Single Nonlinear Ordinary Differential Equation Research Articles

On the existence and uniqueness of a positive solution to a boundary value problem for a nonlinear ordinary differential equation of 4n order

The paper considers a two-point boundary value problem with homogeneous boundary conditions for a single nonlinear ordinary differential equation of order 4n. Using the well-known Krasnoselsky theorem on the expansion (compression) of a cone, sufficient conditions for the existence of a positive solution to the problem under consideration are obtained. To prove the uniqueness of a positive solution, the principle of compressed operators was invoked. In conclusion, an example is given that illustrates the fulfillment of the obtained sufficient conditions for the unique solvability of the problem under study.

Phenomenological model of superconducting optoelectronic loop neurons

Superconducting optoelectronic loop neurons are a class of circuits potentially conducive to networks for large-scale artificial cognition. These circuits employ superconducting components including single-photon detectors, Josephson junctions, and transformers to achieve neuromorphic functions. To date, all simulations of loop neurons have used first-principles circuit analysis to model the behavior of synapses, dendrites, and neurons. These circuit models are computationally inefficient and leave opaque the relationship between loop neurons and other complex systems. Here we introduce a modeling framework that captures the behavior of the relevant synaptic, dendritic, and neuronal circuits at a phenomenological level without resorting to full circuit equations. Within this compact model, each dendrite is discovered to obey a single nonlinear leaky-integrator ordinary differential equation, while a neuron is modeled as a dendrite with a thresholding element and an additional feedback mechanism for establishing a refractory period. A synapse is modeled as a single-photon detector coupled to a dendrite, where the response of the single-photon detector follows a closed-form expression. We quantify the accuracy of the phenomenological model relative to circuit simulations and find that the approach reduces computational time by a factor of ten thousand while maintaining an accuracy of one part in ten thousand. We demonstrate the use of the model with several basic examples. The net increase in computational efficiency enables future simulation of large networks, while the formulation provides a connection to a large body of work in applied mathematics, computational neuroscience, and physical systems such as spin glasses.

Numerical method for determining the coefficient of hydraulic resistance two-phase flow in a gas lift well

The process of stationary movement of a gas-liquid mixture in the lifting pipe of a gas-lift well is considered. To describe this two-phase flow, a mathematical model is proposed that includes the equation of flow motion and the continuity equation for each phase. The presented model is transformed to a single nonlinear ordinary differential equation with respect to pressure. Within the framework of the obtained model, the task is set to determine the hydraulic resistance coefficient of a two-phase flow according to an additionally specified condition with respect to pressure. An additional condition, presented in the form of a nonlinear algebraic equation, is transformed into an ordinary differential equation with respect to an unknown coefficient of hydraulic resistance by applying the method of differentiation by parameter. The solution of the resulting Cauchy problem is determined by the finite difference method. Based on the proposed computational algorithm, numerical experiments were carried out for model data. Keywords: gas lift; two-phase flow; hydraulic resistance coefficient; parameter differentiation method; finite difference method.

Large-Amplitude Vibrations of a Slender Cantilevered Beam

We present a geometrically exact beam model for a cantilevered beam undergoing large-amplitude motions, motivated by recent wind-energy capture systems that exploit vortex-induced vibrations. The derivation of this model leads to a single nonlinear 2nd-order ordinary differential equation that appears similar to a Duffing oscillator with additional nonlinear acceleration terms. For prismatic configurations, the formulation is characterized by its linear natural frequency and a normalization constant of the shape-function. A brief parametric study is conducted to demonstrate the softening effects of the nonlinear acceleration terms. Continuation methods are used to construct the amplitude versus forcing frequency plots to capture the nonlinear response.

