Higher Order 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.

Fast growth and fixed points of solutions of higher-order linear differential equations in the unit disc

<abstract><p>In this paper, we investigate the fast growing solutions of higher-order linear differential equations where $ A_0 $, the coefficient of $ f $, dominates other coefficients near a point on the boundary of the unit disc. We improve the previous results of solutions of the equations where the modulus of $ A_{0} $ is dominant near a point on the boundary of the unit disc, and obtain extensive version of iterated order of solutions of the equations where the characteristic function of $ A_{0} $ is dominant near the point. We also obtain a general result of the iterated exponent of convergence of the fixed points of the solutions of higher-order linear differential equations in the unit disc. This work is an extension and an improvement of recent results of Hamouda and Cao.</p></abstract>

On the Growth of Higher Order Complex Linear Differential Equations Solutions with Entire and Meromorphic Coefficients

We revisit the problem of studying the solutions growth order in complex higher order linear differential equations with entire and meromorphic coefficients of p,q-order, proving how it is related to the growth of the coefficient of the unknown function under adequate assumptions. Our study improves the previous results due to J. Liu - J. Tu - L. Z Shi, L.M. Li - T.B. Cao, and others.

On the best Ulam constant of a higher order linear difference equation

In a Banach space X the linear difference equation with constant coefficients xn+p=a1xn+p−1+…+apxn, is Ulam stable if and only if the roots rk, 1≤k≤p, of its characteristic equation do not belong to the unit circle. If |rk|>1, 1≤k≤p, we prove that the best Ulam constant of this equation is 1|V|∑s=1∞|V1r1s−V2r2s+…+(−1)p+1Vprps|, where V=V(r1,r2,…,rp) and Vk=V(r1,…,rk−1,rk+1,…,rp), 1≤k≤p, are Vandermonde determinants.

Solutions of higher order linear fuzzy differential equations with interactive fuzzy values

In this study, we consider higher order linear differential equations with additional conditions (initial and/or boundary) given by interactive fuzzy numbers. The concept of interactivity arises from the notion of a joint possibility distribution (J). The proposed method for solving fuzzy differential equations is based on an extension of the classical solution via sup-J extension, which is a generalization of Zadeh's extension principle. We prove that under certain conditions, the solution via Zadeh's extension principle is equal to the convex hull of the solutions produced by the sup-J extension. We also show that the solutions based on the Fréchet derivatives of fuzzy functions coincide with the solutions obtained via the sup-J extension. All of the results are illustrated based on a 3rd order fuzzy boundary value problem.

GROWTH OF \(\varphi\)–ORDER SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS WITH MEROMORPHIC COEFFICIENTS ON THE COMPLEX PLANE

In this paper, we study the growth of solutions of higher order linear differential equations with meromorphic coefficients of \(\varphi\)-order on the complex plane. By considering the concepts of \(\varphi\)-order and \(\varphi \)-type, we will extend and improve many previous results due to Chyzhykov–Semochko, Belaïdi, Cao–Xu–Chen, Kinnunen.

Spatial properties of solutions of higher order linear differential equations on unit circle

complex linear differential equation; function space

О дифференциальных уравнениях в банаховых алгебрах

We consider higher-order linear differential equations with constant coefficients in Banach algebras (this is a direct generalization of higher-order matrix differential equations). The presentation is based on higher algebra, differential equations and functional analysis. The results obtained can be used in the study of matrix equations, in the theory of small oscillations in physics, and in the theory of perturbations in quantum mechanics. The presentation is based on the original research of the authors.

Properties of Complex Oscillation of Solutions of a Class of Higher Order Linear Differential Equations

Let A(z) be an entire function with $$\mu (A) < \tfrac{1}{2}$$ such that the equation f(k) + A(z)f = 0, where k ≥ 2, has a solution f with λ(f) < μ(A), and suppose that A1 = A + h, where h ≢ 0 is an entire function with ρ(h) < μ(A). Then g(k) + A1(z)g = 0 does not have a solution g with λ(g) < ∞.

Some properties of meromorphic solutions of higher order linear difference equations

Some properties of meromorphic solutions of higher order linear difference equations

A comparison study for solving systems of high-order ordinary differential equations with constants coefficients by exponential Legendre collocation method

In this article we are interested to study the use of the Legendre exponential (EL) collocation method to solve systems of high order linear ordinary differential equations with constant coefficients. The method transforms the system of differential equations and the conditions given by matrix equations with constant coefficients a new system of equations that corresponds to the system of linear algebraic equations which can be solved . Numerical problems are given to illustrate the validity and applicability of the method. For obtaining the approximate solution Maple software is used.

