Order Linear Differential Equations Research Articles

Two algorithms for computing the matrix cosine function

The computation of matrix trigonometric functions has received remarkable attention in the last decades due to its usefulness in the solution of systems of second order linear differential equations. Several state-of-the-art algorithms have been provided recently for computing these matrix functions. In this work, we present two efficient algorithms based on Taylor series with forward and backward error analysis for computing the matrix cosine. A MATLAB implementation of the algorithms is compared to state-of-the-art algorithms, with excellent performance in both accuracy and cost.

The solution of differential equation with Hulthen potential in curved space

The solution of the Schrödinger equation for a physical system in quantum mechanics is of great importance, because the knowledge of wave- function and energy spectrum contain all possible information about the physical properties of a system. In this paper, we have give solution of the Schrödinger equation in three dimensional curved space with Hulthen potential on the positive constant curvature. Then we achieve the wave-function and energy spectrum for the Hulthen potential. In order to solve the corresponding Schrödinger equation, we use of Nikiforov-Uvarov (N.U) method [1]. The N.U method is based on solving the second- order linear differential equations by reducing to a generalized equation of hypergeometric type.

Entire functions with two radially distributed values

AbstractWe study entire functions whose zeros and one-points lie on distinct finite systems of rays. General restrictions on these rays are obtained. Non-trivial examples of entire functions with zeros and one-points on different rays are constructed, using the Stokes phenomenon for second order linear differential equations.

Solution of second order linear fuzzy ordinary differential equation by Lagrange multiplier method with application in mechanics

In this paper the solution of second order linear fuzzy ordinary differential equation is described. The solution procedure is described by Lagrange multiplier method and extension principle method. Further two mechanics problem with fuzzy initial condition are briefly illustrated. The solutions are defuzzified by a well min of α-cut defuzzification method.

Stokes geometry of higher order linear ordinary differential equations and middle convolution

The middle convolution, introduced by Katz and developed by Dettweiler–Reiter, Oshima and others, defines an operation of reduction of linear ordinary differential equations with polynomial coefficients. In this paper, employing an idea of the exact steepest descent method proposed by Aoki–Kawai–Takei, we study how to determine the complete Stokes geometry and how to obtain the Borel summability of WKB solutions of a higher order linear ordinary differential equation with a large parameter when it is reduced to a second order equation via middle convolution. To show the practical usefulness of the method, we also investigate some concrete examples numerically.

Nonasymptotic Pseudo-State Estimation for a Class of Fractional Order Linear Systems

This paper aims at designing a nonasymptotic and robust pseudo-state estimator for a class of fractional order linear systems which can be transformed into the Brunovsky's observable canonical form of pseudo-state space representation with unknown initial conditions. First, this form is expressed by a fractional order linear differential equation involving the initial values of the fractional sequential derivatives of the output, based on which the modulating functions method is applied. Then, the former initial values and the fractional derivatives of the output are exactly given by algebraic integral formulae using a recursive way, which are used to nonasymptotically estimate the pseudo-state of the system in noisy environment. Second, the pseudo-state estimator is studied in discrete noisy case, which contains the numerical error due to a used numerical integration method, and the noise error contribution due to a class of stochastic processes. Then, the noise error contribution is analyzed, where an error bound useful for the selection of design parameter is provided. Finally, numerical examples illustrate the efficiency of the proposed pseudo-state estimator, where some comparisons with the fractional order Luenberger-like observer and a new fractional order $\mathcal{H}_{\infty}$ -like observer are given.

On one two-point BVP for the fourth order linear ordinary differential equation

Abstract In this article we consider the two-point boundary value problem { u ( 4 ) ⁢ ( t ) = p ⁢ ( t ) ⁢ u ⁢ ( t ) + h ⁢ ( t ) for ⁢ a ≤ t ≤ b , u ( i ) ( a ) = c 1 ⁢ i , u ( i ) ( b ) = c 2 ⁢ i ( i = 0 , 1 ) , \left\{\begin{aligned} &\displaystyle u^{(4)}(t)=p(t)u(t)+h(t)\quad\text{for }% a\leq t\leq b,\\ &\displaystyle u^{(i)}(a)=c_{1i},\quad u^{(i)}(b)=c_{2i}\quad(i=0,1),\end{% aligned}\right. where c 1 ⁢ i , c 2 ⁢ i ∈ R {c_{1i},c_{2i}\in R} , h , p ∈ L ⁢ ( [ a , b ] ; R ) {h,p\in L([a,b];R)} . Here we study the question of dimension of the space of nonzero solutions and oscillatory behaviors of nonzero solutions on the interval [ a , b ] {[a,b]} for the corresponding homogeneous problem, and establish efficient sufficient conditions of solvability for the nonhomogeneous problem.

