1996 1991 MR Subject Classification: 65N30.

Asymptotic Formulas of the Solutions and the Trace Formulas for the Polynomial Pencil of the Sturm-Liouville Operators

This work studies the asymptotic formulas for the solutions of the Sturm-Liouville equation with the polynomial dependence in the spectral parameter. Using these asymptotic formulas it is proved some trace formulas for the eigenvalues of a simple boundary problem generated in a finite interval by the considered Sturm-Liouville equation.

一种光波大气折射率剖面模型构建方法

根据经典光波折射率计算公式和ITU-R建议书中水汽密度经验公式,给出了一种新的光波大气折射率计算公式;借鉴Hopfield折射率静力项剖面模型和ITU-R建议书中水汽密度剖面模型,给出了一种基于历史气象探空数据构建光波大气折射率剖面模型的方法。以青岛地区为例,通过对1986—1995年历史气象探空数据的处理并结合参考标准大气,建立了适合当地的光波大气折射率剖面模型;统计剖面模型预测折射率剖面与实测折射率剖面的均方根误差,结果表明:构建的剖面模型具有较好的预测精度,这对光学外测设备的折光修正数据处理具有很好的参考价值。

空间分数阶偏微分方程非标准有限差分方法的稳定性和收敛性

本文研究空间分数阶偏微分方程非标准有限差分方法数值解的相关问题.采用Grünwald-Letnikov公式和平移Grünwald-Letnikov公式分别对两个空间分数阶导数进行离散.再运用带有时间和空间步长的分母函数构造非标准有限差分方法.进而利用von Neumann分析方法对差分格式的稳定性和收敛性进行研究,获得了一些新的结果.数值例子验证了非标准有限差分方法用于求解空间分数阶偏微分方程的有效性.

非标准大气压下卷烟纸阴燃速率的修正研究

为了修正非标准大气压下与标准大气压下卷烟纸阴燃速率之间的系统性差异,在6家不同大气压下的实验室(5家非标准大气压实验室和1家近似标准大气压实验室)中对4种卷烟纸样品进行阴燃速率测试,分析大气压与卷烟纸阴燃速率之间的相关性,建立非标准大气压下卷烟纸阴燃速率的修正方程,并验证该方程的可行性。结果表明:1)卷烟纸阴燃速率与大气压呈正相关,R^(2)为0.8572~0.9141,且4种卷烟纸样品的阴燃速率与大气压的相关性均达到极显著水平;2)非标准大气压条件下卷烟纸阴燃速率的修正方程为r_(c)=r_(m)×(0.8157×√P_(0)/P_(1)+0.1936);3)实验样品在非标准大气压下阴燃速率的修正值与标准大气压下的阴燃速率无显著差异,且验证样品在阴燃速率的修正值与其在标准大气压下的阴燃速率的最大偏差绝对值仅为0.170,说明所建立的修正方程可用于修正非标准大气压下卷烟纸的阴燃速率,以消除非标准大气压与标准大气压下测试结果之间的系统性差异。

一类非线性Poisson-Nernst-Planck方程的边平均有限元计算

针对一类非线性Poisson-Nernst-Planck数学模型,为提高数值求解效率并保证数值求解稳定性,推导了边平均有限元离散格式,并给出了数值求解的耦合迭代算法。在对网格进行一些适当假设的情况下,边平均有限元离散格式的刚度矩阵是一个M-阵,数值求解比标准有限元方法更稳定。数值结果表明,边平均有限元方法的L^(2)模误差收敛阶达到最优阶,且在自由度相同情况下,边平均有限元方法所用CPU时间大约是标准有限元方法的1/3。

Existence of Positive Solutions of Generalized Sturm-Liouville Boundary Value Problems for a Singular Differential Equation

By employing the fixed point theorem of cone expansion and compression of norm type, we investigate the existence of positive solutions of generalized Sturm-Liouville boundary value problems for a nonlinear singular differential equation with a parameter. Some sufficient conditions for the existence of positive solutions are established. In the last section, an example is presented to illustrate the applications of our main results.