Single Nonlinear Ordinary Differential Equation Research Articles

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.

Towards an understanding of the force-free magnetosphere of rapidly spinning black holes

The ability of a plasma surrounding spinning black holes to extract rotational energy and power energetic emissions has been recognized as a key astrophysical phenomenon. Important insights into the nature of this process are obtained through the analysis of the interplay between a force-free magnetosphere and the black hole. This task involves solving a complicated system of equations, often requiring complex numerical simulations. Recent analytical attempts at tackling this problem have exploited the fact that the near horizon region of extreme Kerr (NHEK) is endowed with an enhanced symmetry group. We continue in this direction and show that for some conformally self-similar solutions, the NHEK force-free equations reduce to a single non-linear ordinary differential equation which is difficult to solve with straightforward integration. We here introduce a new approach specifically tailored to this problem and describe how one can obtain physically meaningful solutions.

Effect of nonlinear elastic foundation on large amplitude free and forced vibration of functionally graded beam

Large amplitude free and forced vibration of FG beam resting on nonlinear elastic foundation containing shearing layer and cubic nonlinearity are investigated. The material properties are assumed to vary continuously according to a simple power law. The theoretical formulations and governing partial deferential equation of motion are derived based on Euler–Bernoulli beam theory and von Karman geometric nonlinearity. Adopting appropriate trial functions for various boundary conditions and employing Galerkin technique and assuming a uniformly distributed harmonic load, single nonlinear ordinary differential equation with quadratic and cubic nonlinearities is obtained. Variational Iteration Method (VIM) is used to derive closed form approximate solutions for both free and forced vibration. Comparison of the acquired results with those of existence literature revealed good agreement with a desired accuracy. The frequency response curves are presented for different coefficients of elastic foundation together with various boundary conditions and the effects of nonlinearities are discussed in detail.

The Solution ofSO(3) through a Single Parameter ODE

In many applications we need to solve an orthogonal transformation tensorQ∈SO(3) from a tensorial equationQ˙ = WQunder a given spin historyW. In this paper, we address some interesting issues about this equation. A general solution ofQis obtained by transforming the governing equation into a new one in the space ofℝP3. Then, we develop a novel method to solveQin terms of a single parameter, whose governing equation is a single nonlinear ordinary differential equation (ODE).

Non-Newtonian fluid flow induced by peristaltic waves in a curved channel

This paper is devoted to the study of peristaltic flow of a non-Newtonian fluid in a curved channel. The constitutive relationship between stress and shear rate for a non-Newtonian third grade fluid is used. The problem is governed by a set of two nonlinear partial differential equations. These equations are then transformed into a single nonlinear ordinary differential equation in the stream function under long wavelength and low Reynolds number assumptions. This nonlinear ordinary differential equation is solved for stream function by the shooting method using Mathematica. The important phenomenon of pumping and trapping is presented graphically and discussed in detail. It is found that for a non-Newtonian third grade fluid an increase in the curvature of the channel helps in reducing the pressure rise over one wavelength in pumping region. This result is in contrast to the previous result obtained for the pressure rise over one wavelength for a Newtonian fluid. For a Newtonian fluid, the pressure rise over one wavelength increases with an increase in the curvature. The trapping phenomenon is also altered with the presence of curvature and as a result the symmetry observed for a bolus of the trapped fluid in the case of a straight channel is destroyed and splits into two asymmetrical parts for the curved channel. The outer bolus suppresses the inner bolus towards the lower wall. It is also noted that an increase in size and circulation of boluses achieve a maximum for large values of the shear thickening parameter β . Moreover, the size of two boluses in a third grade fluid is larger in comparison with their counterparts in a Newtonian fluid. Further, the lower trapping limit of the flow rate Θ is also changed in the curved channel. In fact the lower trapping limit of the curved channel exceeds that of the straight channel.

Isospectral Euler–Bernoulli beams via factorization and the Lie method

We obtain isospectral Euler–Bernoulli beams by using factorization and Lie symmetry techniques. The canonical Euler–Bernoulli beam operator is factorized as the product of a second-order linear differential operator and its adjoint. The factors are then reversed to obtain isospectral beams. The factorization is possible provided the coefficients of the factors satisfy a system of non-linear ordinary differential equations. The uncoupling of this system yields a single non-linear third-order ordinary differential equation. This ordinary differential equation, called the principal equation, is analyzed, reduced or solved using Lie group methods. We show that the principal equation may admit a one-dimensional or three-dimensional symmetry Lie algebra. When the principal system admits a unique symmetry, the best we can do is to depress its order by one. We obtain a one-parameter family of invariant solutions in this case. The maximally symmetric case is shown to be isomorphic to a Chazy equation which is solved in closed form to derive the general solution of the principal equation. The transformations connecting isospectral pairs are obtained by numerically solving systems of ordinary differential equations using the fourth-order Runge–Kutta method.

