Solutions Of Nonlinear Differential Equations Research Articles

Дослідження динаміки завантаження вібраційних машин з вертикальним збуренням

The purpose of research. The main purpose of the work is to develop mathematical software for studying the dynamics of vibration machines of volumetric processing, and calculate the influence of various factors on the efficiency of the process based on applied systems of automated mathematical calculations, including MathCad and MatLab. Method. The research was carried out on the basis of a class of vibrating machines with unbalanced drive type and spring suspension. A mathematical model of loading vibrating processing machines with vertical perturbation is constructed, which is represented by a layering of flat beams that perform vertical oscillations. Methods of nonlinear mechanics were used to build a mathematical model. Results. The dynamic processes at vibration compaction of loading and separation of its fractions for the purpose of increase of intensity of the given technological processes are investigated. The dependences for determining the influence of physical and mechanical properties of loading components on the process dynamics are obtained. Scientific novelty. New approaches for construction of mathematical models of research of processes in machines of vibration processing, in particular vibration consolidation and separation are considered. In the course of research, differential equations were used to determine the change in the amplitude and frequency of loading from the influence of external and internal parameters of the vibration processing process. Practical significance. The obtained mathematical model makes it possible to investigate the dynamics of loading of the vibrating machine, to determine the amplitude and frequency of oscillations of loading of the vibrating machine depending on its properties and the properties of the vibrating machine. The obtained solutions of nonlinear differential equations to describe the motion of loading in vibrating machines allow to automatically determine the amplitudes, frequencies and trajectories of different components of loading depending on the parameters of the system "vibrating machine - loading", by algorithmizing them in applied systems of automated mathematical calculations.

Spherical Symmetric Solitons of Interacting Spinor, Scalar and Gravitational Fields in General Relativity

The concept of soliton as regular localized stable solutions of nonlinear differential equations is being widely utilized in pure science for various aims. In present analysis, the soliton concept is used as a model in order to describe the configurations of elementary particles in general relativity. To this end, our study deals with the spherical symmetric solitons of interacting Spinor, Scalar and Gravitational Fields in General Relativity. Thus, exact spherical symmetric general solutions to the interaction of spinor, scalar and gravitational field equations have been obtained. The Einstein equations have been transformed into a Liouville equation type and solved. Let us emphasize that these solutions are regular with localized energy density and finite total energy. In addition, the total charge and spin are limited. Moreover, the obtained solutions are soliton-like solutions. These solutions can be used in order to describe the configurations of elementary particles.

Integrability via Functional Expansion for the KMN Model

This paper considers issues such as integrability and how to get specific classes of solutions for nonlinear differential equations. The nonlinear Kundu–Mukherjee–Naskar (KMN) equation is chosen as a model, and its traveling wave solutions are investigated by using a direct solving method. It is a quite recent proposed approach called the functional expansion and it is based on the use of auxiliary equations. The main objectives are to provide arguments that the functional expansion offers more general solutions, and to point out how these solutions depend on the choice of the auxiliary equation. To see that, two different equations are considered, one first order and one second order differential equations. A large variety of KMN solutions are generated, part of them listed for the first time. Comments and remarks on the dependence of these solutions on the solving method and on form of the auxiliary equation, are included.

Thermo diffusion aspects in Jeffrey nanofluid over periodically moving surface with time dependent thermal conductivity

Double diffusion flow of Jeffrey fluid in presence of nanoparticles is studied theoretically under time dependent thermal conductivity. The considered nanoparticles are evaporated over convectively heated surface which moves periodically in its own plane. The appropriate dimensionless variables are employed to obtain the dimensionless forms of governing equations. We computed the analytical solution of non-linear differential equations by utilizing homotopy analysis method. The present investigation reveals the features of various emerging parameters like Deborah number, combined parameter, oscillation frequency to stretching rate ratio, Prandtl number, Lewis number, thermophoresis parameter, Brownian motion parameter, nano Lewis number, modified Dufour parameter, and Dufour solutal Lewis number. A useful enhancement in movement of nanoparticles is observed by utilizing the combined magnetic and porosity effects. Unlike traditional studies, present analysis is confined with the unsteady transportation phenomenon from periodically moving surfaces. Such computation may be attributable in flow results from tensional vibrations due to stretching and elastic surfaces. The simulation presented here can be attractable significance in the bioengineered nanoparticles manufacturing. It is observed that the heat transportation of nanoparticles may efficiently enhance through the utilization of variable thermal conductivity. The solutal concentration decreases with increasing Deborah number and Lewis number. It is further noted that the nano Lewis number causes reduction of nanoparticles concentration.

