Approximate Group Analysis and Multiple Time Scales Method for the Approximate Boussinesq Equation

This paper is devoted to investigation of the approximate Boussinesq equation by methods of the approximate symmetry analysis of partial differential equations with a small parameter developed by Baikov, Gazizov and Ibragimov. We combine these methods with the method of multiple time scales to extend the domain of definition of approximate group invariant solutions of the approximate Boussinesq equation.     

Similarity solutions for creeping flow and heat transfer in second grade fluids

The two-dimensional equations of motions for the slowly flowing and heat transfer in second grade fluid are written in cartesian coordinates neglecting the inertial terms. When the inertia terms are simply omitted from the equations of motions the resulting solutions are valid approximately for Re?1. This fact can also be deduced from the dimensionless form of the momentum and energy equations. By employing Lie group analysis, the symmetries of the equations are calculated. The Lie algebra consist of four finite parameter and one infinite parameter Lie group transformations, one being the scaling symmetry and the others being translations. Two different types of solutions are found using the symmetries. Using translations in x and y coordinates, an exponential type of exact solution is presented. For the scaling symmetry, the outcoming ordinary differential equations are more involved and only a series type of approximate solution is presented. Finally, some boundary value problems are discussed.     

SIMILARITY SOLUTIONS FOR CREEPING FLOW AND HEAT TRANSFER IN SECOND GRADE FLUID

IntroductionTheconceptofthesecondgradefluidcanbedevelopedasanexpansionintermsoffadingmemorytotheNewtonianfluid .Insodoing ,higherorderderivativesofthevelocityfieldarerequired.However,secondorderfluidmayprovideonlyanapproximationtorealviscoelasticbehavior.Thephysicalmeaning ,ifany ,ofthehighorderderivativesisunclearnevertheless,theRivlinEricksensecondorderfluidiscommonlyusedandfurtherstudyseemswarranted .TheStokesflowsolutionsandthecreepingsecondgradefluidflowsolutionsarepresentedqualitativel…     

MHD flow of power-law fluid on moving surface with power-law velocity and special injection/blowing

The problem of magnetohydrodynamic （MHD） flow on a moving surface with the power-law velocity and special injection/blowing is investigated. A scaling group transformation is used to reduce the governing equations to a system of ordinary differen- tial equations. The skin friction coefficients of the MHD boundary layer flow are derived, and the approximate solutions of the flow characteristics are obtained with the homotopy analysis method （HAM）. The approximate solutions are easily computed by use of a high order iterative procedure, and the effects of the power-law index, the magnetic parameter, and the special suction/blowing parameter on the dynamics are analyzed. The obtained results are compared with the numerical results published in the literature, verifying the reliability of the approximate solutions.     

Symmetry analysis of modified 2D Burgers vortex equation for unsteady case

In this paper, a symmetry analysis of the modified 2D Burgers vortex equation with a flow parameter is presented. A general form of classical and non-classical symmetries of the equation is derived. These are fundamental tools for obtaining exact solutions to the equation. In several physical cases of the parameter, the specific classical and non-classical symmetries of the equation are then obtained. In addition to rediscovering the existing solutions given by different methods, some new exact solutions are obtained with the symmetry method, showing that the symmetry method is powerful and more general for solving partial differential equations(PDEs).     

Approximate symmetries and conservation laws of the geodesic equations for the Schwarzschild metric

Approximate symmetries have been defined in the context of differential equations and systems of differential equations. They give approximately, conserved quantities for Lagrangian systems. In this paper, the exact and the approximate symmetries of the system of geodesic equations for the Schwarzschild metric, and in particular for the radial equation of motion, are studied. It is noted that there is an ambiguity in the formulation of approximate symmetries that needs to be clarified by consideration of the Lagrangian for the system of equations. The significance of approximate symmetries in this context is discussed.     

Linear interactions affecting the propogation of waves in a linear elastic fluid

Small linear interactions affecting the propogation of waves in a linear elastic fluid are investigated. These linear interactions may occur as a result of impurities on the surface of a linear elastic fluid. These interactions are imposed on the linear wave equations which were investigated in Momoniat (Propogation of waves in a linear elastic fluid, submitted for publication) using the non-classical contact symmetry method. The occurrence of a small parameter in the wave equations under consideration in this paper makes the problem ideal for analysis using an approximate non-classical contact symmetry method. Approximate contact symmetries and approximate solutions are determined and discussed for the problems under consideration. Comparisons are made with the case of no interaction.     

Approximate symmetries of creeping flow equations of a second grade fluid

Creeping flow equations of a second grade fluid are considered. Two current approximate symmetry methods and a modified new one are applied to the equations of motion. Approximate symmetries obtained by different methods and the exact symmetries are contrasted. Approximate solutions corresponding to the approximate symmetries are derived for each method. Symmetries and solutions are compared and advantages and disadvantages of each method are discussed in detail.     