Effects of van der Waals Force and Thermal Stresses on Pull-in Instability of Clamped Rectangular Microplates.

We study the influence of von Karman nonlinearity, van der Waals force, and a athermal stresses on pull-in instability and small vibrations of electrostatically actuated mi-croplates. We use the Galerkin method to develop a tractable reduced-order model for elec-trostatically actuated clamped rectangular microplates in the presence of van der Waals forcesand thermal stresses. More specifically, we reduce the governing two-dimensional nonlineartransient boundary-value problem to a single nonlinear ordinary differential equation. For thestatic problem, the pull-in voltage and the pull-in displacement are determined by solving apair of nonlinear algebraic equations. The fundamental vibration frequency corresponding toa deflected configuration of the microplate is determined by solving a linear algebraic equa-tion. The proposed reduced-order model allows for accurately estimating the combined effectsof van der Waals force and thermal stresses on the pull-in voltage and the pull-in deflectionprofile with an extremely limited computational effort.

A Lie Group Analysis of the Axisymmetric Equations of Finite Elastostatics for Compressible Materials

Many boundary-value problems in nonlinear elastostatics for incompressible isotropic materials have been solved analytically. The situation for compressible materials is quite different. The main difficulty is the absence of the simplified kinematics arising from the incompressibility constraint. Considerable progress in the analytic solution of boundary-value problems for classes of compressible materials has however taken place in recent years and further developments in this direction are considered in this paper. A very special topic in the equilibrium theory for finite deformations of homogeneous isotropic compressible materials is investigated, namely that of spherically symmetric or cylindrically symmetric deformations. Since only axisymmetric deformations are considered, the equilibrium equations reduce to a single second-order nonlinear ordinary differential equation for the radial deformation field. In previous work, two main methods for obtaining analytic solutions for these equations have been developed, the first based on a change of variable that allows one to transform the second-order ODE to a pair of first-order ODEs that may be solved in parametric form. In some cases, the parameter can be eliminated to yield an explicit closed form expression for the deformation. The second method is based on the observation that the second-order ODE is invariant under a stretching transformation of the independent and dependent variables and thus may always be transformed to a first-order ODE. The purpose of this paper is to show how the powerful techniques of Lie group analysis (LGA) for ODEs can be used to unify both the above approaches and how LGA is an invaluable guide in the development of analytic solution methods. It is shown that both of the approaches can be derived from Lie algebra invariance criteria and that the second approach is, in fact, a special case of the first. Results from LGA are then applied to investigate the structure and solution set of the governing ODEs. A systematic procedure for solving the axisymmetric equations of compressible finite elastostatics is thereby proposed. The results are illustrated by application to several specific strain-energy density functions.

Nonlinear oscillations in the molecular dimer: Asymptotic solutions

AbstractNonlinear oscillations in the molecular dimer, composed of two monomers, are investigated, considering an arbitrary number of excitons in the system. The bosonic operators are used to describe the exciton dynamics, that leads to value of the radius of a Bloch sphere that differs from known results. The Hamiltonian of the system is used to reduce the system of equations of motion to a single nonlinear ordinary differential operator equation of the fourth order. Then, the energy of the system is involved in this equation as a parameter. The stability problem is investigated for different energy values. A bifurcation occurs for the energies E > −1/2p. The asymptotic quasiclassical solution for the case E > −1/2p is obtained. The solution contains the number of excitons as a parameter. The quantum solutions can be obtained from the classical solutions using the continual integral technique. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

General brane geometries from scalar potentials: gauged supergravities and accelerating universes

We find broad classes of solutions to the field equations for d-dimensional gravity coupled to an antisymmetric tensor of arbitrary rank and a scalar field with non-vanishing potential. Our construction generates these configurations from the solution of a single nonlinear ordinary differential equation, whose form depends on the scalar potential. For an exponential potential we find solutions corresponding to brane geometries, generalizing the black p-branes and S-branes known for the case of vanishing potential. These geometries are singular at the origin with up to two (regular) horizons. Their asymptotic behaviour depends on the parameters of the model. When the singularity has negative tension or the cosmological constant is positive we find time-dependent configurations describing accelerating universes. Special cases give explicit brane geometries for (compact and non-compact) gauged supergravities in various dimensions, as well as for massive 10D supergravity, and we discuss their interrelation. Some examples lift to give new solutions to 10D supergravity. Limiting cases with a domain wall structure preserve part of the supersymmetries of the vacuum. We also consider more general potentials, including sums of exponentials. Exact solutions are found for these with up to three horizons, having potentially interesting cosmological interpretation. We give several additional examples which illustrate the power of our techniques.

