Fifth-order Differential Equation Research Articles

Oscillatory behavior of a fifth differential equation with unbounded neutral coefficients

"The authors study the oscillatory behavior of solutions to a class of fifth-order differential equations with unbounded neutral coefficients. The results are obtained by a comparison with first order delay differential equations whose oscillatory characters are known. Two examples illustrating the results are provided, one of which is applied to Euler type equations. Keywords: Oscillation, fifth-order, neutral differential equation."

Constructing RKM-Method for Solving Fractional Ordinary Differential Equations of Fifth-Order with Applications

This paper sheds the light on the vital role that fractional ordinary differential equations(FrODEs) play in the mathematical modeling and in real life, particularly in the physical conditions. Furthermore, if the problem is handled directly by using numerical method, it is a far more powerful and efficient numerical method in terms of computational time, number of function evaluations, and precision. In this paper, we concentrate on the derivation of the direct numerical methods for solving fifth-order FrODEs in one, two, and three stages. Additionally, it is important to note that the RKM-numerical methods with two- and three-stages for solving fifth-order ODEs are convenient, for solving class's fifth-order FrODEs. Numerical examples have been analyzed to demonstrate the efficacy of the new methods in comparison to the analytical method. Therefore, the numerical compression is carried out to confirm the efficiency and precision of the modified numerical methods. Significantly, the study demonstrates that the numerical outcomes of the proposed derived and modified numerical applied methods proved to be brilliant. Finally, based on the findings of the study, it could be said that the numerical outcomes of the test-problems using proposed and modified methods agree well with the analytical solutions. Hence, we can conclude that the proposed numerical methods that are derived or modified in the analytic study of this paper are quite efficient.

Periodic solutions for a class of fifth-order differential equations

PurposeThis study aims to provide sufficient conditions for the existence of periodic solutions of the fifth-order differential equation.Design/methodology/approachThe authors shall use the averaging theory, more precisely Theorem $6$.FindingsThe main results on the periodic solutions of the fifth-order differential equation (equation (1)) are given in the statement of Theorem 1 and 2.Originality/valueIn this article, the authors provide sufficient conditions for the existence of periodic solutions of the fifth-order differential equation.

SD-effect for circular plates of lamina cribrosa and optic nerve

Asymptotic equations for stresses are presented and used for numerical modeling and graphical representation of elastoplastic properties of circular SD-plates. A numerical solution of the plate bend is obtained after calculating the system of fifth-order differential equations. Euler difference method and software package COMSOL 5.6 are applied to solve the problem for biological tissue.

The Fourier transform of a function related to cylinder functions and asymptotic expansions with logarithmic terms

We consider a function that is obtained by the first kind Hankel function of order zero by removing its logarithmic singularity. This function, apart from a multiple of the zero-order Bessel function as an additive term, can be considered as the ‘entire part’ of the Hankel function. We constitute the Fourier transform of this function by considering a fifth-order ordinary differential equation which can be considered as a generalization of the differential equation for the zero-order cylinder functions. Since the asymptotics of our entire part of the Hankel function contains logarithmic terms, this new function can be used to derive asymptotic expansions of some other functions whose asymptotics contain logarithmic terms too. In the Fourier image it can be shown that these expansions are not merely asymptotic but actually convergent.

On a fifth-order nonselfadjoint boundary value problem

In this paper we aim to share a way to impose some nonselfadjoint boundary conditions for the solutions of a formally symmetric fifth-order differential equation. Constructing a dissipative operator related with the problem we obtain some informations on spectral properties of the problem. In particular, using coordinate-free approach we construct characteristic matrix-function related with the contraction which is obtained with the aid of the dissipative operator.