Regularization of closed positive currents and intersection theory

Abstract We prove the existence of a closed regularization of the integration current associated to an effective analytic cycle, with a bounded negative part. By means of the King formula, we are reduced to regularize a closed differential form with L1 loc coefficients, which by extension has a test value on any positive current with the same support as the cycle. As a consequence, the restriction of a closed positive current to a closed analytic submanifold is well defined as a closed positive current. Lastly, given a closed smooth differential (qʹ, qʹ)-form on a closed analytic submanifold, we prove the existence of a closed (qʹ, qʹ)-current having a restriction equal to that differential form. After blowing up we deal with the case of a hypersurface and then the extension current is obtained as a solution of a linear differential equation of order 1.

The distributional continuous fractional generalized Hankel-Clifford transformation

The main objective of this paper is to study the fractional generalized Hankel-Clifford transformation and some of their basic properties. The continuous fractional generalized Hankel-Clifford transformation, its inversion formula are also studied. The finite continuous fractional generalized Hankel-Clifford transformation is also studied in Zemanian space. Applications of the finite fractional generalized Hankel-Clifford transformation in solving generalized nth order linear non-homogeneous partial differential equation is given.

Finite element methods for second order linear elliptic partial differential equations in non-divergence form

Finite element methods for second order linear elliptic partial differential equations in non-divergence form

First order linear differential equations with involutive delay and hypergeometric functions

We present an alternative approach to functions satisfying second order linear ordinary differential equations. It turns out that many of them satisfy a first order ordinary differential equation with an involution. The involution acts on the argument as well as on parameters. Basic examples involve the hypergeometric functions and their descendants.

Schwarz problem for higher order linear equations in a polydisc

In this paper, we discuss the Schwarz boundary value problem for higher order linear complex differential equations in the unit polydisc of . After the relevant preparations, we give the solution for a model equation with homogeneous boundary conditions. Then we obtain the solutions of a linear equation involving Schwarz boundary conditions on the unit polydisc.

一阶常微分方程通解中任意常数的分析及教学

本文主要从两个方面分析一阶常微分方程通解中任意常数的含义。一：任意常数不一定取遍所有的实数；二：任意常数和方程的特解不一定是一一对应的，即取定一个任意常数不一定仅仅只确定一个特解，也不是每个特解都可以由通解中的任意常数可以确定。最后为方便使用常数变易法求解非齐次一阶线性微分方程的通解，举例阐明了一阶线性微分方程对应齐次方程通解的规范形式。 The arbitrary constant of the usual solution to first order ordinary differential equation is discussed in this paper. Firstly, the arbitrary constant in the usual solution does not necessarily take over all of the real numbers. Secondly, the arbitrary constant and special solution is not necessarily a one-to-one correspondence, that is, a constant does not determine a unique special solution, also is not that each special solution can be determined by the arbitrary constant in usual solution. In order to use the method of variation of constant to solve the first order linear differential equation, the canonical form of the usual solution to the homogeneous equation of first order linear differential equation is provided.

Nonrectifiable oscillatory solutions of second order linear differential equations

The second order linear differential equation \begin{equation*} (p(x)y^{\prime })^{\prime }+q(x)y=0\,, \quad x \in (0,x_0] \end{equation*} is considered, where $p$, $q \in C^1(0,x_0]$, $p(x)>0$, $q(x)>0$ for $x \in (0,x_0]$. Sufficient conditions are established for every nontrivial solutions to be nonrectifiable oscillatory near $x=0$ without the Hartman–Wintner condition.