A fully nonlinear solitary wave in six-component dusty cometary plasma

The dynamics of fully nonlinear structure of six-component dusty cometary plasma system with the help of Sagdeev pseudo-potential techniques have been investigated. This plasma system is made up of Oxygen (positively and negatively charged) ion pair, negatively charged dust particles, and kappa distributed ions of Hydrogen, hot solar electrons and slightly colder cometary electrons. The linear analysis is carried out and the linear dispersion relation is obtained and investigated with some relevant plasma parameters. As a result, the velocities in the fluid equations are normalized to a general acoustic speed instead of the particular form of acoustic speed. Employing pseudo-potential approach, the basic equations of plasma system were reduced to a single nonlinear ordinary differential equation (energy-balance). By means of Sagdeev potential curves and bifurcation analysis of phase-portrait, the existence and propagation of nonlinear waves are examined. In addition to the periodic solution, it is shown that the localized negative and positive amplitude electrostatic pulses may (co-) exist and propagate in plasma system depending on the Mach number intervals. The impact of some intrinsic plasma parameters on the main features of the solitary wave electrostatic potential profile and the associated electrostatic field are extensively investigated. The results of this research are relevant to plasmas in both laboratory plasmas and exclusively in space plasmas as, comet tails, rings of giant planets, solar wind, interstellar clouds, etc.

Prandtl’s extended mixing length model applied to the two-dimensional turbulent classical far wake

Despite its limitations, Prandtl’s mixing length model is widely applied in modelling turbulent free shear flows. Prandtl’s extended model addresses many of the shortfalls of the original model, but is not so widely used, in part due to additional mathematical complexities that arise in its derivation and implementation. Furthermore, in both models, Prandtl neglects the kinematic viscosity on the basis that it is much smaller in magnitude than the turbulent viscosity. Recent work has shown that including the kinematic viscosity in the original model has both mathematical and physical advantages. In the present work, a novel derivation of the extended model is provided, and it is demonstrated that similar advantages are again obtained when the kinematic viscosity is included. Additionally, through the use of scaling techniques, similarity mean velocity profiles of the extended model are derived, resulting in a single nonlinear ordinary differential equation that is solved numerically with a Hermite spectral method. The computed profiles for the normalized similarity mean velocity and shear stress are compared with experimental observations and shown to be in excellent agreement.

Particle-laden thin-film flow in helical channels with arbitrary shallow cross-sectional shape

Particle-laden flows in helical channels are of interest for their applications in spiral particle separators used in the mining and mineral processing industries. In this paper, we extend the previous work of Lee, Stokes, and Bertozzi [“Behaviour of a particle-laden flow in a spiral channel,” Phys. Fluids 26, 043302 (2014)] by studying thin-film flows of monodisperse particle-laden fluid in helically wound channels of arbitrary centerline curvature and torsion and arbitrary cross-sectional shape. In the case where the particles are uniformly distributed through the depth of the film, significant analytic progress can be made yielding insight into the influence of channel geometry on particle distribution across the channel cross section: the governing equations reduce to a single nonlinear ordinary differential equation, which is readily integrated numerically to obtain the solution subject to appropriate boundary conditions. Motivated by possible application to the design of spiral separators, we consider the effects of changing the channel centerline geometry, the cross-sectional shape and the particle density on the resulting flows, and the radial distribution of particles. Our results support the findings in the work of Arnold, Stokes, and Green [“Thin-film flow in helically wound rectangular channels of arbitrary torsion and curvature,” J. Fluid Mech. 764, 76–94 (2015)] regarding the effect of channel centerline geometry and cross-sectional shape on flows in particle-free regions. In particle-rich regions, similar effects are seen although the primary velocity is lower due to increased effective mixture viscosity. Of key interest is the effect of channel geometry on the focusing of the particles for given fluxes of fluid and particles. We find that introducing a trench into the channel cross section, a feature often used in commercial spiral particle separators, leads to a smaller radial width of the particle-rich region, i.e., sharper focusing of the particles, which is consistent with experiments showing that channel geometry influences particle separation in a spiral separator.