Construction of exact traveling wave solutions of the Bogoyavlenskii equation by [formula omitted]-expansion and [formula omitted]-expansion techniques

In this article, we construct exact solutions of the Bogoyavlenskii equation using (1/G′)-expansion and (G′/G,1/G)-expansion techniques. Both techniques have been successfully implemented to obtain exact solutions including hyperbolic, complex trigonometric, trigonometric and rational solutions of the Bogoyavlenskii equation. 3D, contour and 2D graphics are presented of the solutions obtained for different special values. Further, the advantages and disadvantages of both the techniques have been discussed in this study. The proposed techniques are reliable and applicable for attaining wave solutions of nonlinear differential equations. Also, these techniques can greatly minimize the size of computing work compared to other available techniques.

A new multistage technique for approximate analytical solution of nonlinear differential equations

The article introduces a new multistage technique for solving a polynomial system of nonlinear initial and boundary value problems of differential equations. The radius of convergence R of the series solution to the problem is derived a-priorly in terms of the parameters of the polynomial system. Then guided by the convergence-control parameter h<R, the domain of the problem is split into subintervals. By stepping out in a multistage manner, corresponding subproblems are defined which are then subsequently solved with conventional Parker-Sochacki method to get a piecewise continuous solution with very high accuracy. The method is applied to SIR epidemic model, stiff differential equation modelling combustion, Lorenz chaotic problem, and the Troesch's boundary value problem. The results obtained showed a remarkable accuracy when compared with Runge-Kutta Method of order 4. The article showcased the proposed method as a simple, yet accurate approximate analytical technique for nonlinear differential equations.

Singular solutions of nonlinear differential equations with p(t)-Laplacian

This paper deals with the singular solutions of the nonlinear differential equation (a(t)|x′|p(t)−2x′)′=b(t)|x|q(t)−2x involving p(t)-Laplacian. Sufficient conditions are given for the existence and the nonexistence of singular solutions of the first (second) kind. In addition, we study the existence of proper solutions and the coexistence of proper solutions and singular solutions of the second kind. Finally, some examples are given to illustrate our results.

Positive periodic solution to an indefinite singular equation

In this paper, we prove the existence of a positive periodic solution for nonlinear differential equation with an indefinite singularity by using Green’s function and fixed point theorem in cones. The results are applicable to weak as well as strong singularities. At last, we give one example to show the application of the theorem.

Study Boundary Problem with Integral condition for Fractional Differential Equations

In last many years ago there was a great interest in studying the existence of positive solutions for fractional differential equations. Many authors have considered the existence of positive solutions of non-linear differential equations of non-integer order with integral boundary value conditions using fixed point theorems. G.wang etal(2012)in vest gated the following fractional differential equation 〖^c〗D^α W(t)+f(t,W(t))=0,0 λ is a positive number (0 < λ < 2),〖^C〗D^αis the standard Caputo fractional derivative obtained his results by means of Guo-krosnosel'skii theorem in a cone also A.Cabada etat (2013) established the following non-linear fractional differential equation with integral boundary value conditions D^α W(t)+f(t,W(t))=0 ,00 ,λ≠α ,〗 D^αis Riemann –Liovuville standard fractional derivative and f is a continuous function the results was based on Guo-krasnosel'skii fixed point theorem in a cone . This paper we investigate the existence results of a positive solution for integral boundary value conditions of the following system of equations: 〖^c〗D^β h(t)+k(t,h(t))=0 ,t∈(0,1) h(0)=h^' (0)=h^''' (0)=0 ,h(1)=δ∫_0^1▒h(n)dn where 3< β≤4 ,δ is a positive number , δ≠3 ,〖^C〗D^β denotes Caputo standard derivative and k is a continuous function.Our work based on Banach's and Schauder's theorem.

An extended image encryption with Markov processes in solutions images dynamical system of non-linear differential equations

The process of hiding texts or images into images and signals has been remained contemporary topics over more than decades in Information Sciences. An attempt has been made to hide information or images into the solutions of non-linear differential equations of some especial type like Lorezn and Rossler’s equations, these equations have great importance in almost all practical applications from engineering, geography (weather forecasting) and even in stock markets due to its chaotic behavior of solutions which is dependent on initial conditions. The chaotic nature and the aperiodicity of such solutions have become the main source of attraction for encryptions. The 3D parametric (time) series of solutions of Lorenz and Rossler’s equations are produced phase vectors, and one of these phase vectors is chosen for the purpose of text or image embedding into its error (noise). The error (noise) is a part of chaotic solutions behaved aperiodicity in phase vectors. This attracted the embedding of manipulated text or image into it. Modified or reconstructed phase vector embedded with text or image when reproduced in the form of phase surfaces are quite similar to the original solutions surfaces. So Lorenz and Rossler solutions are used for embedding benchmark images like Lena, Fruits and Vegetables. Having embedded these images into series solutions, we found Lorenz solutions are more appropriate over the Rossler solutions. A thorough statistical analysis together with histograms of both original images and embedded images are reported. Experiments with RGB values of embedded images are also reported. In the present research, we have further extended this technique by using Markov processes to our target images and converted into encrypted images before embedding into solution phase vector of Lorenz and Rossler’s equations. Markov processes are made more confusing/diffusing the target images and has become difficult for intruders to attack for retrieval.