Some Results on the Solutions of Higher-Order Linear Differential Equations

In this paper, we investigate the solutions of certain types of higher-order linear differential equations. By introducing a new concept of n-subnormal solution, we study the existence, growth, and numbers of solutions of this type, and we also estimate the growth of all other solutions.

On the growth of meromorphic solutions of some higher order linear differential equations

On the growth of meromorphic solutions of some higher order linear differential equations

Numerical solution of high order linear complex differential equations via complex operational matrix method

In this work, a numerical method based on a complex operational matrix is utilized to solve high order linear complex differential equations under mixed initial conditions. For this aim, we introduce orthonormal Bernstein polynomials (OBPs), and we obtain their complex operational matrix of differentiation. The main advantage of the proposed method is that by using this method complex differential equations reduce to a linear system of algebraic equations which can be solved by using an appropriate iterative method. To, some results concerning the error analysis associated with the present method are discussed. Finally, we give some numerical examples to reveal accuracy and efficiency of the proposed method. Also, the numerical results obtained by this method are compared with numerical results achieved from other existing methods.

On the growth of solutions of some higher order linear differential equations with meromorphic coefficients

On the growth of solutions of some higher order linear differential equations with meromorphic coefficients

Analytical solution proposal for fast numerical algorithm in special structured higher order differential equations

We suggest a practical method for obtaining the particular solution of non-homogeneous higher order linear differential equations with constant coefficients. The proposed method can be applied directly and simply to such problems. We revealed that is valid for the different type of problem by using sample solutions. This simple analytical solution that we have introduced will help to create a fast numerical algorithm for computers and thus simplify the numerical solutions of higher order physical problems.

General solution to a higher-order linear difference equation and existence of bounded solutions

We present a closed-form formula for the general solution to the difference equation xn+k−qnxn=fn,n∈N0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$x_{n+k}-q_{n}x_{n}=f_{n},\\quad n\\in \\mathbb {N}_{0}, $$\\end{document} where kin mathbb {N}, (q_{n})_{nin mathbb {N}_{0}}, (f_{n})_{nin mathbb {N}_{0}}subset mathbb {C}, in the case q_{n}=q, nin mathbb {N}_{0}, qin mathbb {C}setminus{0}. Using the formula, we show the existence of a unique bounded solution to the equation when |q|>1 and sup_{nin mathbb {N}_{0}}|f_{n}|<infty by finding a solution in closed form. By using the formula for the bounded solution we introduce an operator that, together with the contraction mapping principle, helps in showing the existence of a unique bounded solution to the equation in the case where the sequence (q_{n})_{nin mathbb {N}_{0}} is real and nonconstant, which shows that, in this case, there is an elegant method of proving the result in a unified way. We also obtain some interesting formulas.

Li(e)nearity

This article brings to light the fact that linearity is by itself a meaningful symmetry in the senses of Lie and Noether. First, the role played by that ‘linearity symmetry’ in the quadrature of linear second-order differential equations is revisited through the use of adapted variables and the identification of a conserved quantity as Lie invariant. Second, the celebrated Caldirola–Kanai Lagrangian—from which the differential equation is deducible—is shown to be naturally generated by a Jacobi last multiplier inherited from the linearity symmetry. Then, the latter is recognised to be also a Noether one. Finally, the study is extended to higher-order linear differential equations, derivable or not from an action principle. Incidentally, this work can serve as an introduction to the central question of continuous symmetries in physics and mathematics. It has the advantage of being approachable to undergraduate students since the linearity symmetry is by its very nature sufficiently simple to be treatable without any use of Lie generators.

On the hyper-order of transcendental meromorphic solutions of certain higher order linear differential equations

In this paper, we investigate the growth of meromorphic solutions of the linear differential equation \[f^{(k)}+h_{k-1}(z)e^{P_{k-1}(z)}f^{(k-1)}+\ldots +h_{0}(z)e^{P_{0}(z)}f=0,\] where \(k\geq 2\) is an integer, \(P_{j}(z)\) (\(j=0,1,\ldots ,k-1\)) are nonconstant polynomials and \(h_{j}(z)\) are meromorphic functions. Under some conditions, we determine the hyper-order of these solutions. We also consider nonhomogeneous linear differential equations.

Dissipative Switched Linear Differential Systems

The authors develop a dissipativity theory for switched systems whose dynamical modes are described by systems of higher order linear differential equations. They give necessary and sufficient conditions for dissipativity based on systems of linear matrix inequalities (LMIs), constructed from the coefficient matrices of the differential equations describing the modes. The relationship between dissipativity and stability is also discussed, and an application to the stabilization of power converters is provided.