Comparison and Combination of Techniques for Determining the Parameters of a Magnetic Hysteresis Model

The Jiles-Atherton scalar hysteresis model presents five parameters used to represent the material tested and used to calculate the magnetic losses. This article presents a comparative analysis of the performance of two methods of identifying these parameters. In the first method, the equations of Jiles-Atherton were assembled into a single non-linear ordinary differential equation as a function of the variables of interest. An algebraic system of five equations with five unknowns is obtained by evaluating the non-linear ordinary differential equation in five points belonging to the branch of the experimental hysteresis loop. The parameters are obtained by solving this system of equations using the method of Non-Linear Least Squares (NLLS). In the second method, the inverse model of Jiles-Atherton is used to calculate the magnetic field H from the experimental values of magnetic induction B. Using the method of genetic algorithms (MGA), the objective function given by the sum of the relative error of calculated magnetic field and experimental magnetic field along the hysteresis loop is minimized. To validate methods the experimental curves were compared with calculated ones. When applying the methods, it was verified that NLLS besides providing more accurate results, it is faster when compared to MGA. In the MGA the convergence of the calculated magnitudes to the experimental magnitudes improves when one of the chromosomes of the initial population is the solution obtained applying NLLS.

A Boundary Layer Approach to Multibody Systems Involving Single Frictional Impacts

A systematic theoretical approach is presented, revealing dynamics of a class of multibody systems. Specifically, the motion is restricted by a set of bilateral constraints, acting simultaneously with a unilateral constraint, representing a frictional impact. The analysis is carried out within the framework of Analytical Dynamics and uses some concepts of differential geometry, which provides a foundation for applying Newton's second law. This permits a successful and illuminating description of the dynamics. Starting from the unilateral constraint, a boundary is defined, providing a subspace of allowable motions within the original configuration manifold. Then, the emphasis is focused on a thin boundary layer. In addition to the usual restrictions imposed on the tangent space, the bilateral constraints cause a correction of the direction where the main impulse occurs. When friction effects are negligible, the dominant action occurs along this direction and is described by a single nonlinear ordinary differential equation (ODE), independent of the number of the original generalized coordinates. The presence of friction increases this to a system of three ODEs, capturing the essential dynamics in an appropriate subspace, arising by bringing the image of the friction cone from the physical to the configuration space. Moreover, it is shown that the classical Darboux–Keller approach corresponds to a special case of the new method. Finally, the theoretical results are complemented by a selected set of numerical results for three examples.

Results of Single Sculling Technique Analysis Using 1D Mathematical Model

Sports results in rowing depend on two the most important factors: the athlete physical features and the motion’s techniques. Both assessment and optimisation of rowing techniques are possible only when one disposes the reliable mathematical model predicting the regatta results, that is the time to cover the assumed distance. A single scull participating in the 2000 meters distance’s regatta is our subject. The purpose of this study is to create a simplified mathematical model to simulate the rowing boat dynamics. The boat-rower system is treated here as a material point here. The oar has a prescribed angular motion vs. oarlock depending upon the time. The way of hydrodynamic force developed on the oar’s blade was modelled here. Then the boat motion, with the influence both of wind and water current, was described by a single nonlinear ordinary differential equation (NODE). The used simple model gives the possibilities of fast and reliable simulation of the single sculling technique and of forecasting the result of rowing regattas. The parametric sensitivity coefficient was defined here, as well the efficiency coefficient of sculling.

Analytical solutions to nonlinear problems by the generalized form of HAM : A note

The objective of this article is to obtain analytical solutions for a set of nonlinear problems by using 'further generalisation of HAM'. In comparison to the Homotopy analysis method (HAM) solutions, more accurate solutions are obtained by introducing an extra term in the frame of HAM. We consider a set of three nonlinear problems of which first two are governed by single nonlinear ordinary differential equation (they are two cases of the forced Van der Pol Duffing oscillator) and third one is governed by a system of four coupled nonlinear ordinary differential equations. A maximum reduction of approximately 25% in the square residual error is obtained by using the generalised form of HAM compared to the square residual error without the generalised form.

