Homogeneous Ordinary Differential Equations Research Articles

Higher order contribution to ion-acoustic Korteweg–de Vries (KdV) soliton in unmagnetized quantum plasmas with arbitrary degeneracy of electrons

The propagation of nonlinear ion-acoustic Korteweg–de Vries (KdV) solitons and dressed solitons (with higher order contribution term) in unmagnetized quantum plasmas with arbitrary degeneracy of electrons is investigated using quantum hydrodynamic model. The reductive perturbation method is used to derive the first order nonlinear ion-acoustic KdV equation and also higher order contribution of inhomogeneous differential equation is obtained for the dressed ion-acoustic wave soliton in arbitrary degenerate electron plasmas. Moreover, the higher order inhomogeneous differential equation is solved (nonsecular solution is obtained) using the renormalization method. The conditions for the validity of the higher order correction are also discussed in detail for ion-acoustic solitons in unmagnetized arbitrary degenerate electrons plasma case. The effects of chosen values of electron temperature, equilibrium fugacity and corresponding electron density of a physical quantum plasma system and quantum diffraction parameter H (for two conditions i.e., H < 2 or H > 2) on the amplitude and width of the KdV and dressed ion-acoustic wave solitons for both electrostatic potential hump (compressive) or dip (rarefactive) structures are investigated. The numerical plots are also presented for illustration with numerical values of a physical partially degenerate electrons plasma system exist in the astrophysical or laboratory conditions.

Analytic solutions to nonlinear ODEs via spectral power series

Solutions to most nonlinear ordinary differential equations (ODEs) rely on numerical solvers, but this gives little insight into the nature of the trajectories and is relatively expense to compute. In this paper, we derive analytic solutions to a class of nonlinear, homogeneous ODEs with linear and quadratic terms on the right-hand side. We formulate a power series expansion of each state variable, whose function depends on the eigenvalues of the linearized system, and solve for the coefficients using some linear algebra and combinatorics. Various experiments exhibit quickly decaying coefficients, such that a good approximation to the true solution consists of just a few terms.

Solution homogeneous and non-homogeneous fractional ordinary differential equations by ZMA-transform

In this view, a modern vision to discuss homogeneous and non-homogeneous fractional ordinary differential equations (HFODEs)- (NHFODEs) with initial condition (IC) by employment ZMA-transform (ZMAT). Out of exhibit basic properties for theorems of ZMA-transform (ZMAT) and fractional calculus(FC). Moreover presented ZMA-transform (ZMAT) of Riemann-Liovill derivative the consequent application of ZMA-transform (ZMAT) of fractional derivatives and to understand the task more, some applications has been solved.

Atmospheric Ekman flows of the Northern Hemisphere with latitude and height in ellipsoidal coordinates

In this paper, we study the atmospheric Ekman flows of the Northern Hemisphere related to latitude and height in ellipsoidal coordinates. Firstly, by using a suitable transformation, we convert the atmospheric Ekman equation into a second order elliptic partial differential equation. Further, we prove it is equivalent to a second order homogeneous ordinary differential equation with variable coefficients. Finally, we construct the explicit solution and give the laws of motion of the atmospheric Ekman flows.

ELASTIC TRANSFORMATION AND ITS INVERSE TRANSFORMATION FOR SOLVING FIRST- AND THIRD-ORDER NONLINEAR VARIABLE COEFFICIENT ORDINARY DIFFERENTIAL EQUATIONS

Here elastic transformation and its inverse transformation are used to solve two kinds of first- and third-order nonlinear ordinary differential equations (ODEs) with variable coefficients. First, elastic transformations are used to upgrade and reduce the first- and third-order nonlinear ODEs to a class of second-order linear homogeneous ODEs. Then the general solutions of the first- and third-order ODEs are obtained separately by using the general solution of the obtained second-order linear homogeneous ODEs, elastic transformation, inverse transformation of elastic transformation and related operation rules. Second, the steps for solving these two kinds of ODEs are summarized. Finally, examples and parametric analysis are given to prove that it is effective, simple and feasible to solve two kinds of first- and third-order nonlinear ODEs by elastic transformation and its inverse transformation. Elastic transformation and its inverse transformation provide a new method for solving ODEs. The research in this paper expands the solvable classes of ODEs.

