Order Homogeneous Linear Differential Equations Research Articles

Meromorphic solutions to linear differential equations with entire coefficients of [p,q]-order

ABSTRACTIn this paper, we investigate the growth and value distribution of meromorphic solutions to higher order homogeneous and nonhomogeneous linear differential equations in which the coefficients are entire functions of finite [p, q]-order. We get the results about [p, q]-order and the [p, q]-convergence exponent of solutions for such equations.

The absence of residual property for total hyper-frequencies of solutions to third order differential equations

It is established that total hyper-frequencies considered as functionals on the set of solutions to third order homogeneous linear differential equations with continuous coefficients bounded on the semiaxis are not residual (i.e. can be changed when changing the solution on a finite interval).

A New Method and Applications of the Boundary Value Problem of Differential Equation

The similar structure of solution for the boundary value problem of second order linear homogeneous differential equation has been studied. Based on the analysis of the relationship between similar structure of solution, its kernel function, the equation and boundary conditions, similar constructive method (shortened as SCM) of solution is obtained. According to the SCM, the similar structure of solution and its kernel function are constructed for the mathematical model of homogeneous reservoir which considers the influence of bottom-hole storage and skin effect under the infinite outer boundary condition. The SCM is a new and innovative way to solve boundary value problem of differential equation and seepage flow theory, which is especially used in Petroleum Engineering.

Riccati equations for holographic 2-point functions

Any second order homogeneous linear ordinary differential equation can be transformed into a first order non-linear Riccati equation. We argue that the Riccati form of the linearized fluctuation equations that determine the holographic 2-point functions simplifies considerably the numerical computation of such 2-point functions and of the corresponding transport coefficients by computing directly the response functions, eliminating the arbitrary source from the start. Moreover, it provides a neat criterion for the infrared regularity of the fluctuations. In particular, it is shown that the infrared regularity conditions for scalar and tensor fluctuations coincide, and hence they are either both regular or both singular. We demonstrate our numerical recipe based on the Riccati equations by computing the holographic 2-point functions for the stress tensor and a scalar operator in a number of asymptotically anti de Sitter backgrounds of bottom up scalar-gravity models. Analytical results are obtained for the 2-point function of the transverse traceless part of the stress tensor in two confining geometries, including a geometry that belongs to the class of IHQCD. We find that in this background the spin-2 spectrum is linear and, as expected, the position space 2-point function decays exponentially at large distances at a rate proportional to the confinement scale.

一类高阶线性微分方程亚纯解与小函数的关系

本文研究了一类亚纯函数系数高阶齐次和非齐次线性微分方程的亚纯解的增长性，并进一步研究了它们的亚纯解与小函数的关系。其中某一个系数具有有限亏值或满足一定条件，其余系数也满足相应的条件。 In this paper, we consider the growth of the meromorphic solutions of a class of higher order homogeneous and non-homogeneous linear differential equations with meromorphic coefficients, and further consider the relation between their meromorphic solutions and functions of small growth, where one of these coefficients has a finite deficient value or satisfies some conditions, others satisfy corresponding conditions.

线性微分方程亚纯解的零点和增长级的定量估计

本文研究了高阶齐次和非齐次线性微分方程无穷极亚纯解的增长性问题，使方程的零点和增长性得到了精确估计。 In this paper, we investigate the growth of linear order meromorphic solution of higher order homogeneous and no-homogeneous linear differential equation, and we obtain some precise estimates for their zeros and hyper-orders.

On zeros of solutions of higher order homogeneous linear differential equations

In this article, we investigate the exponent of convergence of zeros of solutions for some higher-order homogeneous linear differential equation, and prove that if Ak−1 is the dominant coefficient, then every transcendental solution f(z) of equation f(k)+Ak-1 f(k-1)+⋯+A0 f=0satisfies λ(f) = ∞, where λ(f) denotes the exponent of convergence of zeros of the meromorphic function f(z).

First Order Linear Homogeneous Fuzzy Ordinary Differential Equation Based on Lagrange Multiplier Method

In this paper the First Order Linear Fuzzy Ordinary Differential Equations are described. Here coefficients and /or initial condition of said differential equation are taken as the Generalized Triangular Fuzzy Numbers (GTFNs).The solution procedure of this Fuzzy Differential Equation is developed by Lagrange Multiplier Method. An imprecise barometric pressure problem is described.

GENERAL SOLUTIONS OF SECOND ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS

GENERAL SOLUTIONS OF SECOND ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS

System of First Order Linear Homogeneous Differential Equations (FLHE) and Riccati Equation

We present a simple method of solving a system of two first order linear homogeneous differential equations with variable co-efficients and demonstrate a correspondence of the system of equations with non-linear Riccati equation. By using appropriate substitution, this non-linear equation is converted into second order linear homogeneous differential equation, the solution of which is used to find the solution of the system of equations. It is seen that different sets of the variable co-efficients of the system of equations correspond to second order linear homogeneous differential equations of different nature. We have illustrated the method with one example. We have shown that the method can be applied to obtain the approximate analytical solutions of the spin dependent coupled Altarelli–Parisi evolution equation in Quantum Chromo Dynamics under plausible conditions.

