Solutions Of Complex Differential Equations Research Articles

Investigation of solution behavior of Differential Equations by Sumudu methods of random complex Partial Differential Equations

In this study, solutions of random complex partial differential equations were found using the two-dimensional Sumudu transformation method(STM). The initial conditions of a deterministic equation or the non-homogeneous part of the equation are transformed into random variables to obtain a random complex partial differential equation. With the help of the properties of two-dimensional Sumudu and inverse Sumudu transformation, an approximate analytical solution of a complex partial differential equation with random constant coefficients was obtained by selecting a random variable with an initial condition of Normal and Gamma distribution. The probability characteristics of the resulting solutions, such as expected value and variance, were obtained and graphically shown with the help of the Maple package program.

On the Application of the Emad-Sara Transform Technique to the Solution of Complex Differential Equations

When attempting to express a differential equation of the first order, it is usual practice to use the equation represented by dy/dx= f(x,y). The above equation contains a function denoted by the notation embodied by f(x,y), defined on a portion of the xy-plane and dependent on two variables, x, and y, which are independent variables. The equation considered to be of the first order is the one that has just the first derivative, which is denoted by the notation dy/dx since there are no higher-order derivatives present in the equation. It is usual practice to make use of the differential equation when attempting to explain the connection that exists between a function and the derivatives of that function. Using this technique to identify functions inside a given domain in physics, chemistry, and many other fields of science is feasible. In order to do so, it is necessary to have previous knowledge about functions and the derivatives of those functions. In this article, the Emad-Sara transformation is presented as a method for identifying the general solution of complex differential equations of the first order that have constant coefficients. This transformation may be used to get the general solution of these equations. This approach, which is both practical and economical, may be used to find solutions to a wide variety of linear operator equations. These equations can range from simple to complex

Limit Directions of Julia Sets of Entire Functions and Complex Differential Equations

The limiting directions of Julia sets of infinite order entire functions are studied by combining the theory of complex dynamic system and the theory of complex differential equations, in which the lower bound of the measure of limiting direction of Julia set of entire solutions of complex differential equations is obtained.

Growth properties of solutions of complex differential equations with entire coefficients of finite (alpha,beta,gamma)-order

In this article, we investigate the complex higher order linear differential equations in which the coefficients are entire functions of (α, β, γ)-order and obtain some results which improve and generalize some previous results of Tu et al. [29] as well as Belaidi [1, 2, 3].

On Petrenko's deviations and the Julia limiting directions of solutions of complex differential equations

The Julia set J(f) of a transcendental meromorphic function f has chaotic behavior in the complex dynamics. The ray arg⁡z=θ∈[0,2π) is said to be a limiting direction of J(f) if there is an unbounded sequence {zn}⊆J(f) such that limn→∞⁡arg⁡zn=θ. In this paper, we mainly investigated the Julia limiting directions of entire solutions of complex differential equations, of which coefficients are associated with Petrenko's deviations. In fact, we proved the lower bounds of the sets of Julia limiting directions of these solutions have closed relations with Petrenko's deviations of the coefficients.

Growth of solutions of complex differential equations with entire coefficients having a multiply-connected Fatou component

In this study, we show that all non-trivial solutions of $$f''+A(z)f'+B(z)f=0$$ have infinite order, provided that the entire coefficient A(z) has certain restrictions and B(z) has multiply-connected Fatou component. We also extend these results to higher order linear differential equations.

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>

Solution of complex differential equations with variable coefficients by using reduced differential transform

In this article, solution of complex partial derivative equations with variable coefficients from the first and second order have been investigated. For this solution, an iteration relation was obtained using the reduced differential transform method. This method also was been applied for ordinary complex differential equations which examined in the literature. The solution which has been obtained has been seen compatible with the literature.

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.

Julia Limiting Directions of Entire Solutions of Complex Differential Equations

Julia Limiting Directions of Entire Solutions of 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>

Order and Hyper-order of Solutions of Second Order Linear Differential Equations

We show that all non-trivial solutions of complex differential equation $$f''+ A(z)f'+B(z)f = 0$$ are of infinite order if coefficients A(z) and B(z) are of special type and establish a relation between the hyper-order of these solutions and the orders of coefficients A(z) and B(z). We have also extended these results to higher order complex differential equations.