Oscillation of a First Order Linear Impulsive Delay Differential Equation with Continuous and Piecewise Constant Arguments

A class of first order linear impulsive delay differential equation with continuous and piecewise constant arguments is studied. Using a connection between impulsive delay differential equations and non- impulsive delay differential equations sufficient conditions for the oscillation of the solutions are obtained.

Dynamic instability of super-long elastic rod in viscous fluid

The external environment affects the structural form of biological system. Many biological systems are surrounded by cell solutions, such as DNA and bacteria. The solution will offer a viscous resistance as the biological system moves in the viscous fluid. How does the viscous resistance affect the stability of biological system and what mode will be selected after instability? In this paper, we establish a super-long elastic rod model which contains the viscous resistance to model this phenomenon. The stability and instability of the super-long elastic rod in the viscous fluid are studied. The dynamic equations of motion of the super-long elastic rod in viscous fluid are given based on the Kirchhoff dynamic analogy. Then a coordinate basis vector perturbation scheme is reviewed. According to the new perturbation method, we obtain the first order perturbation representation of super-long elastic rod dynamic equation in the viscous fluid, which is a group of the second order linear partial differential equations. The stability of the super-long elastic rod can be determined by analyzing the solutions of the second order linear partial differential equations. The results are applied to a twisted planar DNA ring. The stability criterion of the twisted planar DNA ring and its critical region are obtained. The results show that the viscous resistance has no effect on the stability of super-long elastic rod dynamics, but affects its instability. The mode selection and the influence of the viscous resistance on the instability of DNA ring are discussed. The amplitude of the elastic loop becomes smaller under the influence of the viscous resistance, and a bifurcation occurs. The mode number of instability of DNA loop becomes bigger with the increase of viscous resistance.

Comparison on Sufficient Conditions for the Stability of Hill Equation: An Arnold’s Tongues Approach

It is known that the solutions of a second order linear differential equation with periodic coefficients are almost always analytically impossible to obtain and in order to study its properties we often require a computational approach. In this paper we compare graphically, using the Arnold Tongues, some sufficient criteria for the stability of periodic differential equations. We also present a brief explanation on how the authors, of each criterion, obtained them. And a comparison between four sufficient stability criteria and the stability zones found by perturbation methods is presented.

单位圆内二阶线性微分方程解的导数的不动点

本文研究了一类二阶线性微分方程 的解及其导数的不动点性质，其中A(z)和B(z)是单位圆 内的解析函数，得到了解及其一阶、二阶导数的不动点的精确估计。 The properties of the fixed points of the solutions and their derivatives of a type of second order linear differential equation , where A(z) and B(z) are analytic functions in the unit disc , are investigated. We obtain some precise estimates of the fixed points of the solutions and their 1st, 2nd derivatives of the equations.

Inherent Numerical Instability in Computing Invariant Measures of Markov Chains

Invariant measures of Markov chains in discrete or continuous time with a countable set of states are characterized by its steady state recurrence relations. Exemplarily, we consider transition matrices and Q-matrices with upper bandwidth n and lower bandwidth 1 where the invariant measures satisfy an (n + 1)-order linear difference equation. Markov chains of this type arise from applications to queueing problems and population dynamics. It is the purpose of this paper to point out that the forward use of this difference equation is subject to some hitherto unobserved aspects. By means of the concept of generalized continued fractions (GCFs), we prove that each invariant measure is a dominated solution of the difference equation such that forward computation becomes numerically unstable. Furthermore, the GCF-based approach provides a decoupled recursion in which the phenomenon of numerical instability does not appear. The procedure results in an iteration scheme for successively computing approximants of the desired invariant measure depending on some truncation level N. Increasing N leads to the desired solution. A comparison study of forward computation and the GCF-based approach is given for Q-matrices with upper bandwidth 1 and 2.

Oscillation Conditions for a Type of Second Order Neutral Differential Equations with Impulses

In this paper, we study a certain type of second order linear neutral differential equation with constant impulsive jumps. This type of equation is known always to possess an unbounded non-oscillatory solution. The method and technique of impulse imposition used here is due to studies by Bainov and Simeonov [1]. By assuming, amongst other conditions, that the constant coefficient of the equation in question lies between zero and one and the delay function is non-decreasing, it is shown that all bounded solutions of the said neutral impulsive equation are oscillatory.