Modified Variational Iteration Method Approach for Fuzzy Initial Value Problem

In this paper, an application of&nbsp; modified variational iteration method (MVIM) for solving homogeneous and nonhomogeneous ordinary differential equations with fuzzy initial value conditions (FDEs) are examined. This modified form is about a procedure some changes in the general formula of the variational iteration method (VIM) . MVIM showed a qualitative and excellent improvement over the standard VIM. To illustrate how this modified form is works we picked some numerical examples and solved them. The efficacy and validity of MVIM have been confirmed through results obtained. &nbsp;

An analytical solution for vibration response of CNT/GPL/fibre/polymer hybrid composite micro/nanoplates

In the present article, a closed-form solution is carried out for the vibration response of CNT/GPL/fiber/polymer hybrid composite macro/micro/nanoplates resting on elastic support. Multi-layered fiber/polymer hybrid plates with either functionally graded carbon nanotube reinforced composite (FG-CNTRC) material or functionally graded graphene platelets reinforced composite (FG-GPLRC) material are two different types of plate considered here. The distribution of reinforcing nanocomposites can be uniform or functionally graded through the plate layers. The governing vibration equations are developed in the framework of the nonlocal elasticity theory of Eringen and using the first-order shear deformation plate theory. In the analytical solution procedure, the partial differential equations of motion are simplified to a series of homogeneous ordinary differential equations using a displacement field for Levy-type of the boundary conditions. The general solution of the obtained equations is expressed in terms of exponential functions. Using that the displacements should satisfy the remaining boundary conditions, the vibrational frequencies are derived. Finally, the influences of different parameters including geometry and material properties such as nonlocal parameter, number and angle of layers, volume fraction, and type of distribution of reinforcing nanocomposites are investigated and discussed. Natural frequencies for various boundary conditions are illustrated. The numerical results illustrate that the effects of the nonlocal parameter on the frequency decrease when one of the boundaries is free, and these effects increase when one of the boundaries is clamped for both FG-CNTRC and FG-GPLRC plates.

Self-excited and hidden attractors in a multistable jerk system

In this paper we investigate a jerk system which is modeled by a homogeneous third-order ordinary differential equation, with four parameters that control the dynamics. The proper choice of one of these parameters allows the system to display real or non-real equilibrium points. This implies that we can choose such a parameter so that the associated attractors are either self-excited or hidden. We consider the two situations, and investigate the dynamics of the two versions of this jerk system, in cross-sections of the three-dimensional parameter-space generated by the other three parameters. We show that both versions of the jerk system display multistability, with coexistence of periodic–periodic, chaotic–chaotic, and periodic–chaotic attractors in the phase-space, regardless of whether the attractors are self-excited or hidden. We also show that basins of attraction and attractors occupy smaller volumes in the case of the system with no equilibrium points.

Mittag–Leffler stability of numerical solutions to time fractional ODEs

The asymptotic stable region and long-time decay rate of solutions to linear homogeneous Caputo time fractional ordinary differential equations (F-ODEs) are known to be completely determined by the eigenvalues of the coefficient matrix. Very different from the exponential decay of solutions to classical ODEs, solutions of F-ODEs decay only polynomially, leading to the so-called Mittag-Leffler stability, which was already extended to semi-linear F-ODEs with small perturbations. This work is mainly devoted to the qualitative analysis of the long-time behavior of numerical solutions. By applying the singularity analysis of generating functions developed by Flajolet and Odlyzko (SIAM J. Disc. Math. 3(2), 216–240, 1990), we are able to prove that both $$\mathcal {L}$$ 1 scheme and strong A-stable fractional linear multistep methods (F-LMMs) can preserve the numerical Mittag-Leffler stability for linear homogeneous F-ODEs exactly as in the continuous case. Through an improved estimate of the discrete fractional resolvent operator, we show that strong A-stable F-LMMs are also Mittag-Leffler stable for semi-linear F-ODEs under small perturbations. For the numerical schemes based on $$\alpha$$ -difference approximation to Caputo derivative, we establish the Mittag-Leffler stability for semi-linear problems by making use of properties of the Poisson transformation. The new results and analyses provide not only the rigorous justifications and explanations of the Mittag-Leffler stability of numerical solutions with exact decay rate, but also establish some close connection between the continuous and discrete F-ODEs. Numerical experiments are presented for several typical time fractional evolutional equations, including time fractional sub-diffusion equations and semi-linear F-ODEs. All the numerical results exhibit the typical long-time polynomial decay rate, which is fully consistent with our theoretical predictions.