Existence of Two Solutions for Nonlinear Difference Equations Involving p(k)-Laplacian Operator and Boundary Value Conditions

In this paper, we deal with the existence of at least&nbsp; two solutions for an anisotropic discrete non-linear problem involving p(k)-Laplacian with Dirichlet boundary value conditions. The technical approach&nbsp;is based on&nbsp; a two critical points theorem for differentiable functionals. Two&nbsp;examples are inserted to illustrate the importance of main results../files/site1/files/62/6Abstract.pdf

Entanglement and symmetry resolution in two dimensional free quantum field theories

We present a thorough analysis of the entanglement entropies related to different symmetry sectors of free quantum field theories (QFT) with an internal U(1) symmetry. We provide explicit analytic computations for the charged moments of Dirac and complex scalar fields in two spacetime dimensions, both in the massive and massless cases, using two different approaches. The first one is based on the replica trick, the computation of the partition function on Riemann surfaces with the insertion of a flux α, and the introduction of properly modified twist fields, whose two-point function directly gives the scaling limit of the charged moments. With the second method, the diagonalisation in replica space maps the problem to the computation of a partition function on a cut plane, that can be written exactly in terms of the solutions of non-linear differential equations of the Painlevé V type. Within this approach, we also derive an asymptotic expansion for the short and long distance behaviour of the charged moments. Finally, the Fourier transform provides the desired symmetry resolved entropies: at the leading order, they satisfy entanglement equipartition and we identify the subleading terms that break it. Our analytical findings are tested against exact numerical calculations in lattice models.

Multiple Soliton Solutions for Nonlinear Differential Equations with a New Version of Extended F-Expansion Method

In this article, a new version of extended F-expansion method is suggested. A multiple Jacobi elliptic functions are presented in the solution function. We achieve analytical solutions of the (1 + 1)-dimensional dispersive long wave (DLW) equation and (2 + 1)-dimensional Kadomtsev–Petviashvili (KP) equation using the new version of extended F-expansion method. Then, the single and the combined non-degenerative Jacobi elliptic function solutions and their degenerate solutions to the above-mentioned class of nonlinear evolution equations are obtained.

On the Periodicity of General Class of Difference Equations

In this paper, we are interested in studying the periodic behavior of solutions of nonlinear difference equations. We used a new method to find the necessary and sufficient conditions for the existence of periodic solutions. Through examples, we compare the results of this method with the usual method.

Semi-Implicit Multistep Extrapolation ODE Solvers

Multistep methods for the numerical solution of ordinary differential equations are an important class of applied mathematical techniques. This paper is motivated by recently reported advances in semi-implicit numerical integration methods, multistep and extrapolation solvers. Here we propose a novel type of multistep extrapolation method for solving ODEs based on the semi-implicit basic method of order 2. Considering several chaotic systems and van der Pol nonlinear oscillator as examples, we implemented a performance analysis of the proposed technique in comparison with well-known multistep methods: Adams–Bashforth, Adams–Moulton and the backward differentiation formula. We explicitly show that the multistep semi-implicit methods can outperform the classical linear multistep methods, providing more precision in the solutions for nonlinear differential equations. The analysis of stability regions reveals that the proposed methods are more stable than explicit linear multistep methods. The possible applications of the developed ODE solver are the long-term simulations of chaotic systems and processes, solving moderately stiff differential equations and advanced modeling systems.

Exploring the Novel Continuum-Cancellation Leal-Method for the Approximate Solution of Nonlinear Differential Equations

