Complex Differential Equations Research Articles

Weierstrass Integrability of Complex Differential Equations

We characterize the complex differential equations of the form $${{dy} \over {dx}} = {a_n}(x){y^n} + {a_{n - 1}}(x){y^{n - 1}} + \cdots + {a_1}(x)y + {a_0}(x),$$ where aj(x) are meromorphic functions in the variable x for j = 0,…,n that admit either a Weierstrass first integral or a Weierstrass inverse integrating factor.

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.

New Iterative Method of Solving Nonlinear Equations in Fluid Mechanics

This paper presents the results of applying a new iterative method to linear and nonlinear fractional partial differential equations in fluid mechanics. A numerical analysis was performed to find an exact solution of the fractional wave equation and fractional Burgers’ equation, as well as an approximate solution of fractional KdV equation and fractional Boussinesq equation. Fractional derivatives of the order α are described using Caputo's definition with <i>0</i> < α ≤ <i>1</i> or <i>1</i> < α ≤ <i>2</i>. A comparative analysis of the results obtained using a new iterative method with those obtained by the Adomian decomposition method showed the first method to be more efficient and simple, providing accurate results in fewer computational operations. Given its flexibility and ability to solve nonlinear equations, the iterative method can be used to solve more complex linear and nonlinear fractional partial differential equations.

Complex uncertain differential equations with application to time integral

In this paper, we propose complex uncertain differential equations (CUDEs) based on uncertainty theory. In order to describe the evolution of complex uncertain phenomenon related to belief degrees, we apply the complex Liu process to CUDEs. Firstly, we pose a concept of a linear CUDE and prove that homogeneous linear CUDE and general linear CUDE have solutions. Then, we prove existence and uniqueness theorem of a special CUDE. Further, we design a numerical algorithm to obtain inverse uncertainty distribution of the solution. Finally, as an application, we analyse the inverse uncertainty distributions of time integral of CUDEs and design numerical algorithms to obtain inverse uncertainty distributions of time integral.

Torsional load-bearing capacity of half-shell segments for prestressed concrete towers

This article deals with the torsional load-bearing behaviour of concrete half-shell structures. The analogy between the bending stressed tension members according to second order theory and torsional members with warping hindered cross-section leads to a significantly simplified calculation of the additionally occurring warping stresses. This eliminates the need for the laborious solution of complex differential equations of the classical torsional warping theory to determine the load-bearing behaviour of warped cross-sections. The load redistribution approach projects the global torsion load of the segment tower onto the individual half-shell. The combination of the two approaches enables the analytical calculation of the torsional load-bearing behaviour of the concrete half-shell construction as a function of the global torsional moment and the system variables of the half-shell.The objective is to investigate and describe the torsional load-bearing behaviour of segmented half-shell structures in an engineering model. Compared to closed circular ring segments, the vertical joints have a significant influence, especially on the torsional rigidity. In order to determine the horizontal joint bearing capacity of these innovative constructions, it is essential to understand the load transfer of the torsional load. The presented approach allows to separate the individual effects of the load transfer in half-shell constructions from each other and thus to identify the factors influencing the torsional load-bearing behaviour.

Meromorphic solutions of one certain type of non-linear complex differential equation

In this paper, we are mainly concerned with one certain type of non-linear complex differential equation where are non-zero constants and is a differential polynomial in f of degree d at most n−1. For this kind of equation, if it admits a meromorphic solution such that , then firstly we give a positive answer to Li's question with a weaker restriction on and than Liu's when ; secondly, we show when .

Julia Limiting Directions of Entire Solutions of Complex Differential Equations

Julia Limiting Directions of Entire Solutions of Complex Differential Equations

Influence assessment of inlet parameters on thermal risk and productivity: Application to the epoxidation of vegetable oils

The influence of inlet parameters on the production and thermal risk of complex chemical systems can be cumbersome to evaluate. To determine the optimum safe operating conditions, one needs to solve complex differential equations derived from energy and material balances. This robust approach cannot be made on-site, and it is essential to propose simplest tools to evaluate rapidly the performance and safety of some operating conditions. This is the aim of this paper that establishes explicit relationships between the production and thermal risk parameters, and the inlet parameters. In addition, it also proposes a Pareto chart that can be used to make the tradeoff between safety and performance. Such relationships and chart were developed for the production of epoxidized cottonseed oil under isoperibolic and semi-batch mode. The kinetic model developed by Zheng et (Zheng et al., 2016). was used. First, a numerical approach, i.e., least square method, was used to find explicit relationships between thermal risk parameters, production parameters and six inlet parameters. The use of such an approach allows a better understanding of this process. Second, safety and performance indicators are proposed and discussed to evaluate the operating conditions thanks to a simple and intuitive schema. Besides, this approach can be used to find the optimum conditions more rapidly.