Определение параметров модели системы стабилизации ракеты в процессе полета

Динамические характеристики системы объект управления и регулятор в значительной степени зависят от выбора закона регулирования, определяемого номинальными значениями параметров математической модели процесса стабилизации и его приоритетным показателем. Из-за отклонения параметров ракеты и, соответственно, модели от номинальных величин проектанты устанавливают коэффициенты запаса, исходя из наиболее неблагоприятных условий, что отрицательно сказывается на общих показателях, в частности на относительном весе полезной нагрузки. Поэтому возникает потребность в разработке алгоритмов уточнения - идентификации в процессе полета параметров модели с помощью сигналов устройств измерения и возможности бортовых вычислительных машин. Это повысит эффективность использования методов выбора закона регулирования с точки зрения таких показателей, как запас устойчивости, точность стабилизации и мощность исполнительного устройства. Целью статьи является разработка методов уточнения параметров модели системы стабилизации движения ракеты в плоскости риска, опирающихся на использование текущих данных измерительных устройств части координат вектора состояния, и проверка эффективности уточнения с точки зрения указанных показателей. Принята линейная стационарная в окрестности определенной точки траектории модель системы стабилизации возмущенного движения ракеты с учетом инерции исполнительного устройства в виде обычных дифференциальных уравнений пятого порядка. Для приближения параметров модели к их фактическим значениям предложено два подхода: 1) в пространстве параметров модели находится минимум интеграла расстояния между точками траектории согласно сигналам измерительных устройств и траектории, полученной путем моделирования процесса компенсации возмущения; 2) фактические значения параметров являются результатом решения системы нелинейных уравнений, включающих данные измерительных устройств и соответствующие данные, полученные путем моделирования. На примере параметров ракеты космического назначения показано, что выбор закона регулирования, исходя из фактических коэффициентов модели, приводит к существенному уменьшению отклонений от заданного значения запаса устойчивости системы, погрешности стабилизации и мощности исполнительного устройства.

A modified Lorenz system: Definition and solution

Based on the approximations of the Lorenz system of differential equations from the papers [B. Zlatanovska and D. Dimovski, Systems of difference equations approximating the Lorentz system of differential equations, Contributions Sec. Math. Tech. Sci. Manu. XXXIII 1–2 (2012) 75–96, B. Zlatanovska and D. Dimovski, Systems of difference equations as a model for the Lorentz system, in Proc. 5th Int. Scientific Conf. FMNS, Vol. I (Blagoevgrad, Bulgaria, 2013), pp. 102–107, B. Zlatanovska, Approximation for the solutions of Lorenz system with systems of differential equations, Bull. Math. 41(1) (2017) 51–61], we define a Modified Lorenz system, that is a local approximation of the Lorenz system. It is a system of three differential equations, the first two are the same as the first two of the Lorenz system, and the third one is a homogeneous linear differential equation of fifth order with constant coefficients. The solution of this system is based on the results from [D. Dimitrovski and M. Mijatovic, A New Approach to the Theory of Ordinary Differential Equations (Numerus, Skopje, 1995), pp. 23–33].

Solvability of initial-boundary value problem of a multiple characteristic fifth-order operator-differential equation

In this study, we establish existence-uniqueness of a vector function in appropriate Sobolev-type space for a boundary value problem of a fifth-order operator differential equation. Proper conditions are obtained for the given problem to be well-posed. Much effort is devoted to develop the association between these conditions and the operator coefficients of the investigated equation. In this paper, accurate estimates of the norms of the intermediate derivatives operators are presented and used to determine the solvability conditions.

Linearization of Fifth-Order Ordinary Differential Equations by Generalized Sundman Transformations

In this article, the linearization problem of fifth-order ordinary differential equation is presented by using the generalized Sundman transformation. The necessary and sufficient conditions which allow the nonlinear fifth-order ordinary differential equation to be transformed to the simplest linear equation are found. There is only one case in the part of sufficient conditions which is surprisingly less than the number of cases in the same part for order 2, 3, and 4. Moreover, the derivations of the explicit forms for the linearizing transformation are exhibited. Examples for the main results are included.

Reduction of Fifth-Order Ordinary Differential Equations to Linearizable form by Contact Transformations

This paper is devoted to the study of the linearization problem of fifth-order ordinary differential equations by means of contact transformations. The necessary and sufficient conditions for linearization are obtained. The procedure for obtaining the linearizing transformations is provided in explicit form. Examples demonstrating the procedure of using the linearization theorems are presented.

Intelligent computing to solve fifth-order boundary value problem arising in induction motor models

In this study, biologically inspired intelligent computing approached based on artificial neural networks (ANN) models optimized with efficient local search methods like sequential quadratic programming (SQP), interior point technique (IPT) and active set technique (AST) is designed to solve the higher order nonlinear boundary value problems arise in studies of induction motor. The mathematical modeling of the problem is formulated in an unsupervised manner with ANNs by using transfer function based on log-sigmoid, and the learning of parameters of ANNs is carried out with SQP, IPT and ASTs. The solutions obtained by proposed methods are compared with the reference state-of-the-art numerical results. Simulation studies show that the proposed methods are useful and effective for solving higher order stiff problem with boundary conditions. The strong motivation of this research work is to find the reliable approximate solution of fifth-order differential equation problems which are validated through strong statistical analysis.

