Perturbed Nonlinear Oscillations

We consider one-phase (formal) asymptotic solutions in the Kuzmak-Whitham form for the nonlinear Klein-Gordon equation and for the Korteweg-de Vries equation. In this case, the leading asymptotic expansion term has the form X(S(x, t)/h+Φ(x, t), I(x, t), x, t) +O(h), where h ≪ 1 is a small parameter and the phase S}(x, t) and slowly changing parameters I(x, t) are to be found from the system of “averaged” Whitham equations. We obtain the equations for the phase shift Φ(x, t) by studying the second-order correction to the leading term. The corresponding procedure for finding the phase shift is then nonuniform with respect to the transition to a linear (and weakly nonlinear) case. Our observation, which essentially follows from papers by Haberman and collaborators, is that if we incorporate the phase shift Φ into the phase and adjust the parameter Ĩ by setting \( \tilde S \) = S +hΦ+O(h 2),Ĩ = I + hI 1 + O(h 2), then the functions \( \tilde S \)(x, t, h) and Ĩ(x, t, h) become solutions of the Cauchy problem for the same Whitham system but with modified initial conditions. These functions completely determine the leading asymptotic term, which is X(\( \tilde S \)(x, t, h)/h, Ĩ(x, t, h), x, t) + O(h).

Singular perturbation of boundary value problem for quasilinear third-order ordinary differential equations involving two small parameters

A uniformly valid asymptotic expansion for integrals of the form \[ F(t,\nu ) = e^{\nu h(t)} \int_t^\infty {e^{ - \nu h(x)} } g(x)x^{\alpha - 1} dx, \] where $0 < \alpha \leqq 1$ and $h(x)$ has a zero of order $r \geqq 1$ at $x = 0$, is established. The result, which generalizes a well-known one for $h(x) = x$, also confirms the formal matched asymptotic expansion solution of the equivalent singular perturbation problem $ - y' + \nu h'(t)y = g(t)t^{\alpha - 1} $, $y(\infty ) = 0$. Comparable matched expansion results are derived for related integrals, including one from Bessel function theory, which do not satisfy differential equations.

Modelling physical problems in mathematical form yields the governing equations that may be linear or nonlinear for known and unknown boundaries. The exact solution for those equations may or may not be obtained easily. Hence we seek an analytical approximation solution in terms of asymptotic expansion. In this study, we focus on a singular perturbation in second order ordinary differential equations. Solutions to several perturbed ordinary differential equations are obtained in terms of asymptotic expansion. The aim of this work is to find an approximate analytical solution using the classical method of matched asymptotic expansion (MMAE). The Mathematica computer algebra system is used to perform the algebraic computations. The details procedures will be discussed and the underlying concepts and principles of the MMAE will be clarified. Perturbation problem for linear equation that occurs at one boundary and two boundary layers are discussed. Approximate analytical solution obtained for both cases are illustrated by graph using selected parameter by showing the outer, inner and composite solution separately. Then, the composite solution will be compare to the exact solution to show their accuracy by graph. By comparison, MMAE is found to be one of the best methods to solve singular perturbation problems in second order ordinary differential equation since the results obtained are very close to the exact solution.

Averaging of the system of equations of motion of a viscous fluid in a porous medium