Trajectory tracking control for mobile robots considering position of mass center

AbstractIn this study, the control problem of trajectory tracking of mobile robots considering the difference between the center of mass and the geometric center is presented. In vehicles, sometimes the mass and geometric centers are not the same, which may be due to a load. In this case, the tracking error model is obtained for the first time. Then, this model is linearized around the reference trajectory. The control law consists of feedforward and feedback actions. The feedforward part is calculated from the reference trajectory, which leads to solving a complex differential equation. On the other hand, based on the linearized tracking error model, a model predictive controller is utilized to compute the feedback control part. Finally, the simulation results illustrate the modeling accuracy and efficiency of the proposed method.

Optimal system, invariance analysis of fourth-Order nonlinear ablowitz-Kaup-Newell-Segur water wave dynamical equation using lie symmetry approach

Nonlinear science is very common and important natural phenomenon in our surroundings. It is quite possible to find a large number of phenomena, which become a cause of the formulation of a non-linear partial differential equation. Due to the presence of non-linear partial differential equations in each branch of science, these equations have become a useful tool to deal with complex natural phenomena. It is interesting to investigate any complex non-linear partial differential equations for different exact solutions and examine the behavior of the solutions. Many effective approaches are developed to obtain the explicit exact solutions of the NLPDEs. Lie symmetry analysis is also one of the significant approaches to investigate the NLPDEs. Based on the Lie group analysis, we investigate a very famous and important equation, which is named as fourth-order Ablowitz-Kaup-Newell-Segur water wave dynamical equation. The symmetry groups, Commutator Tables and Adjoint of infinitesimals are constructed for this equation. Further, using the adjoint table, the optimal system is obtained. According to the optimal system, we tried to find the possible exact solution using symmetry reduction and presented a brief study of the properties of different solutions. We found some new exact solutions described with graphical representation showing solution wave structure, contour plot and wave propagation of the solution profile. The results are often helpful for studying the interaction of waves in many new localized structures and high-dimensional models.

Real-Time Path Planning With Virtual Magnetic Fields

Humans and animals have learned or evolved to use magnetic fields for navigation. Knowing how to model and estimate these fields can be used for motion planning. However, computing the propagation of electromagnetic fields in a given environment requires solving complex differential equations with advanced numerical methods, and therefore it is not suitable for real-time decision making. In this latter, we present a real-time approximator for Maxwell's equations based on deep neural networks that predicts the distribution of a virtual magnetic field. We show how our approximator can be used to perform autonomous 2D navigation tasks, outperforming state-of-the-art navigation algorithms, ensuring completeness, and providing a near-optimal path up to 200 times per second without any post processing stage. We demonstrate the effectiveness of our method with physics-based simulations of an unmanned aerial vehicle, an autonomous car, as well as real-world experiments using a small off-road autonomous racing vehicle. Furthermore, we show how the approach can be applied to multi-robot systems, video game technology, and can be extended to 3D problems.

Growth of solutions of non-homogeneous linear differential equations and its applications

In this paper, we investigate the growth properties of solutions of the non-homogeneous linear complex differential equation $L\left(f\right)=b\left(z\right)f+c\left(z\right)$, where $L\left(f\right)$ is a linear differential polynomial and $b\left( z\right) $, $c\left( z\right) $ are entire functions and give some of its applications on sharing value problems.

The twin properties of rogue waves and homoclinic solutions for some nonlinear wave equations

Abstract The parameter limit method on the basis of Hirota’s bilinear method is proposed to construct the rogue wave solutions for nonlinear partial differential equations (NLPDEs). Some real and complex differential equations are used as concrete examples to illustrate the effectiveness and correctness of the described method. The rogue waves and homoclinic solutions of different structures are obtained and simulated by three-dimensional graphics, respectively. More importantly, we find that rogue wave solutions and homoclinic solutions appear in pairs. That is to say, for some NLPDEs, if there is a homoclinic solution, then there must be a rogue wave solution. The twin phenomenon of rogue wave solutions and homoclinic solutions of a class of NLPDEs is discussed.