The Existence of Weak Solutions for the Vorticity Equation Related to the Stratosphere in a Rotating Spherical Coordinate System

In this paper, based on the Euler equation and mass conservation equation in spherical coordinates, the ratio of the stratospheric average width to the planetary radius and the ratio of the vertical velocity to the horizontal velocity are selected as parameters under appropriate boundary conditions. We establish the approximate system using these two small parameters. In addition, we consider the time dependence of the system and establish the governing equations describing the atmospheric flow. By introducing a flow function to code the system, a nonlinear vorticity equation describing the planetary flow in the stratosphere is obtained. The governing equations describing the atmospheric flow are transformed into a second-order homogeneous linear ordinary differential equation and a Legendre’s differential equation by applying the method of separating variables based on the concepts of spherical harmonic functions and weak solutions. The Gronwall inequality and the Cauchy–Schwartz inequality are applied to priori estimates for the vorticity equation describing the stratospheric planetary flow under the appropriate initial and boundary conditions. The existence and non-uniqueness of weak solutions to the vorticity equation are obtained by using the functional analysis technique.

Generalized Hukuhara Derivatives of Solving of Second Order Linear Homogeneous ODEs by Generalized Triangular Fuzzy Number Using Fuzzy Laplace Transform

Second order linear homogeneous ordinary differential equations are solved using fuzzy Laplace transform method with generalized Hukuhara differentiability concept. The results obtained is in this study is in the form of generalized triangular fuzzy number. Existence and uniqueness of solution are also obtained. However, based on the cases presented, the results have shown that relationship existed between the FLT of second order and its kth derivative for k ≥1.

New integral “Kuffi-Abbas-Jawad” KAJ transform and its application on ordinary differential equations

In this work, a new integral transform called the “Kuffi-Abbas-Jawad” KAJ integral transform has been proposed and applied to solve ordinary differential equations. The proposed KAJ integral transform is a modification of the well-known (Sadiq-Emad-Eman) SEE integral transform. It is a novel method that could be used to solve ordinary differential equations. The KAJ integral transform has been tested and proven for its excellent ability to solve homogeneous and non-homogeneous higher-order ordinary differential equations.

3D anisotropic TEM modeling with loop source using model reduction method

Abstract For model reduction techniques, there have been relatively few studies performed regarding the forward modeling of anisotropic media in comparison to transient electromagnetic (TEM) forward modeling of isotropic media. The transient electromagnetic method (TEM) responses after the current has been turned off can be represented as a homogeneous ordinary differential equation (ODE) with an initial value, and the ODE can be solved using a matrix exponential function. However, the order of the matrix exponential function is large and solving it directly is challenging, thus this study employs the Shift-and-Invert (SAI-Krylov) subspace algorithm. The SAI-Krylov subspace technique is classified as a single-pole approach compared to the multi-pole rational Krylov subspace approach. It only takes one LU factorization of the coefficient matrix, along with hundreds of backward substitutions. The research in this paper shows that the anisotropic medium has little effect on the optimal shift ${\gamma _{opt}}$ and subspace order m. Furthermore, as compared to the mimetic finite volume method (SAI-MFV) of the SAI-Krylov subspace technique, the method proposed in this paper (SAI-FEM) can further improve the computing efficiency by roughly 13%. In contrast to the standard implicit time step iterative technique, the SAI-FEM method does not require discretization in time, and the TEM response at any moment within the off-time period can be easily computed. Next, the accuracy of the SAI-FEM algorithm was verified by 1D solutions for an anisotropic layer model and a 3D anisotropic model. Finally, the electromagnetic characteristics of the anisotropic anomalous body of the center loop device and separated device of the airborne transient electromagnetic method were analyzed, and it was found that horizontal conductivity has a considerable influence on the TEM response of the anisotropic medium.

The Application of Differential Equations to Industrial Mathematics

The application of ordinary differential equations in the mathematical description of observable quantities (such as position, temperature, population, concentration, electrical current, etc.) underscores the significance of differential equations in science and engineering (ODEs). Using ordinary differential equations, we calculate the heat transported by convection in fluids. Consequently, we first explore the solution of ordinary homogeneous differential equations, followed by an application of first-order ODEs to heat transfer, specifically in fluid.

Concept of gH Differentiability in Solving Second Order Linear Homogeneous ODEs Based on the Relation between FLT and Its kth Derivative for the Case in Fuzzy Environment

In this study, a fuzzy Laplace transform is used to solve second order linear homogeneous ordinary differential equations. The solution obtained is based on the concept of gH differentiability and the relation between the fuzzy Laplace transform and its derivative for is obtained. Examples are constructed for the existence and uniqueness of solutions of second order FODE.