INSTABILITY OF CERTAIN NONLINEAR DIFFERENTIAL EQUATIONS OF FIFTH ORDER

This paper establishes certain sufficient conditions to guarantee the non-existence of periodic solutions for a class of nonlinear vector differential equations of fifth order. With this work, we extend and improve two related results in the literature from scalar cases to vectorial cases. An example is given to illustrate the theoretical analysis made in this paper.DOI : http://dx.doi.org/10.22342/jims.22.1.249.83-91

О совместном решении уравнения КДВ и дифференциального уравнения пятого порядка

О совместном решении уравнения КДВ и дифференциального уравнения пятого порядка

On Positive Periodic Solutions to Nonlinear Fifth-Order Differential Equations with Six Parameters

We study the existence and multiplicity of positive periodic solutions to the nonlinear differential equation:u5(t)+ku4(t)-βu3-ξu″(t)+αu'(t)+ωu(t)=λh(t)f(u), in 0≤t≤1, ui(0)=ui(1), i=0,1,2,3,4, wherek,α,ω,λ>0, β,ξ∈R,h∈C(R,R)is a 1-periodic function. The proof is based on the Krasnoselskii fixed point theorem.

AN ORDER SEVEN CONTINUOUS EXPLICIT METHOD FOR DIRECT SOLUTION OF GENERAL FIFTH ORDER ORDINARY DIFFERENTIAL EQUATIONS

This paper provides an explicit continuous method of order seven for direct integration of fifth-order general differential equations without reduction to systems of lower order equations. The explicit method requires no main predictor as a starting value for implementation. In the derivation of the method, collocation and interpolation procedures are adopted using power series as the basis function. The purpose of this work is to produce a high order explicit method that could be used as a predictor for the implementation of implicit methods in predictor-corrector mode in order to enhance the accuracy and efficiency of such methods. The method is symmetric, normalized, consistent and zero-stable. It is suitable for both non-stiff and mildly-stiff problems.

On the Fiber Preserving Transformations for the Fifth-Order Ordinary Differential Equations

This paper is devoted to the study of the linearization problem of fifth-order ordinary differential equations by means of fiber preserving transformations. The necessary and sufficient conditions for linearization are obtained. The procedure for obtaining the linearizing transformations is provided in explicit form. Examples demonstrating the procedure of using the linearization theorems are presented.

Efficient spectral-Petrov-Galerkin methods for third- and fifth-order differential equations using general parameters generalized Jacobi polynomials

Two new families of general parameters generalized Jacobi polynomials are introduced. Some efficient and accurate algorithms based on these families are developed and implemented for solving third- and fifth-order differential equations in one variable subject to homogeneous and nonhomogeneous boundary conditions using a dual Petrov-Galerkin method. The use of general parameters generalized Jacobi polynomials leads to simplified analysis, more precise error estimates and well conditioned algorithms. The method leads to linear systems with specially structured matrices that can be efficiently inverted. Numerical results indicating the high accuracy and effectiveness of the proposed algorithms are presented.

Efficient spectral-Petrov–Galerkin methods for the integrated forms of third- and fifth-order elliptic differential equations using general parameters generalized Jacobi polynomials

This article analyzes some algorithms for solving numerically the integrated forms of third- and fifth-order differential equations in one variable subject to homogenous and nonhomogeneous boundary conditions using a dual Petrov–Galerkin method. Two new families of general parameters generalized Jacobi polynomials are introduced and used for this purpose. Numerical results indicating the high accuracy and effectiveness of the proposed algorithms are presented.

Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations

A shifted Jacobi Galerkin method is introduced to get a direct solution technique for solving the third‐ and fifth‐order differential equations with constant coefficients subject to initial conditions. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. A quadrature Galerkin method is introduced for the numerical solution of these problems with variable coefficients. A new shifted Jacobi collocation method based on basis functions satisfying the initial conditions is presented for solving nonlinear initial value problems. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms. The algorithms are easy to implement and yield very accurate results.