Localization and approximation of attractors for the Ginzburg-Landau equation

This paper studies the long-term behavior of solutions to the Ginzburg-Landau partial differential equation. For each positive integerm we explicitly produce a sequence of approximate inertial manifolds _m,j ,j = 1, 2,..., of dimensionm and associate with each manifold a thin neighborhood into which the orbits enter with an exponential speed and in a finite time. Of course this neighborhood contains the universal attractor which embodies the large time dynamics of the equations. The thickness of these neighborhoods decreases with increasingm for a fixed orderj; however, for a fixedm no conclusion can be made about the thickness of the neighborhoods associated to two differentj's. The neighborhoods associated to the manifolds localize the universal attractor and provide computabie large time approximations to solutions of the Ginzburg-Landau equation.     

Singular perturbation of linear algebraic equations with application to stiff equations

In this paper the singular perturbation problem of linear algebraic equations with a small parameter is presented by an example in practice. The existence and uniqueness theorem of its solution is proved by the perturbation method and the estimation of error for its approximate solution is given. Finally, the example mentioned above explaining how to apply the theory to solve the stiff equations is shown.     

The asymptotic solving equations of thick ring shell and its solution under moment Mo

In this paper, from the fundamental equations of three dimensional elastic mechanics, we have found a sequence of asymptotic solving equations of thick ring shell (or body) applied arbitrary loads by the perturbation method based upon a geometric small parameter a=r_o/R_o, which may be divided into two independent equation groups which are similar to the equation groups for plane strain and torsional problems. Using these equations, we have also found first order and second order approximate solutions of thick ring shell applied moment M_o.     

Construction of approximate analytical solutions to strongly nonlinear damped oscillators

An analytical approximate method for strongly nonlinear damped oscillators is proposed. By introducing phase and amplitude of oscillation as well as a bookkeeping parameter, we rewrite the governing equation into a partial differential equation with solution being a periodic function of the phase. Based on combination of the Newton’s method with the harmonic balance method, the partial differential equation is transformed into a set of linear ordinary differential equations in terms of harmonic coefficients, which can further be converted into systems of linear algebraic equations by using the bookkeeping parameter expansion. Only a few iterations can provide very accurate approximate analytical solutions even if the nonlinearity and damping are significant. The method can be applied to general oscillators with odd nonlinearities as well as even ones even without linear restoring force. Three examples are presented to illustrate the usefulness and effectiveness of the proposed method.     

On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions

We study the nonlinear hyperbolic partial differential equation, (u _t+uu_x)_x=1/2u _x ² . This partial differential equation is the canonical asymptotic equation for weakly nonlinear solutions of a class of hyperbolic equations derived from variational principles. In particular, it describes waves in a massive director field of a nematic liquid crystal.Global smooth solutions of the partial differential equation do not exist, since their derivatives blow up in finite time, while weak solutions are not unique. We therefore define two distinct classes of admissible weak solutions, which we call dissipative and conservative solutions. We prove the global existence of each type of admissible weak solution, provided that the derivative of the initial data has bounded variation and compact support. These solutions remain continuous, despite the fact that their derivatives blow up.There are no a priori estimates on the second derivatives in any L ^p space, so the existence of weak solutions cannot be deduced by using Sobolev-type arguments. Instead, we prove existence by establishing detailed estimates on the blowup singularity for explicit approximate solutions of the partial differential equation.We also describe the qualitative properties of the partial differential equation, including a comparison with the Burgers equation for inviscid fluids and a number of illustrative examples of explicit solutions. We show that conservative weak solutions are obtained as a limit of solutions obtained by the regularized method of characteristics, and we prove that the large-time asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kink-wave.     

插值摄动法及其在力学中的应用

本文研究了最高阶导数乘以小参数，或出现奇点的微分方程的定解问题，用插值摄动法求得了一级近似解，它和通常的奇异摄动法（匹配法、多尺度法）的一级近似解的精度相同。     

Study of differential control method for solving chaotic solutions of nonlinear dynamic system

In this paper, we focus on the need to solve chaotic solutions of high-dimensional nonlinear dynamic systems of which the analytic solution is difficult to obtain. For this purpose, a Differential Control Method (DCM) is proposed based on the Mechanized Mathematics-Wu Elimination Method (WEM). By sampling, the computer time of the differential operator of the nonlinear differential equation can be substituted by the differential quotient of solving the variable time of the sample. The main emphasis of DCM is placed on substituting the differential quotient of a small neighborhood of the sample time of the computer system for the differential operator of the equations studied. The approximate analytical chaotic solutions of the nonlinear differential equations governing the high-dimensional dynamic system can be obtained by the method proposed. In order to increase the computational efficiency of the method proposed, a thermodynamics modeling method is used to decouple the variable and reduce the dimension of the system studied. The validity of the method proposed for obtaining approximate analytical chaotic solutions of the nonlinear differential equations is illustrated by the example of a turbo-generator system. This work is applied to solving a type of nonlinear system of which the dynamic behaviors can be described by nonlinear differential equations.     

