Solutions Of Linear Differential Equations Research Articles

Growth Analysis of Meromorphic Solutions of Linear Difference Equations with Entire or Meromorphic Coefficients of Finite φ-Order

Many researchers’ attentions have been attracted to various growth properties of meromorphic solution f (of finite φ-order) of the following higher order linear difference equation Anzfz+n+...+A1zfz+1+A0zfz=0, where Anz,…,A0z are entire or meromorphic coefficients (of finite φ-order) in the complex plane (φ:[0,∞)→(0,∞) is a non-decreasing unbounded function). In this paper, by introducing a constant b (depending on φ) defined by lim̲r→∞logrlogφ(r)=b<∞, and we show how nicely diverse known results for the meromorphic solution f of finite φ-order of the above difference equation can be modified.

On [p, q]-order of growth of solutions of linear differential equations in the unit disc

<abstract><p>The $ [p, q] $-order of growth of solutions of the following linear differential equations $ (**) $ is investigated,</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f^{'}+A_{0}(z)f = 0, (**) $\end{document} </tex-math></disp-formula></p> <p>where $ A_{i}(z) $ are analytic functions in the unit disc, $ i = 0, 1, ..., k-1 $. Some estimations of $ [p, q] $-order of growth of solutions of the equation $ (\ast*) $ are obtained when $ A_{j}(z) $ dominate the others coefficients near a point on the boundary of the unit disc, which is generalization of previous results from S. Hamouda.</p></abstract>

On transcendental directions of entire solutions of linear differential equations

<abstract><p>This paper is devoted to studying the transcendental directions of entire solutions of $ f^{(n)}+A_{n-1}f^{(n-1)}+...+A_0f = 0 $, where $ n(\geq 2) $ is an integer and $ A_i(z)(i = 0, 1, ..., n-1) $ are entire functions of finite lower order. With some additional conditions, the set of common transcendental directions of non-trivial solutions, their derivatives and their primitives must have a definite range of measure. Moreover, we obtain the lower bound of the measure of the set defined by the common transcendental directions of Jackson difference operator of non-trivial solutions.</p></abstract>

О коградиентности и контрградиентности линейных дифференциальных уравнений и систем

The Siegel-Shidlovskii method remains one of the basic methods in the theory of transcendental numbers. By using this method, it is possible to prove the transcendence and algebraic independence of the values of entire functions of a certain class (so-called E-functions). A necessary condition for applying this method is that all of the considered functions must constitute a solution of a system of linear differential equations and were algebraically independent over . The question about algebraic independence of the solutions of linear differential equations and systems of such equations is of great importance in differential algebra, analytical theory of differential equations, theory of special functions, and calculus (in the broad sense of the word). As is shown in papers by E. Kolchin, F. Beukers, W.D. Brownawell, and G. Heckman, this question boils down in many instances to verification of the cogredience and contragredience condition. Two systems of 1st order linear homogeneous differential equations with coefficients from are said to be cogredient (or, respectively, contragredient), if for arbitrary fundamental matrices and Ψ of these systems one of the equations Φ=gBΨC, Φ (ΨC)^*=gB, is fulfilled, where C∈GL(C), B∈GL(C(z)), g=g(z) is a function with the condition g^'/g∈C(z), and A^* is the matrix transposed to . The notions of cogredience and contragredience for linear homogeneous differential equations of arbitrary order are defined similarly. Another, more restricted definitions of cogredience and contragredience were in fact used in some papers of the author, devoted to generalized hypergeometric functions. According to these definitions, the function in the presented equalities is the product of a power function and an exponential function of some kind. The conditions for equivalence of these definitions are found.

Extraction of Some New Solutions of Linear Differential Equations by Dual Space

In this study, the solutions of second order linear homogeneous ordinary differential equations being variable and constant coefficients have been obtained by using power series method which is used frequently in dual space. The solutions of second order homogeneous ordinary differential equations being constant and variable coefficients in dual space by using power series have given and some applications have implemented.