Nonlinear Pantograph-Type Diffusion PDEs: Exact Solutions and the Principle of Analogy

We study nonlinear pantograph-type reaction–diffusion PDEs, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments of the form w=u(px,t), w=u(x,qt), and w=u(px,qt), where p and q are the free scaling parameters (for equations with proportional delay we have 0<p<1, 0<q<1). A brief review of publications on pantograph-type ODEs and PDEs and their applications is given. Exact solutions of various types of such nonlinear partial functional differential equations are described for the first time. We present examples of nonlinear pantograph-type PDEs with proportional delay, which admit traveling-wave and self-similar solutions (note that PDEs with constant delay do not have self-similar solutions). Additive, multiplicative and functional separable solutions, as well as some other exact solutions are also obtained. Special attention is paid to nonlinear pantograph-type PDEs of a rather general form, which contain one or two arbitrary functions. In total, more than forty nonlinear pantograph-type reaction–diffusion PDEs with dilated or contracted arguments, admitting exact solutions, have been considered. Multi-pantograph nonlinear PDEs are also discussed. The principle of analogy is formulated, which makes it possible to efficiently construct exact solutions of nonlinear pantograph-type PDEs. A number of exact solutions of more complex nonlinear functional differential equations with varying delay, which arbitrarily depends on time or spatial coordinate, are also described. The presented equations and their exact solutions can be used to formulate test problems designed to evaluate the accuracy of numerical and approximate analytical methods for solving the corresponding nonlinear initial-boundary value problems for PDEs with varying delay. The principle of analogy allows finding solutions to other nonlinear pantograph-type PDEs (including nonlinear wave-type PDEs and higher-order equations).

Nehari’s univalence criteria, pre-Schwarzian derivative and applications

In this paper we study sharp estimates of pre-Schwarzian derivatives of functions belonging to the Nehari-type classes by using techniques from differential equations. In the sequel, we also see that a solution of a complex differential equation has a special form in terms of ratio of hypergeometric functions resulting to an integral representation. Finally, we attempt to study those univalent functions in the unit disk for which the image domain is an unbounded John domain.

On zeros and growth of solutions of complex difference equations

Let f be an entire function of finite order, let ngeq 1, mgeq 1, L(z,f)not equiv 0 be a linear difference polynomial of f with small meromorphic coefficients, and P_{d}(z,f)not equiv 0 be a difference polynomial in f of degree dleq n-1 with small meromorphic coefficients. We consider the growth and zeros of f^{n}(z)L^{m}(z,f)+P_{d}(z,f). And some counterexamples are given to show that Theorem 3.1 proved by I. Laine (J. Math. Anal. Appl. 469:808–826, 2019) is not valid. In addition, we study meromorphic solutions to the difference equation of type f^{n}(z)+P_{d}(z,f)=p_{1}e^{alpha _{1}z}+p_{2}e^{alpha _{2}z}, where ngeq 2, P_{d}(z,f)not equiv 0 is a difference polynomial in f of degree dleq n-2 with small mromorphic coefficients, p_{i}, alpha _{i} (i=1,2) are nonzero constants such that alpha _{1}neq alpha _{2}. Our results are improvements and complements of Laine 2019, Latreuch 2017, Liu and Mao 2018.

Application of müntz-legendre polynomials for solving complex differential equations

Application of müntz-legendre polynomials for solving complex differential equations

On properties of solutions of complex differential equations in the unit disc

<abstract> The properties of solutions of the following differential equation <p class="disp_formula">$ f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f = F(z) $ are studied, where $ A_{j}(z) $ and $ F(z) $ are analytic in the unit disc $ \mathbb{D} = \{z:|z| &lt; 1\} $, $ j = 0, 1, \ldots, k-1 $. First, the growth of solutions of the equation is estimated. Second, some coefficient's conditions such that the solution of the equation belong to Hardy type spaces are showed. Finally, some related question are studied in this paper. </abstract>

Binomial series and complex difference equations

In this study, we consider the properties of binomial series ∑n=0∞anzn_, where zn_=z(z−1)⋯(z−n+1), and the convergence of binomial series in the complex domain. We discuss the order of growth for the entire and meromorphic solutions of some difference equations represented by binomial series. Some examples are provided. As an application, we construct a difference Riccati equation that possesses a transcendental meromorphic solution of order 1/2.

Разработка «Умной теплицы» на основе модели «растение–среда–ситуация–управление»

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.