Linear Differential Polynomials Research Articles

English

We study the uniqueness of entire functions which share a polynomial with their linear differential polynomials.

Differential Resultants

We review classical concepts of resultants of algebraic polynomials, and we adapt some of these concepts to objects in differential algebra, such as linear differential operators and differential polynomials.

Uniqueness of entire functions sharing a small function with linear differential polynomials

Uniqueness of entire functions sharing a small function with linear differential polynomials

The new types of wave solutions of the Burger's equation and the Benjamin–Bona–Mahony equation

In this article, we suggest the two variable (G′/G, 1/G)-expansion method for extracting further general closed form wave solutions of two important nonlinear evolution equations (NLEEs) that model one-dimensional internal waves in deep water and the long surface gravity waves of small amplitude propagating uni-directionally. The method can be regarded as an extension of the(G′/G)-expansion method. The ansatz of this extension method to obtain the solution is based on homogeneous balance between the highest order dispersion terms and nonlinearity which is similar to the (G′/G) method whereas the auxiliary linear ordinary differential equation (LODE) and polynomial solution differs. We applied this method to find explicit form solutions to the Burger's and Benjamin–Bona–Mahony (BBM) equations to examine the effectiveness of the method and tested through mathematical computational software Maple. Some new exact travelling wave solutions in more general form of these two nonlinear equations are derived by this extended method. The method introduced here appears to be easier and faster comparatively by means of symbolic computation system.

Brück Conjecture and Linear Differential Polynomials

In the paper we prove a uniqueness theorem for meromorphic functions that share a function of slower growth with linear differential polynomials. Our result is closely related to a conjecture of Bruck.

AN ENTIRE FUNCTION SHARING TWO VALUES WITH ITS LINEAR DIFFERENTIAL POLYNOMIAL

We consider the uniqueness of an entire function and a linear differential polynomial generated by it. One of our results improves a result of Li and Yang [‘Value sharing of an entire function and its derivatives’, J. Math. Soc. Japan51(4) (1999), 781–799].

Entire functions sharing a linear polynomial with linear differential polynomials

In the paper we study the uniqueness of entire functions sharing a linear polynomial with linear differential polynomials generated by them. The results of the paper improves the corresponding results of P. Li (Kodai Math J. 22: 446-457, 1999), Lahiri-Present author(G. K. Ghosh) (Analysis (Munich)31: 331-340,2011) and Lahiri-Mukherjee(Bull. Aust. Math. Soc. 85: 295-306, 2012).

Brück Conjecture for a linear differential polynomial

Brück Conjecture for a linear differential polynomial

Uniqueness of Entire Functions that Share an Entire Function of Smaller Order with One of Their Linear Differential Polynomials

Uniqueness of Entire Functions that Share an Entire Function of Smaller Order with One of Their Linear Differential Polynomials

Nonhomogeneous linear differential polynomials generated by solutions of complex differential equations in the unit disc

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English

The problem of uniqueness of an entire or a meromorphic function when it shares a value or a small function with its derivative became popular among the researchers after the work of Rubel and Yang (1977). Several authors extended the problem to higher order derivatives. Since a linear differential polynomial is a natural extension of a derivative, in the paper we study the uniqueness of a meromorphic function that shares one small function CM with a linear differential polynomial, and prove the following result: Let $f$ be a nonconstant meromorphic function and $L$ a nonconstant linear differential polynomial generated by $f$. Suppose that $a = a(z)$ ($\not \equiv 0, \infty $) is a small function of $f$. If $f-a$ and $L-a$ share $0$ CM and \[ (k+1)\overline N(r, \infty ; f)+ \overline N(r, 0; f')+ N_{k}(r, 0; f')< \lambda T(r, f')+ S(r, f') \] for some real constant $\lambda \in (0, 1)$, then $ f-a=(1+ {c}/{a})(L-a)$, where $c$ is a constant and $1+{c}/{a} \not \equiv 0$.

Meromorphic Functions Sharing Three Values with Their Linear Differential Polynomials in Some Angular Domains

We study the uniqueness question of transcendental meromorphic functions sharing three distinct finite values with their linear differential polynomials in some angular domains instead of the whole complex plane, and study the uniqueness question of transcendental entire functions sharing two distinct finite values with their linear differential polynomials in some angular domains instead of the whole complex plane. The results in this paper improve the corresponding results from Frank and Weissenborn (Complex Var 7:33–43, 1986), Frank and Schick (Results Math 22:679–684, 1992), Bernstein et al. (Forum Math 8:379–396, 1996) and improve Theorem 3 in Zheng (Can Math Bull 47:152–160, 2004).

Unicity of meromorphic functions and their linear differential polynomials

This paper studies the unicity of meromorphic functions that share an arbitrary nonzero small function with their general linear differential polynomials. The result derived here extends one by Brück in 1996 and others. MSC:30D35, 30D30.

A normality criterion for meromorphic functions having multiple zeros

We prove a normality criterion for a family of meromorphic functions having multiple zeros which involves sharing of a non-zero value by the product of functions and their linear differential polynomials.

A Non-zero Value Shared by an Entire Function and its Linear Differential Polynomials

Abstract.In this paper we study uniqueness of entire functions sharing a non-zero finite value with linear differential polynomials and address a result of W.Wang and P. Li.

Numerical analytical method of studying some linear functional differential equations

This paper presents the results of studying a scalar linear functional differential equation of a delay type ẋ(t) = a(t)x(t − 1) + b(t)x(t/q) + f(t), q > 1. Primary attention is given to the original problem with the initial point, when the initial condition is specified at the initial point, and the classical solution, whose substitution into the original equation transforms it into an identity, is sought. The method of polynomial quasi-solutions, based on representation of an unknown function x(t) as a polynomial of degree N, is applied as the method of investigation. Substitution of this function into the original equation yields a residual Δ(t) = O(tN), for which an accurate analytical representation is obtained. In this case, the polynomial quasi-solution is understood as an exact solution in the form of a polynomial of degree N, disturbed because of the residual of the original initial problem. Theorems of existence of polynomial quasi-solutions for the considered linear functional differential equation and exact polynomial solutions have been proved. Results of a numerical experiment are presented.

An entire function sharing one nonzero value with its linear differential polynomials

We study the uniqueness of an entire function when it shares a nonzero finite value with its first derivative and two linear differential polynomials.

Linear sparse differential resultant formulas

Let P be a system of n linear nonhomogeneous ordinary differential polynomials in a set U of n−1 differential indeterminates. Differential resultant formulas are presented to eliminate the differential indeterminates in U from P. These formulas are determinants of coefficient matrices of appropriate sets of derivatives of the differential polynomials in P, or in a linear perturbation Pε of P. In particular, the formula ∂FRes(P) is the determinant of a matrix M(P) having no zero columns if the system P is “super essential”. As an application, if the system P is sparse generic, such formulas can be used to compute the differential resultant ∂Res(P) introduced by Li et al. (2011) [19].

Non linear differential polynomials sharing fixed points with finite weights

Abstract We employ the notion of weighted sharing to investigate the uniqueness of meromorphic functions when two nonlinear differential polynomials share fixed points. The results of the paper improve and generalize the recent results due to Xu–Lu–Yi [10].

Meromorphic functions that share a small function with one of its linear differential polynomials

In this paper, we investigate meromorphic functions that share a small function with one of its linear differential polynomials and prove several theorems which generalize and improve the main results given by J. L. Zhang and L. Z. Yang. Some examples are provided to show that the results in this paper are sharp.