Higher Order Linear Differential Equations Research Articles

Zeros of solutions of certain higher order linear differential equations

Zeros of solutions of certain higher order linear differential equations

Subnormal solutions of differential equations with periodic coefficients

In this article, we apply the concept of hyper-order to higher order linear differential equations with periodic coefficients, investigate the existence and the form of its subnormal solution, and estimate the growth of all other solutions, and answer the question raised by Gundersen and Steinbart for more general periodic differential equations.

On Subnormal Solutions of Periodic Differential Equations

We investigate the existence and the form of subnormal solutions of higher‐order linear periodic differential equations, and precisely estimate the growth of all solutions.

A boundary value problem on the half-line for higher-order nonlinear differential equations

This article is devoted to the study of the existence of solutions as well as the existence and uniqueness of solutions to a boundary-value problem on the half-line for higher-order nonlinear ordinary differential equations. An existence result is obtained by the use of the Schauder–Tikhonov theorem. Furthermore, an existence and uniqueness criterion is established using the Banach contraction principle. These two results are applied, in particular, to the specific class of higher-order nonlinear ordinary differential equations of Emden–Fowler type and to the special case of higher-order linear ordinary differential equations, respectively. Moreover, some (general or specific) examples demonstrating the applicability of our results are given.

ON THE COMPLEX OSCILLATION OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS

In this paper, we investigate the growth of solutions and the existence of subnormal solutions for a class of higher order linear differential equations. We obtain some results which improve and extend the results of Chen-Shon [2] and Gundersen-Steinbart [5].

GROWTH AND FIXED POINTS OF MEROMORPHIC SOLUTIONS OF HIGHER-ORDER LINEAR DIFFERENTIAL EQUATIONS

In this paper, we investigate the growth and fixed points of meromorphic solutions of higher order linear differential equations with meromorphic coefficients and their derivatives. Because of the restriction of differential equations, we obtain that the properties of fixed points of meromorphic solutions of higher order linear differential equations with meromorphic coefficients and their derivatives are more interesting than that of general transcendental meromorphic functions. Our results extend the previous results due to M. Frei, M. Ozawa, G. Gundersen, and J. K. Langley and Z. Chen and K. Shon.

Solving high‐order linear differential equations by a Legendre matrix method based on hybrid Legendre and Taylor polynomials

AbstractA numerical method for solving the high‐order linear differential equations with variable coefficients under the mixed conditions is presented. The method is based on the hybrid Legendre and Taylor polynomials. The solution is obtained in terms of Legendre polynomials. Comparison of the present solution is made with the existing solution and excellent agreement is noted. Illustrative examples are included to demonstrate the validity and applicability of the technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

Oscillation of complex high order linear differential equations with coefficients of finite iterated order

In this paper, we investigate the growth of solutions of complex high order linear differential equations with entire or meromorphic coefficients of finite iterated order and we obtain some results which improve and extend some previous results of Z. X. Chen and L. Kinnunen.

Oscillation of solutions of some higher order linear differential equations

In this paper, we deal with the order of growth and the hyper order of solutions of higher order linear differential equations f (k) + Bk−1f (k−1)

Growth of meromorphic solutions of higher-order linear differential equations

In this paper, we investigate the higher-order linear differential equations with meromor- phic coefficients. We improve and extend a result of M.S. Liu and C.L. Yuan, by using the estimates for the logarithmic derivative of a transcendental meromorphic function due to Gundersen, and the extended Winman-Valiron theory which proved by J. Wang and H.X. Yi. In addition, we also consider the nonhomogeneous linear differential equations.

Growth of solutions of certain higher order linear differential equations

The article is devoted to investigating the growth of solutions of certain higher order linear differential equations, extending some of the preceding results in the second-order case.

Solutions of higher order linear differential equations in an angle

In this paper, we discuss the higher order linear differential equation f ( k ) + A k − 1 f ( k − 1 ) + ⋯ + A 1 f ′ + A 0 f = 0 on an angle. We apply Nevanlinna sectorial theory to estimate the growth of solutions of the equation restricted in the angle and prove the existence of asymptotic value of the derivative of solutions of the equation in the angle. By using these results we can improve some known results in the plane.