On the asymptotic solutions of the coupled quasiparticle-oscillator system

A coupled quasiparticle-oscillator system is considered for an arbitrary number of excitons. The exciton dynamics is described in terms of the second quantization (i.e. by means the bosonic operators). As a consequence a radius of a Bloch sphere is obtained different to the previous results. Some integrals of motion are obtained that allowed to reduce the system of equations of motion to a single nonlinear ordinary differential equation of the fourth order. This equation contains the energy of the system as a parameter. The fixed points are found as a functions of the energy of the system, and its stability properties are investigated. It is demonstrated that a bifurcation is presented for the energies H<−1/2 p. An asymptotic quasiclassical solution around fixed point for the case H>−1/2 p is obtained. The solutions around other stable fixed points can be obtained analogously. The expression for the evolution operator of the quasiparticle-oscillator system is obtained as a functional on the classical solutions.

A LOCK-FREE MATERIAL FINITE ELEMENT FOR NON-LINEAR OSCILLATIONS OF LAMINATED PLATES

The objective of the present paper is to propose an efficient, accurate and robust four-node shear flexible composite plate element with six degrees of freedom per node to investigate the non-linear oscillatory behavior of unsymmetrical laminated plates. The degrees of freedom considered are three displacement (u, v, w) along x-, y- and z -axis, two rotations (θx, θy) abouty - and x -axis and twist θxy. The elementc employs coupled displacement field, which is derived using moment-shear equilibrium and in-plane equilibrium of composite strips along the x - andy -axis. The displacement field so derived not only depend on the element co-ordinates but are a function of extensional, bending–extensional, bending and transverse shear stiffness coefficients as well. A bi-cubic polynomial distribution with 16 generalized undetermined coefficients for the transverse displacement is assumed. The element stiffness and mass matrices are computed numerically by employing 3×3 Gauss Legendre product rules. The element is found to be free of shear locking and does not exhibit any spurious modes. The element is found to be free of shear locking and does not exhibit any spurious modes. In order to compute the non-linear frequencies, linear mode shape corresponding to fundamental frequency is assumed as the spatial distribution and non-linear finite element equations are reduced to a single non-linear second order ordinary differential equation. This equation is solved by employing direct numerical integration method. A series of numerical examples is solved to demonstrate the efficacy of the proposed material finite element.

Mathematical possibility of certain systems in geometrical optics

A system of functional differential equations arising from Fermat’s principle is used to study qualitative questions concerning the mathematical possibility of certain rotationally symmetric piecewise homogeneous optical systems. In particular, it is shown that, given two pairs of points on the optical axis, there exist precisely two systems of single-element lenses having prescribed axial thickness and index of refraction such that these points are perfect foci. This finding sharpens an earlier result. Embedding the solution in a one-parameter family permits the construction of an asymptotic solution that requires the solution of a single nonlinear ordinary differential equation. The leading-order solution corresponds to an optical system satisfying the Herschel condition. The existence and uniqueness results are extended to optical systems having three lens boundaries such as achromatic doublets.

Nonlinear Finite Deformation Analysis of Beams and Columns

A method for solving the finite-displacement problem of a curved elastic beam with axial, shear, and flexural deformation subject to distributed and point loads is presented. Within the context of the kinematic assumptions of the Timoshenko theory, a Lagrangian formulation of the problem is developed. In terms of three cross-sectional stress resultants, three Euler equations of equilibrium for the beam are derived with the aid of a variational principle for finite deformation. Upon linearization to small strains and the adoption of a linear elastic constitutive relation between the stress and strain tensors, it is shown that the problem is reducible to a single second-order nonlinear ordinary differential equation. Subject to appropriate boundary conditions, the resulting two-point boundary-value problem is solved by a finite-element method. By virtue of a continuation algorithm, accurate solutions of the system of nonlinear equations can be obtained for a variety of bifurcation and buckling problems. Comprehensive results are presented for two cantilever beams as illustrations.