Solution of the Fractional Form of Unsteady Squeezing Flow through Porous Medium

We propose two friendly analytical techniques called Adomian decomposition and Picard methods to analyze an unsteady axisymmetric flow of nonconducting, Newtonian fluid. This fluid is assumed to be squeezed between two circular plates passing through porous medium channel with slip and no-slip boundary conditions. A single fractional order nonlinear ordinary differential equation is obtained by means of similarity transformation with the help of the fractional calculus definitions. The resulting fractional boundary value problems are solved by the proposed methods. Convergence of the two methods’ solutions is confirmed by obtaining various approximate solutions and various absolute residuals for different values of the fractional order. Comparison of the results of the two methods for different values of the fractional order confirms that the proposed methods are in a well agreement and therefore they can be used in a simple manner for solving this kind of problems. Finally, graphical study for the longitudinal and normal velocity profiles is obtained for various values of some dimensionless parameters and fractional orders.

Macroscopic Einstein equations for a cosmological model with a λ term

Through averaging the Einstein equations over transverse gravitational perturbations it is obtained a closed system of two ordinary differential equations describing macroscopic cosmological evolution of the isotropic space-flat Universe filled with gravitational radiation. It is found an asymptotic solution of evolution equation for gravitational perturbation amplitude. Making the substitution of this solution into Einstein equation averaged over gravitational perturbations, the single evolution non-linear ordinary differential second-order equation relative to macroscopic scale factor is obtained. It is also found a solution of evolution equation for scale factor in WKB-approximation which analytically describes the process of transformation from ultrarelativistic regime of cosmological extension to inflationary one.

Iterative Methods for Solving the Fractional Form of Unsteady Axisymmetric Squeezing Fluid Flow with Slip and No-Slip Boundaries

An unsteady axisymmetric flow of nonconducting, Newtonian fluid squeezed between two circular plates is proposed with slip and no-slip boundaries. Using similarity transformation, the system of nonlinear partial differential equations of motion is reduced to a single fourth-order nonlinear ordinary differential equation. By using the basic definitions of fractional calculus, we introduced the fractional order form of the fourth-order nonlinear ordinary differential equation. The resulting boundary value fractional problems are solved by the new iterative and Picard methods. Convergence of the considered methods is confirmed by obtaining absolute residual errors for approximate solutions for various Reynolds number. The comparisons of the solutions for various Reynolds number and various values of the fractional order confirm that the two methods are identical and therefore are suitable for solving this kind of problems. Finally, the effects of various Reynolds number on the solution are also studied graphically.

Hydrodynamics, resurgence, and transasymptotics

The second-order hydrodynamical description of a homogeneous conformal plasma that undergoes a boost- invariant expansion is given by a single nonlinear ordinary differential equation, whose resurgent asymptotic properties we study, developing further the recent work of Heller and Spalinski [Phys. Rev. Lett. 115, 072501 (2015)]. Resurgence clearly identifies the non-hydrodynamic modes that are exponentially suppressed at late times, analogous to the quasi-normal-modes in gravitational language, organizing these modes in terms of a trans-series expansion. These modes are analogs of instantons in semi-classical expansions, where the damping rate plays the role of the instanton action. We show that this system displays the generic features of resurgence, with explicit quantitative relations between the fluctuations about different orders of these non-hydrodynamic modes. The imaginary part of the trans-series parameter is identified with the Stokes constant, and the real part with the freedom associated with initial conditions.

