Radial distributions of Julia sets of difference operators of entire solutions of complex differential equations

<abstract><p>In this paper, we mainly investigate the radial distribution of Julia sets of difference operators of entire solutions of complex differential equation $ F(z)f^{n}(z)+P(z, f) = 0 $, where $ F(z) $ is a transcendental entire function and $ P(z, f) $ is a differential polynomial in $ f $ and its derivatives. We obtain that the set of common limiting directions of Julia sets of non-trivial entire solutions, their shifts have a definite range of measure. Moreover, an estimate of lower bound of measure of the set of limiting directions of Jackson difference operators of non-trivial entire solutions is given.</p></abstract>

Further study on solutions of some non-linear homogeneous differential equations in connection to Brück conjecture

In this paper, using the theory of complex differential equations, we study the solution of some non-linear complex differential equations in connection to Brück conjecture which generalized some earlier results due to Pramanik, D. C. and Biswas, M., On solutions of some non-linear differential equations in connection to Bruck conjecture, Tamkang J. Math., 48 (2017), No. 4, 365–375; and Wang, H., Yang, L-Z. and Xu, H-Y., On some complex differential and difference equations concerning sharing function, Adv. Diff. Equ., 2014, 2014:274.

Development of a “smart greenhouse” based on the model “plant–environment–situation–control”

The greenhouse is a closed-type agroecological system in which energy processes are strictly determined by the technological process of growing plants, taking into account the influence of the environment. As you know, greenhouse models are divided into two types: white box models and black box models. The well-known model of the “Soil-Plant-Atmosphere” system belongs to the first type, built on the physical principles of thermo-, hydro- and gas dynamics. They consist of several complex differential equations that use numerous coefficients and parameters that are known in advance. Such models are cumbersome and require large computational resources and time-consuming. The proposed model of the system “Plant-Environment-Situation-Management” is a practical analogue of the well-known model “Soil-Plant-Atmosphere”. The main difference of this model is that it refers to a black box model, which is an approximation of the observed processes and allows you to describe processes based on experimental data. On the basis of the “Plant-Environment-Situation-Management” model, the software and hardware system “Smart Greenhouse” was developed, which is a human-machine system with a rational separation of the functions of preparation (computer) and decision-making (Man). It allows you to control and control the growth and development of the plant during the growing season, taking into account the influence of environmental conditions. The system is implemented and used in the greenhouse of the al-Farabi Kazakh National University.

Further study on the Brück conjecture and some non-linear complex differential equations

PurposeThe purpose of this current paper is to deal with the study of non-constant entire solutions of some non-linear complex differential equations in connection to Brück conjecture, by using the theory of complex differential equation. The results generalize the results due to Pramanik et al.Design/methodology/approach39B32, 30D35.FindingsIn the current paper, we mainly study the Brück conjecture and the various works that confirm this conjecture. In our study we find that the conjecture can be generalized for differential monomials under some additional conditions and it generalizes some works related to the conjecture. Also we can take the complex number a in the conjecture to be a small function. More precisely, we obtain a result which can be restate in the following way: Let f be a non-constant entire function such that σ2(f)<∞, σ2(f) is not a positive integer and δ(0, f)>0. Let M[f] be a differential monomial of f of degree γM and α(z), β(z)∈S(f) be such that max{σ(α), σ(β)} <σ(f). If M[f]+β and fγM−α share the value 0 CM, then M[f]+βfγM−α=c,where c≠0 is a constant.Originality/valueThis is an original work of the authors.

On some higher order equations admitting meromorphic solutions in a given domain

This paper relates to a recent trend in complex differential equations which studies solutions in a given domain. The classical settings in complex equations were widely studied for meromorphic solutions in the complex plane. For functions in the complex plane, we have a lot of results of general nature, in particular, the classical value distributions theory describing numbers of a-points. Many of these results do not work for functions in a given domain. A recent principle of derivatives permits us to study the numbers of Ahlfors simple islands for functions in a given domain; the islands play, to some extend, a role similar to that of the numbers of simple a-points. In this paper, we consider a large class of higher order differential equations admitting meromorphic solutions in a given domain. Applying the principle of derivatives, we get the upper bounds for the numbers of Ahlfors simple islands of similar solutions.

On the Exact Solutions of Two (3+1)-Dimensional Nonlinear Differential Equations

In this article, exact solutions of two (3+1)-dimensional nonlinear differential equations are derived by using the complex method. We change the (3+1)-dimensional B-type Kadomtsev-Petviashvili (BKP) equation and generalized shallow water (gSW) equation into the complex differential equations by applying traveling wave transform and show that meromorphic solutions of these complex differential equations belong to class W, and then, we get exact solutions of these two (3+1)-dimensional equations.