Entire functions of finite order as solutions to certain complex linear differential equations

When is an entire function of finite order a solution to a complex 2nd order homogeneous linear differential equation with polynomial coefficients? In this paper we will give two (equivalent) answers to this question. The starting point of both answers is the Hadamard product representation of a given entire function of finite order. While the first answer involves certain Stieltjes-like relations associated to the function, the second one requires the vanishing of all but finitely many suitable expressions constructed via the Gil’ sums of the zeros of the function. Applications of these results will also be given, most notably to the spectral theory of one-dimensional Schrödinger operators with polynomial potentials.

Research on Numerical Computation of Natural Frequency of Transverse Vibration for Simply Supported Beam of Non-Uniform

Recursive and inexplicit differential equation of the second order with variable coefficients is derived from the fourth order linear homogeneous differential equation with variable coefficients of transverse vibration of non-uniform beam, which is about deflection and bending moment according to boundary conditions and order reduction. By finite difference method, numerical computation and accuracy are studied for natural frequency of transverse vibration for simply supported beam of non-uniform. Theoretical analysis and orthogonal computation examples show that numerical computation algorithm is very simple, and accuracy of computation depends on variety rate of gradually changed cross section in vertical direction and numbers of computation step, which is independent of width and length of beam; numerical accuracy of computation is estimable for given length or numbers of computation step; and reasonable length or numbers of computation step is determinable for given accuracy demand.

On the hyper order of solutions of linear differential equations with entire coefficients

In this paper, we investigate higher order homogeneous linear differential equations with entire coefficients of finite order. We improve and extend the results due to the second author and Hamouda by introducing the concept of hyper-order. We also consider non homogeneous linear differential equations. .

Iterated order of solutions of linear differential equations with entire coefficients

In this paper, we investigate the iterated order of solutions of higher order homogeneous linear differential equations with entire coef- ficients. We improve and extend some results of Bela¨odi and Hamouda by using the concept of the iterated order. We also consider nonhomogeneous linear differential equations.

Uniqueness intervals and two–point boundary value problems

Abstract Consider a linear nth order differential equation with continuous coefficients and continuous forcing term. The maximal uniqueness interval for a classical 2-point boundary value problem will be calculated by an algorithm that uses an auxiliary linear system of differential equations, called a Mikusinski system. This system always has higher order than n. The algorithm leads to a graphical representation of the uniqueness profile and to a new method for solving 2-point boundary value problems. The ideas are applied to construct a graphic for the conjugate function associated with the nth order linear homogeneous differential equation. Details are given about how to solve classical 2-point boundary value problems, using auxiliary Mikusinski systems and Green’s function.

A symbolic algorithm for exact power series solutions of nth order linear homogeneous differential equations with polynomial coefficients near an ordinary point

We developed an algorithm in Kıymaz and Mirasyedioğlu [O. Kıymaz and Ş. Mirasyedioğlu, An algorithmic approach to exact power series solutions of second order linear homogeneous differential equations with polynomial coefficients, Appl. Math. Comp. 139 (1) (2003) 165–178] for computing exact power series solutions of second order linear homogeneous differential equations with polynomial coefficients, near a point x = x 0. In this paper we present a symbolic algorithm to compute the exact power series solutions of nth order linear homogeneous differential equations with polynomial coefficients, near an ordinary point.

Integrability of Planar Polynomial Differential Systems through Linear Differential Equations

In this work, we consider rational ordinary differential equations dy/dx = Q(x,y)/P(x,y), with Q(x,y) and P(x,y) coprime polynomials with real coefficients. We give a method to construct equations of this type for which a first integral can be expressed from two independent solutions of a second-order homogeneous linear differential equation. This first integral is, in general, given by a non Liouvillian function. We show that all the known families of quadratic systems with an irreducible invariant algebraic curve of arbitrarily high degree and without a rational first integral can be constructed by using this method. We also present a new example of this kind of families. We give an analogous method for constructing rational equations but by means of a linear differential equation of first order.

Growth of solutions of three order differential equations

In this paper, the precise estimation of the order and hyper-order of solutions of a class of three order homogeneous and non-homogeneous linear differential equations are obtained. The results of M. Ozawa (1980), G. Gundersen (1988) and J. K. Langley (1986) are improved.

ON FACTORIZATION OF SOLUTIONS TO SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS

If a meromorphic solution of second order homogeneous linear differential equation is factorizable, then the right factor of the factorization of the solution has order not more than the coefficient's. And some asympotic properties of solutions are studied.

An algorithmic approach to exact power series solutions of second order linear homogeneous differential equations with polynomial coefficients

In 1992, Koepf [J. Symb. Comp. 13 (1992) 581] introduced an algorithm for computing a Formal Power Series of a given function using generalized hypergeometric series and a recurrence equation of hypergeometric type. The main aim of this paper is to develop a new algorithm for computing exact power series solutions of second order linear differential equations with polynomial coefficients, near a point x=x0, if its recurrence equation is hypergeometric type. The algorithm, which has been implemented in MAPLE, is based on symbolic computation.