Asymptotics of Solutions of Linear Differential Equations with Holomorphic Coefficients in the Neighborhood of an Infinitely Distant Point

This study is devoted to the description of the asymptotic expansions of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhood of an infinitely distant singular point. This is a classical problem of analytical theory of differential equations and an important particular case of the general Poincare problem on constructing the asymptotics of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhoods of irregular singular points. In this study we consider such equations for which the principal symbol of the differential operator has multiple roots. The asymptotics of a solution for the case of equations with simple roots of the principal symbol were constructed earlier.

Exponentially Dichotomous Difference Equations with Piecewise Constant Operator Coefficients

We establish necessary and sufficient conditions for the exponential dichotomy of the solutions of linear difference equations with piecewise constant operator coefficients.

Existence, uniqueness and continuous dependence of solutions to conformable stochastic differential equations

Abstract In this paper, we study conformable stochastic differential equations. Firstly, the Ito formula is established and used to discuss the explicit expression of solutions of linear differential equations. Secondly, the existence and uniqueness of solutions of nonlinear conformable stochastic differential equations are proved by the Picard iteration method, and the continuous dependence of solutions on initial values is proved by the Gronwall inequality, the exponential estimation of solutions is also given. Finally, some examples are given to illustrate the theoretically results and we compare the simulation results for the conformable stock model with different ρ.

Finite-Order Entire Solutions of a Class of Algebraic Differential Equations

We study the properties of finite-order entire solutions of a class of algebraic differential equations. It is shown that, under certain conditions, all such solutions are solutions of linear differential equations whose coefficients are polynomials in the independent complex variable.

On the growth analysis of meromorphic solutions of finite ϕ-order of linear difference equations

Abstract In this paper, we investigate some growth properties of meromorphic solutions of higher-order linear difference equation A n ⁢ ( z ) ⁢ f ⁢ ( z + n ) + … + A 1 ⁢ ( z ) ⁢ f ⁢ ( z + 1 ) + A 0 ⁢ ( z ) ⁢ f ⁢ ( z ) = 0 , A_{n}(z)f(z+n)+\dots+A_{1}(z)f(z+1)+A_{0}(z)f(z)=0, where A n ⁢ ( z ) , … , A 0 ⁢ ( z ) {A_{n}(z),\dots,A_{0}(z)} are meromorphic coefficients of finite φ-order in the complex plane where φ is a non-decreasing unbounded function. We extend some earlier results of Latreuch and Belaidi [Z. Latreuch and B. Belaïdi, Growth and oscillation of meromorphic solutions of linear difference equations, Mat. Vesnik 66 2014, 2, 213–222].

Stability of fractional-order systems with Prabhakar derivatives

Fractional derivatives of Prabhakar type are capturing an increasing interest since their ability to describe anomalous relaxation phenomena (in dielectrics and other fields) showing a simultaneous nonlocal and nonlinear behaviour. In this paper we study the asymptotic stability of systems of differential equations with the Prabhakar derivative, providing an exact characterization of the corresponding stability region. Asymptotic expansions (for small and large arguments) of the solution of linear differential equations of Prabhakar type and a numerical method for nonlinear systems are derived. Numerical experiments are hence presented to validate theoretical findings.

On the influence of integral perturbations to the asymptotic stability of solutions of a second-order linear differential equation

Sufficient conditions for the asymptotic stability of the solutions of a second-order linear integro-differential equation of the Volterra type are established in the case where the solutions of the corresponding second-order linear differential equation may have no property under study. Thus, the influence of integral perturbations on the asymptotic stability of solutions of linear differential equations of the second order is revealed. For this purpose, the method of auxiliary kernels is developed. An illustrative example is given.

On Representation Formulas for Solutions of Linear Differential Equations with Caputo Fractional Derivatives

Abstract In the paper, a linear differential equation with variable coefficients and a Caputo fractional derivative is considered. For this equation, a Cauchy problem is studied, when an initial condition is given at an intermediate point that does not necessarily coincide with the initial point of the fractional differential operator. A detailed analysis of basic properties of the fundamental solution matrix is carried out. In particular, the Hölder continuity of this matrix with respect to both variables is proved, and its dual definition is given. Based on this, two representation formulas for the solution of the Cauchy problem are proposed and justified.