Solution of complex differential equations by using reduced differential transform method

Solution of complex differential equations by using reduced differential transform method

Growth of Solutions of Complex Differential Equations with Solutions of Another Equation as Coefficients

We study the growth of solutions of $$f''+A(z)f'+B(z)f=0$$ , where A(z) and B(z) are non-trivial solutions of another second-order complex differential equations. Some conditions guaranteeing that every non-trivial solution of the equation is of infinite order are obtained, in which the notion of accumulation rays of the zero sequence of entire functions is used.

A unit disc study for (p, q) – order meromorphic solutions of complex differential equations

In this paper, we study the following non-homogeneous complex linear differential equation f(k) + Ak–1 f(k–1) + … + A0f = F, under the conditions: k ≥ 2, F and Af are meromorphic in Δ = {z ϵ ℂ :|z|<1} with finite iterated (p, q) – order.

Solutions of Complex Differential Equation Having Pre-given Zeros in the Unit Disc

Behavior of solutions of $f''+Af=0$ is discussed under the assumption that $A$ is analytic in $\mathbb{D}$ and $\sup_{z\in\mathbb{D}}(1-|z|^2)^2|A(z)|<\infty$, where $\mathbb{D}$ is the unit disc of the complex plane. As a main result it is shown that such differential equation may admit a non-trivial solution whose zero-sequence does not satisfy the Blaschke condition. This gives an answer to an open question in the literature. It is also proved that $\Lambda\subset\mathbb{D}$ is the zero-sequence of a non-trivial solution of $f''+Af=0$ where $|A(z)|^2(1-|z|^2)^3\, dm(z)$ is a Carleson measure if and only if $\Lambda$ is uniformly separated. As an application an old result, according to which there exists a non-normal function which is uniformly locally univalent, is improved.

Numerical solution for high‐order linear complex differential equations with variable coefficients

In this paper, we have obtained the numerical solutions of complex differential equations with variable coefficients by using the Legendre Polynomials and we have performed it on two test problems. Then, we applied with different technical of error analysis to the test problems. When we compared exact solutions and numerical solutions of tables and graphs, we realized that our method is reliable, practical, and functional.

Limiting directions of julia sets of entire solutions to complex differential equations

In this paper, we mainly investigate the dynamical properties of entire solutions of complex differential equations. With some conditions on coefficients, we prove that the set of common limiting directions of Julia sets of solutions, their derivatives and their primitives must have a definite range of measure.

Limiting direction and baker wandering domain of entire solutions of differential equations

In this paper, we mainly investigate entire solutions of complex differential equations with coefficients involving exponential functions, and obtain the dynamical properties of the solutions, their derivatives and primitives. With some conditions on coefficients, for the solutions, their derivatives and their primitives, we consider the common limiting directions of Julia set and the existence of Baker wandering domain.

A new trial equation method for finding exact chirped soliton solutions of the quintic derivative nonlinear Schrödinger equation with variable coefficients

In this work, we propose an efficient generalization of the trial equation method introduced recently by Liu [Appl. Math. Comput. 217 (2011) 5866] to construct exact chirped traveling wave solutions of complex differential equations with variable coefficients. The effectiveness of the proposed method has been tested by applying it successfully to the quintic derivative nonlinear Schrödinger equation with variable coefficients. As a result, a class of chirped soliton-like solutions including bright and kink solitons is derived for the first time. Compared with previous work of Liu in which unchirped solutions were given, we obtain exact chirped solutions which have nontrivial phase that varies as a function of the wave intensity. These localized structures characteristically exist due to a balance among the group-velocity dispersion, self-steepening and competing cubic-quintic nonlinearity. Parametric conditions for the existence of envelope solutions with nonlinear chirp are also presented. It is shown that the chirping can be effectively controlled through the variable parameters of group-velocity dispersion and self-steepening.