This work presents the novel continuum-cancellation Leal-method (CCLM) for the approximation of nonlinear differential equations. CCLM obtains accurate approximate analytical solutions resorting to a process that involves the continuum cancellation (CC) of the residual error of multiple selected points; such CC process occurs during the successive derivatives of the differential equation resulting in an accuracy increase of the inner region of the CC-points and, thus, extends the domain of convergence and accuracy. Users of CCLM can propose their own trial functions to construct the approximation as long as they are continuous in the CC-points, that is, it can be polynomials, exponentials, and rational polynomials, among others. In addition, we show how the process to obtain the approximations is straightforward and simple to achieve and capable to generate compact, and easy, computable expressions. A convergence control is proposed with the aim to establish a solid scheme to obtain optimal CCLM approximations. Furthermore, we present the application of CCLM in several examples: Thomas–Fermi singular equation for the neutral atom, magnetohydrodynamic flow of blood in a porous channel singular boundary-valued problem, and a system of initial condition differential equations to model the dynamics of cocaine consumption in Spain. We present a computational convergence study for the proposed approximations resulting in a tendency of the RMS error to zero as the approximation order increases for all case studies. In addition, a computation time analysis (using Fortran) for the proposed approximations presents average times from 3.5 nanoseconds to 7 nanoseconds for all the case studies. Thence, CCML approximations can be used for intensive computing simulations.

Some convergence results of waveform relaxation for a class of second-order quasilinear parabolic equations

Waveform relaxation method at the PDE level can simplify the original differential system by several decoupling approaches, which is very useful for the iterative solution of nonlinear differential equations. In this paper, the convergence of waveform relaxation method at the PDE level for quasilinear equations is analysed for the first time. We mainly consider a class of general quasilinear parabolic equations both in divergence form and in non-divergence form. Some superlinear and linear convergence results under several convergence conditions for these quasilinear equations are given, and numerical experiments illustrated these superlinear or linear convergence rates, which claim that waveform relaxation method is available as an iterative framework for nonlinear problems.

МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ ВАКУУМНОЇ ДЕГАЗАЦІЇ СТАЛІ У КОВШІ З АРГОННОЮ ПРОДУВКОЮ

У статті подано математичну модель нестаціонарного процесу деазотації і дегідрогенізація розплаву сталі у вакууматорі камерного типу з аргонною продувкою. Дегазація сталі за допомогою вакууму — поширена серед металургійних підприємств технологія, яка дає можливість досягати надзвичайно низької концентрації водню та азоту у металевому розплаві, що необхідно для підвищення якості сталевих виробів. За відомою гіпотезою спочатку атоми газу знаходяться у розплаві у розчиненому стані. Бульбашки водню і азоту формуються з розчину на поверхні ковшової футерівки при умові достатньо низького феростатичного тиску металевого розплаву. Тиск, необхідний для появи бульбашки, визначається відповідно закону Сівертса. Значною мірою на дегазацію впливає і продувка аргоном, бульбашки якого збирають водень і азот на своєму шляху, спливаючи через розплав. Також важливим завданням є зменшення тривалості дегазації для зберігання температури розплаву на достатньо високому рівні. Проведення чисельних досліджень означеного вище процесу на математичній моделі зменшує витрати часових і фінансових ресурсів, тому побудова моделі є актуальним завданням. Опис плину розплаву і газів у ковші здійснюється на основі законів збереження маси та вектору кількості руху суцільного середовища, що виправдано через дрібний розмір бульбашок і їх велику кількість. З огляду на складність пошуку аналітичного розв’язку нелінійних диференціальних рівнянь у часткових похідних у тривимірній постановці, пропонується використовувати метод центральних різниць, який має достатню точність і широко використовується для подібних задач. Обчислювати математичну модель пропонується у комп’ютерній програмі на мові C#, яка має широкі можливості по програмуванню алгоритмів. Програмний додаток дозволить оцінити вплив інтенсивності аргонної продувки, а також глибини розплаву, на ступінь його дегазації, що може бути використано при впровадженні технологічних рекомендацій у виробництво сталі.

Lucas polynomial solution of nonlinear differential equations with variable delays

In this study, a novel matrix method based on Lucas series and collocation points has been used to solve nonlinear differential equations with variable delays. The application of the method converts the nonlinear equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown Lucas coefficients. The method is tested on three problems to show that it allows both analytical and approximate solutions.

A General Exact Closed-Form Solution for Nonlinear Differential Equation of Pendulum

In the present paper, the nonlinear differential equation of pendulum is investigated to find an exact closed form solution, satisfying governing equation as well as initial conditions. The new concepts used in the suggested method are introduced. Regarding the fact that the governing equation for any arbitrary system represents its inherent properties, it is shown that the nonlinear term causes that the system to have a variable identity. Hence, the original function is included as a variable in the solution to can take into account the variation of governing equation. To find the exact closed form solution, the variation of the nonlinear differential equation tends to zero, where in this case the system with a local linear differential equation has a definite identity with a definite local answer. It is shown that the general answer is an arbitrary curve on a surface, a newly developed concept known as super function, and different initial conditions give different curves as particular solutions. The comparison of the results with those of finite difference shows an exact agreement for any arbitrary amplitude and initial conditions.