On the influence of integral perturbations to the asymptotic stability of solutions of a second-order linear differential equation

Sufficient conditions for the asymptotic stability of the solutions of a second-order linear integro-differential equation of the Volterra type are established in the case where the solutions of the corresponding second-order linear differential equation may have no property under study. Thus, the influence of integral perturbations on the asymptotic stability of solutions of linear differential equations of the second order is revealed. For this purpose, the method of auxiliary kernels is developed. An illustrative example is given.

Експоненціально дихотомічні різницеві рівняння з кусково-сталими операторними коефіцієнтами

УДК 517.988.6 Отримано необхiднi i достатнi умови експоненцiальної дихотомiї розв’язкiв лiнiйних рiзницевих рiвнянь iз кусковосталими операторними коефiцiєнтами.

Numerical solution of linear differential equations by Walsh polynomials approach

AbstractIn 1975 C. F. Chen and C. H. Hsiao established a new procedure to solve initial value problems of systems of linear differential equations with constant coefficients by Walsh polynomials approach. However, they did not deal with the analysis of the proposed numerical solution. In a previous article we study this procedure in case of one equation with the techniques that the theory of dyadic harmonic analysis provides us. In this paper we extend these results through the introduction of a new procedure to solve initial value problems of differential equations with not necessarily constant coefficients.

On the Lyapunov Type Inequality

A.M. Lyapunov proved inequality, which enables us to estimate distance between two consecutive zeros a and b of the solution of linear differential equation of the second order x″(t) + q(t)x(t) = 0, where q(t) is a continuous for t ∈ [a, b] function. In the present paper we solve analogous problem for the linear differential equation x″(t) + p(t)x′(t) + q(t)x(t) = 0. The obtained inequality is applicable for estimations of the periods of periodic solutions of non-linear differential equations such as A. Lienard and B. Van der Pol equations.

Deficient Values of Solutions of Linear Differential Equations

Differential equations of the form f'' + A(z)f' + B(z)f = 0 (*) are considered, where A(z) and B(z) not equiv 0 are entire functions. The Lindelöf function is used to show that for any rho in (1/2, infty ), there exists an equation of the form (*) which possesses a solution f with a Nevanlinna deficient value at 0 satisfying rho =rho (f)ge rho (A)ge rho (B), where rho (h) denotes the order of an entire function h. It is known that such an example cannot exist when rho le 1/2. For smaller growth functions, a geometrical modification of an example of Anderson and Clunie is used to show that for any rho in (2, infty ), there exists an equation of the form (*) which possesses a solution f with a Valiron deficient value at 0 satisfying rho =rho _{log }(f)ge rho _{log }(A)ge rho _{log }(B), where rho _{log }(h) denotes the logarithmic order of an entire function h. This result is essentially sharp. In both proofs, the separation of the zeros of the indicated solution plays a key role. Observations on the deficient values of solutions of linear differential equations are also given, which include a discussion of Wittich’s theorem on Nevanlinna deficient values, a modified Wittich theorem for Valiron deficient values, consequences of Gol’dberg’s theorem, and examples to illustrate possibilities that can occur.

Numerical Solution of Linear Differential Equations with Nonlocal Nonlinear Conditions

The numerical solution of systems of linear ordinary differential equations with nonlocal nonlinear conditions depending on the values of the desired function at intermediate points is investigated. Conditions for the existence of a solution to the problem under consideration are given. For the numerical solution, an approach is proposed that reduces the problem to two auxiliary linear systems of differential equations with linear conditions and to a single nonlinear algebraic system with a dimension depending only on the number of given intermediate points. The proposed approach is illustrated by solving two problems. The auxiliary problems are solved analytically in one of them and numerically in the other.

Procedures for Searching Laurent and Regular Solutions of Linear Differential Equations with the Coefficients in the Form of Truncated Power Series

Procedures for Searching Laurent and Regular Solutions of Linear Differential Equations with the Coefficients in the Form of Truncated Power Series