On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane

We investigate the growth of the Nevanlinna characteristic of f(z+η) for a fixed η∈C in this paper. In particular, we obtain a precise asymptotic relation between T(r,f(z+η)) and T(r,f), which is only true for finite order meromorphic functions. We have also obtained the proximity function and pointwise estimates of f(z+η)/f(z) which is a discrete version of the classical logarithmic derivative estimates of f(z). We apply these results to give new growth estimates of meromorphic solutions to higher order linear difference equations. This also allows us to solve an old problem of Whittaker (Interpolatory Function Theory, Cambridge University Press, Cambridge, 1935) concerning a first order difference equation. We show by giving a number of examples that all of our results are best possible in certain senses. Finally, we give a direct proof of a result in Ablowitz, Halburd and Herbst (Nonlinearity 13:889–905, 2000) concerning integrable difference equations.

Growth order of Meromorphic Solutions of Higher-order Linear Differential Equations

Abstract. In this paper, we investigate higher-order linear differential equations withentire coefficients of iterated order. We improve and extend the result of L. Z. Yangby using the estimates for the logarithmic derivative of a transcendental meromorphicfunction due to Gundersen and the extended Wiman-Valiron theory by Wang and Yi. Wealso consider the nonhomogeneous linear differential equations. 1. Introduction and main resultsIn this paper, we shall assume that the reader is familiar with the fundamentalresults and the standard notation of the Nevanlinna value distribution theory ofmeromorphic functions (see [12], [8]). The term “meromorphic function” will meanmeromorphic in the whole complex plane C.The linear measure of a set E ⊂ [0,+∞) is defined as m(E) =R +∞0 χ E (t)dt.ThelogarithmicmeasureofasetE ⊂ [1,+∞)isdefinedbylm(E) =R +∞1 χ E (t)/tdt,where χ E (t) is the characteristic function of E. The upper and lower densities ofE aredensE = limsup r→+∞ m(E ∩[0,r])r, densE = liminf r→+∞ m(E ∩[0,r])r.For k ≥ 2, we consider a linear differential equation(1.1) A

On the dependent property of solutions for higher order periodic differential equations

This article investigates the property of linearly dependence of solutions f( z) and f( z + 2π i) for higher order linear differential equations with entire periodic coefficients.

On the growth of solutions of a class of higher order linear differential equations with coefficients having the same order

In this paper, the authors investigate the growth of solutions of a class of higher order linear differential equations f ( k ) + A k − 1 f ( k − 1 ) + ⋯ + A 0 f = 0 when most coefficients in the above equations have the same order with each other, and obtain some results which improve previous results due to K.H. Kwon [K.H. Kwon, Nonexistence of finite order solutions of certain second order linear differential equations, Kodai Math. J. 19 (1996) 378–387] and Z.-X. Chen [Z.-X. Chen, The growth of solutions of the differential equation f ″ + e − z f ′ + Q ( z ) f = 0 , Sci. China Ser. A 31 (2001) 775–784 (in Chinese); Z.-X. Chen, On the hyper order of solutions of higher order differential equations, Chinese Ann. Math. Ser. B 24 (2003) 501–508 (in Chinese); Z.-X. Chen, On the growth of solutions of a class of higher order differential equations, Acta Math. Sci. Ser. B 24 (2004) 52–60 (in Chinese); Z.-X. Chen, C.-C. Yang, Quantitative estimations on the zeros and growth of entire solutions of linear differential equations, Complex Var. 42 (2000) 119–133].

On the Complex Oscillation of Higher Order Linear Differential Equations with Meromorphic Coefficients

In this paper, we investigate the complex oscillation of higher order homogenous and non-homogeneous linear differential equations with meromorphic coefficients of iterated order, and obtain some results which improve and extend those given by Z. X. Chen, L. Kinnunen, etc.

The growth of meromorphic solutions for a class of higher-order linear differential equations†

In this article, the growth of meromorphic solutions for a class of higher-order linear differential equation with meromorphic coefficients is investigated by applying the method of value distribution, and some estimates for the hyper-order of their solutions are obtained. †Dedicated to Professor Wei Lin on the occasion of his 70th birthday.

A Taylor collocation method for the numerical solution of complex differential equations with mixed conditions in elliptic domains

An approximate method for solving higher-order linear complex differential equations in the elliptic domains is proposed. The approach is based on a Taylor collocation method, which consists of the matrix representation of expressions in the differential equation and the collocation points defined in the elliptic domain. Illustrative examples are included to demonstrate the validity and applicability of the technique, and performed on the computer using a program written in Maple9.

A matrix method for solving high-order linear difference equations with mixed argument using hybrid legendre and taylor polynomials

A numerical method for solving the higher order linear difference equations with variable coefficients and mixed argument under the mixed conditions is presented. The method is based on the hybrid Legendre and Taylor polynomials. The solution is obtained in terms of Legendre polynomials. IIIustrative examples are included to demonstrate the validity and applicability of the technique.