Spiral waves in dispersive media

The Lighthill kinematic approach is applied for the description of standing waves generated at circular motion of a point source in a dispersive medium. A single nonlinear ordinary differential equation is derived. Numerical calculations are performed for different dispersion relations. The spiral shape of waves and the existence of special points for wave crests are the main peculiarities of solutions obtained. A possible application of the kinematic approach to the description of spiral galaxies and other physical phenomena is demonstrated.

TypeN twisting vacuum gravitational fields

For vacuum, typeN, twisting gravitational fields the Einstein field equations reduce to partial differential equations for two functions, one real, the other complex, which may be regarded as initial data on a localJ+. If it is assumed that in some frame these initial data take on a certain product form, one factor involving only a spatial variable, the other only a retarded time variable, then these equations become relatively tractable and reduce further to two ordinary differential equations. Rejecting all solutions which lead to Minkowski space or to zero twist leaves just two possibilities. One corresponds to the only explicitly known spacetime of the kind, namely that of Hauser. The other leads to new typeN twisting metrics. However, these metrics can be constructed explicitly only once a single nonlinear third-order ordinary differential equation has been solved.

Non-linear free vibration analysis of composite plates with initial imperfections and in-plane loading

In this paper the non-linear response of layered composite plates with initial imperfections and subjected to in-plane preload is considered. The basic equations used in the analysis are those corresponding to an accurate shear deformation theory which employs parabolic shear strain variations across the thickness and requires no correction factors. The five governing non-linear equilibrium equations of the problem are reduced to a single non-linear ordinary differential equation (ODE) using a single mode approach in conjunction with the Galerkin method. Numerical results are obtained tor two problems: 1.(i) large amplitude vibration of imperfect plates and2.(ii) small oscillations in the vicinity of a static buckled position of an in-plane loaded two-layered (0 90) plate.

Global aspects of shear-free perfect fluids in general relativity

In an earlier investigation of the class of shear-free expanding (or contracting) irrotational perfect fluids obeying a barotropic equation of state p=p( μ), and satisfying the field equations of general relativity, it was shown that the space-times form three distinct classes. In one class, the fluid acceleration is zero (i.e., the flow is geodesic), and the space-times are the well-known spatially homogeneous and isotropic Friedmann–Robertson–Walker (FRW) models. In the other two classes, the acceleration is nonzero, and the space-times are spatially anisotropic and are less familiar. One of these classes consists of the spherically symmetric Wyman solutions, whereas models that are plane symmetric, and either spatially or temporally homogeneous, constitute the final class. Analytic forms for these anisotropic space-time metrics were given, although in each case their exact determination would depend upon the solution of a single nonlinear ordinary differential equation, which has not been achieved. The purpose of the present article is to provide a pictorial description of the solutions of these equations, to depict qualitatively similar and distinct subclasses of solution, and hence to discover in some detail all possible global behaviors of the associated space-times. In all cases, it is found that, when the space-time is sufficiently extended, the fluid exhibits unphysical properties. The conclusion is that shear-free expanding (or contracting) relativistic perfect fluids that obey an equation of state p=p( μ) must be FRW, or else must be restricted to ‘‘local’’ regions, by means of a suitable extension in which at least one of the conditions defining the entire family is relaxed.

On the equations of facilitated diffusion

THE PROCESS in which a permeant molecule crosses a solution barrier both by means of passive diffusion and by being carried on a molecular carrier which itself diffuses passively is called facilitated diffusion. The process has been invoked as a model in chemical engineering [l, 7, 191, and in biology [20, 211. We are interested in the mathematics of the facilitated diffusion of oxygen and carbon monoxide with the carriers hemoglobin and myoglobin. The mathematical equations governing the simultaneous chemical reaction and diffusion of several molecular species have been known for a long time [3], and are commonly referred to as reaction-diffusion equations. When applied to the steady-state facilitated diffusion of a substrate reacting reversibly with and diffusing in a solution slab of carrier molecules, they may be reduced to a single nonlinear ordinary differential equation for one molecular species [l]. We shall call such an equation a Wyman’s Equation [24]. Consider the reaction

Fate of the false vacuum: Semiclassical theory

It is possible for a classical field theory to have two homogeneous stable equilibrium states with different energy densities. In the quantum version of the theory, the state of higher energy density becomes unstable through barrier penetration; it is a false vacuum. This is the first of two papers developing the qualitative and quantitative semiclassical theory of the decay of such a false vacuum for theories of a single scalar field with nonderivative interactions. In the limit of vanishing energy density between the two ground states, it is possible to obtain explicit expressions for the relevant quantities to leading order in $h$; in the more general case, the problem can be reduced to solving a single nonlinear ordinary differential equation.