Development of a Fast Thermal Model for Calculating the Temperature of the Interior PMSM

A 40 kW–4000 rpm interior permanent magnet synchronous machine (IPMSM) applied to an electric vehicle (EV) is introduced as the study object in this paper. The main work of this paper is theoretical derivation and validation of the first-order and multi-order transient lumped-parameter thermal network (LPTN) for the development of a fast thermal model. Based on the first-order LPTN built, the study finds that the heat transfer coefficient of fluid and thickness of the air gap layer are the main influencing factors for the final temperature and time of reaching the steady state. The larger the heat transfer coefficient of fluid is, the lower the steady node temperature is. The smaller the air layer thickness is, the lower the steady node temperature is. The multi-order LPTN theory is further deduced based on the extension of the first-order LPTN. For the constant load and rectangular periodic load, transient node temperatures of the IPMSM are obtained by modeling and solving the first order inhomogeneous differential equations. Temperature rise curves and efficiency maps of the IPMSM under load conditions are realized on a dynamometer platform. The FLUKE infrared-thermal imager and the thermocouple PTC100 are used to validate the mentioned method. The experiment shows that the LPTN of the IPMSM can accurately predict the node temperature.

THE PROBLEM OF ADAPTING THE METHODOLOGY OF TEACHING DIFFERENTIAL EQUATIONS AT SCHOOL

The general properties of the solution of a homogeneous equation are given. Solutions of a homogeneous linear ordinary differential equation of the second order with constant coefficients in three cases related to the coefficients of the equation are investigated. The obtained results are justified in the form of a theorem. These conclusions are proved in the framework of the methods of high school mathematics. This theory, known in general mathematics, is fully adapted to the implementation in secondary school mathematics and developed with the help of new elementary techniques that are understandable to the student. The main purpose of the work was to develop methods for solving a linear homogeneous differential equation of the second-order at a level that a schoolboy can master. The result will be the creation of a special course program on the basics of ordinary differential equations in secondary schools of the natural-mathematical direction, the preparation of appropriate content material, and providing them with a simple teaching method.

Extraction of Some New Solutions of Linear Differential Equations by Dual Space

In this study, the solutions of second order linear homogeneous ordinary differential equations being variable and constant coefficients have been obtained by using power series method which is used frequently in dual space. The solutions of second order homogeneous ordinary differential equations being constant and variable coefficients in dual space by using power series have given and some applications have implemented.

Homogeneous and Non Homogeneous Ordinary Differential Equations with the Second Order

Since Newton time, Differential Equations have been used to comprehend most of physical, geometrical, and vital science in addition to their participation in the study of arithmetic analysis. Furthermore, they have been used in economical and social aspects. They have been developed and got significant position in various sciences. This paper presents for the non-homogeneous ordinary differential equations with the second order. This idea starts in chapter one which talks about the notion of those equations, their orders, in addition to the study of the linear differential equation with the first order and its solution. Whereas the second chapter studies the ordinary differential equations (homogeneous and non-homogeneous) and their solution. Each chapter is supported by solved examples that cover most of the aspects of the topic and we hope that this paper will make a good reference for those who would like to study this topic furtherly.

On computing the analytic-signal backbone of the unforced harmonic oscillator

Backbone curves are usually obtained by analyzing nonlinear oscillators (e.g., the Duffing equation) whereas the backbones of linear oscillators are incidental by-products of such studies. Nevertheless, we focus on the harmonic oscillator, a linear, homogeneous constant-coefficient second-order ordinary differential equation. The backbone of the sinusoidally forced harmonic oscillator is well known, but the backbone of the unforced harmonic oscillator seems to have received essentially no attention. Since the 1980s, various computerized methods of backbone extraction have been under development, one of which is based on analytic-signal theory, a subfield of digital signal processing that involves Hilbert transform computations. By focusing on linear-theory backbones, a prerequisite to nonlinear analysis, we aim to pinpoint some of the theoretical and computational issues that arise when dealing with analytic-signal backbones. This paper examines the Mirror Periodic Method (MPM) as a practical way to reduce end effects in Hilbert transform computations and approximate analytic-signal backbones of unforced harmonic oscillators with light damping (less than 10 percent of critical). Using numerical examples, we demonstrate that MPM works well with both displacement and acceleration data, even for short signals, meaning just one or two damped cycles of oscillation. We then apply MPM techniques to accelerometer data that were recorded as part of earthquake-engineering teaching demonstrations involving a small-scale laboratory model of a linear single-degree-of-freedom system.