Peristaltic Transport of Visco-Elasto-Plastic Fluids in a Planar Channel

Abstract We numerically investigate peristaltic transport of incompressible visco-elasto-plastic fluids in a two-dimensional symmetric channel. The constitutive equation used for extra stress tensor is of more general form as it includes a number of well-known models like Maxwell A and B, Johnson–Segalman, Oldroyd-B, and Bingham models as its special cases. A detailed mathematical modelling of the problem is presented. The flow equations in the wave frame reduce to a single nonlinear ordinary differential equation in stream function by the implication of widely taken assumptions of long wavelength and low Reynolds number. The solution of the problem is obtained by two ways; namely, shooting method and Matlab built in routine bvp4c, and their comparison shows an excellent agreement. A parametric study based on bvp4c solution is performed to see the effects of parameters on velocity profile, pressure rise per wavelength, frictional forces, and trapping phenomenon.

Large amplitude free vibration of symmetrically laminated magneto-electro-elastic rectangular plates on Pasternak type foundation

Nonlinear free vibration of symmetrically laminated magneto-electro-elastic rectangular plate resting on an elastic foundation is studied analytically. The plate is considered to be simply supported on all edges. It is also assumed that the magneto-electro-elastic body is poled along the z direction and subjected to electric and magnetic potentials between the upper and lower surfaces. To model the motion of the plate, the first order shear deformation theory along with the Gauss's equations for electrostatics and magnetostatics are used. Then equations of motion are reduced to a single nonlinear ordinary differential equation which is solved analytically by multiple scales method. The results are compared with the published results and good agreement is found. Some numerical examples are presented to investigate the effects of several parameters on the linear and nonlinear behavior of these plates.

Simultaneous effects of viscoelasticity and curvature on peristaltic flow through a curved channel

We have analyzed peristaltic flow of an Oldroyd-B fluid in a curved channel. Assuming the flow to be incompressible, laminar and two-dimensional, the governing partial differential equations are reduced under long wavelength and low Reynolds number approximations into a single nonlinear ordinary differential equation in the stream function. Matlab built-in routine bvp4c is utilized to solve this nonlinear ordinary differential equation. The solution thus obtained is used to investigate the effects of curvature of the channel and Weissenberg number on important phenomena of pumping and trapping associated with peristaltic motion. It is found that for small values of Weissenberg number, the effects of curvature are dominant. However, for large values of Weissenberg number, viscoelastic effects counteract the effects of curvature and help the flow velocity and circulating bolus of fluid to regain their symmetry.

Linear and nonlinear free vibration of a multilayered magneto-electro-elastic doubly-curved shell on elastic foundation

Nonlinear and linear free vibration of symmetrically laminated magneto-electro-elastic doubly-curved thin shell resting on an elastic foundation is studied analytically. The shell is considered to be simply-supported on all edges and the magneto-electro-elastic body is poled along the z direction and subjected to electric and magnetic potentials between the upper and lower surfaces. To obtain the equations of motion, the Donnell shell theory in the presence of rotary inertia effect is used. Moreover, Gauss' laws for electrostatics and magnetostatics are used to model the electric and magnetic behavior. The nonlinear partial differential equations of motion are reduced to a single nonlinear ordinary differential equation by introducing a force function and using the single-term Galerkin method. The resulting equation is solved analytically by Lindstedt-Poincare perturbation method. After validation of the present study, several numerical studies are done to investigate the effects of foundation parameters, geometrical properties of the shell, and electric and magnetic potentials on the linear and nonlinear behavior of these smart shells.

Modeling and analysis of unsteady axisymmetric squeezing fluid flow through porous medium channel with slip boundary.

The aim of this article is to model and analyze an unsteady axisymmetric flow of non-conducting, Newtonian fluid squeezed between two circular plates passing through porous medium channel with slip boundary condition. A single fourth order nonlinear ordinary differential equation is obtained using similarity transformation. The resulting boundary value problem is solved using Homotopy Perturbation Method (HPM) and fourth order Explicit Runge Kutta Method (RK4). Convergence of HPM solution is verified by obtaining various order approximate solutions along with absolute residuals. Validity of HPM solution is confirmed by comparing analytical and numerical solutions. Furthermore, the effects of various dimensionless parameters on the longitudinal and normal velocity profiles are studied graphically.