Weakly perturbed boundary-value problems for differential equations in a Banach space

We consider boundary-value problems for a system of ordinary differential equations with a small parameter ε in the equations and boundary conditions. We establish conditions for the bifurcation of solutions of a weakly perturbed linear boundary-value problem in a Banach space.     

Unsteady three‐dimensional boundary layer flow due to a stretching surface in a micropolar fluid

In this paper, the unsteady three‐dimensional boundary layer flow due to a stretching surface in a viscous and incompressible micropolar fluid is considered. The partial differential equations governing the unsteady laminar boundary layer flow are solved numerically using an implicit finite‐difference scheme. The numerical solutions are obtained which are uniformly valid for all dimensionless time from initial unsteady‐state flow to final steady‐state flow in the whole spatial region. The equations for the initial unsteady‐state flow are also solved analytically. It is found that there is a smooth transition from the small‐time solution to the large‐time solution. The features of the flow for different values of the governing parameters are analyzed and discussed. The solutions of interest for the skin friction coefficient with various values of the stretching parameter c and material parameter K are presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Hydromagnetic flow of a dusty fluid over a stretching sheet

Analysis of hydromagnetic flow of a dusty fluid over a stretching sheet is carried out with a view to throw adequate light on the effects of fluid-particle interaction, particle loading, and suction on the flow characteristics. The equations of motion are reduced to coupled non-linear ordinary differential equations by similarity transformations. These coupled non-linear ordinary differential equations are solved numerically on an IBM 4381 with double precession, using a variable order, variable step-size finite-difference method. The numerical solutions are compared with their approximate solutions, obtained by a perturbation technique. For small values of β the exact (numerical) solution is in close agreement with that of the analytical (approximate) solution. It is observed that, even in the presence of a transverse magnetic field and suction, the transverse velocity of both the fluid and particle G phases decreases with an increase in the fluid-particle interaction parameter, β, or the particle-loading parameter, k. Moreover, the particle density is maximum at the surface of the stretching sheet, and the shearing stress increases with an increase in β or k.     

Travelling Breathers in Klein-Gordon Lattices as Homoclinic Orbits to <Emphasis Type="BoldItalic">p</Emphasis>-Tori

Klein-Gordon chains are one-dimensional lattices of nonlinear oscillators in an anharmonic on-site potential, linearly coupled with their first neighbors. In this paper, we study the existence in such networks of spatially localized solutions, which appear time periodic in a referential in translation at constant velocity. These solutions are called travelling breathers. In the case of travelling wave solutions, the existence of exact solutions has been obtained by Iooss and Kirchgässner. Formal multiscale expansions have been used by Remoissenet to derive approximate solutions of travelling breathers in the form of modulated plane waves. James and Sire have studied the existence of specific travelling breather solutions, consisting in pulsating travelling waves which are exactly translated of 2 lattice sites after a fixed propagation time T. In this paper, we generalize this approach to pulsating travelling waves which are exactly translated of p≥ 3 sites after a given time T p being arbitrary. By formulating the problem as a dynamical system, one is able to reduce the system locally to a finite dimensional set of ordinary differential equations (ODE), whose dimension depends on the parameter values of the problem. We prove that the principal part of this system of ODE admits homoclinic connections to p-tori under general conditions on the potential. One can obtain leading order approximations of these homoclinic connections and these orbits should correspond, for the oscillator chain, to small amplitude travelling breather solutions superposed on an exponentially small quasi-periodic tail.     

Application of Uniform Asymptotics to the Second Painlevé Transcendent

In this work we propose a new method for investigating connection problems for the class of nonlinear second‐order differential equations known as the Painlevé equations. Such problems can be characterized by the question as to how the asymptotic behaviours of solutions are related as the independent variable is allowed to pass towards infinity along different directions in the complex plane. Connection problems have been previously tackled by a variety of methods. Frequently these are based on the ideas of isomonodromic deformation and the matching of WKB solutions. However, the implementation of these methods often tends to be heuristic in nature and so the task of rigorising the process is complicated. The method we propose here develops uniform approximations to solutions. This removes the need to match solutions, is rigorous, and can lead to the solution of connection problems with minimal computational effort. Our method relies on finding uniform approximations of differ ential equations of the generic form as the complex‐valued parameter . The details of the treatment rely heavily on the locations of the zeros of the function F in this limit. If they are isolated, then a uniform approximation to solutions can be derived in terms of Airy functions of suitable argument. On the other hand, if two of the zeros of F coalesce as , then an approximation can be derived in terms of parabolic cylinder functions. In this paper we discuss both cases, but illustrate our technique in action by applying the parabolic cylinder case to the “classical” connection problem associated with the second Painlevé transcendent. Future papers will show how the technique can be applied with very little change to the other Painlevé equations, and to the wider problem of the asymptotic behavio ur of the general solution to any of these equations. (Accepted May 15, 1997)