This paper uses the method of matched asymptotic expansions to derive asymptotic solutions to various problems in nonlinear acoustics. Model equations, generalizing the well known Burgers equation to include effects of cylindrical or spherical spreading and of non-equilibrium relaxation, are given and regarded as governing the propagation. Solutions are sought for initial or boundary conditions of N-wave or harmonic wave form. For a thermoviscous medium, the small parameter upon which the asymptotic expansions are predicated is an inverse acoustic Reynolds number; for a relaxing medium it is the product of wave frequency and relaxation time. The complete asymptotic solution for N-waves in a thermoviscous fluid is known, in the case of plane motion, from the Cole—Hopf solution of Burgers’s equation. Here a similarly complete solution is found for spherical N-waves, with the exception of one region of space—time in which an irreducible nonlinear problem remains unsolved. In this region the outer limiting behaviour is, nevertheless, determined, so that the solutions in all other regions are completely fixed. For cylindrical N-waves an irreducible problem again results, but the motion can be followed right through into its ‘old age’ phase aside from an undetermined purely numerical constant. Correct results are obtained here for the ‘correction due to diffusivity' to the weak-shock theory prediction of shock centre location for plane, cylindrical and spherical N-waves. These results indicate a non-uniformity in weak shock theory at large times, and also, in the case of spherical N-waves, reveal a large time non-uniformity in the Taylor shock solution. Harmonic waves, plane, cylindrical and spherical, in thermoviscous fluids and relaxing fluids are considered, and the asymptotic solutions are found to leading order in most of the many overlapping asymptotic regions of space-time. A single dimensionless function remains undetermined in the important case of spherical harmonic waves. We have also been unable to find scalings and differential equations describing precisely how a discontinuity is formed at the front of a partly dispersed shock in a relaxing gas, though the shock centre is located for both fully and partly dispersed shocks. The harmonic wave solutions unify and extend certain solutions (the Fay, Fubini and old-age solutions) which are well known in the nonlinear acoustics literature, and the amplitude saturation and scaling laws for the old age regime are in accord with experiments on high amplitude spherical waves in water.

Dynamics of a prestressed stiff layer on an elastic half space: filtering and band gap characteristics of periodic structural models derived from long-wave asymptotics

The Gevrey asymptotics in the initial value problem for singularly perturbed nonlinear differential equations

Asymptotic solutions for mixed mode loading of cracks and wedges in power law hardening materials

The asymptotic homogenization method is applied to obtain formal asymptotic solution and the homogenized solution of a Dirichlet boundary-value problem for an elliptic equation with rapidly os- cillating coefficients. The proximity of the formal asymptotic solution and the homogenized solution to the exact solution is proved, which provides the mathematical justification of the homogenization pro- cess. Preservation of the symmetry and positive-definiteness of the effective coefficient in the homogenized problem is also proved. An example is presented in order to illustrate the theoretical results.

This paper reports the first formal asymptotic solution to the line contact problem of elastohydrodynamic lubrication (EHL), a fundamental problem describing the elastic deformation of lubricated rolling elements such as roller bearings, gear teeth and other contacts of similar geometry. The asymptotic régime considered is that of small λ, a dimensionless parameter proportional to rolling speed, viscosity and the elastic modulus. The solution is shown to possess four regions: a zone where the lubricating film is both thin and slowly narrowing and which is closely related to the contact area that occurs in the absence of lubricant, an upstream inlet zone of low pressure, and two thin layers on either side of the contact zone. The solutions in the first two just-mentioned zones are given by simple analytical expressions. The solutions in the two thin layers are obtained from two universal functions obtained by Bissett &amp; Spence (Proc. R. Soc. Lond. A 424, 409 (1989)). Although these two functions, related to the local film thickness, are obtained by numerical techniques by Bissett &amp; Spence, it should be emphasized that all cases in the asymptotic régime considered are hereby solved definitively without recourse to further computation. Although some features of this structure have been suggested by other solution approaches, generally, these are numerical or ad hoc approximations. See the texts by Johnson (Contact Mechanics, pp. 328 (1985)) and Dowson &amp; Higginson (Elasto-hydrodynamic lubrication (1977)), this work provides a formal mathematical basis for understanding most of the principal features of EHL. The solution provides a simple formula for minimum film thickness and displays the sharp narrowing of the lubricating film in the thin layer near the exit. In the basic asymptotic solution provided here, the dimensionless pressure-viscosity coefficient, α, is assumed to be O(1), and in this parameter régime, no pressure spike will occur. By comparing with the work of Hooke (J. mech. Engng Sci. 19(4), 149 (1977)), we can show that an incipient pressure spike occurs when α becomes as large as O(λ-1/5). However, asymptotic solutions in this latter parameter régime require new numerical solutions for each case of interest and are not pursued here.

We address the homogenisation of a linear viscoelastic and hierarchical composite material in a one-dimensional (1D) framework via a three-scale asymptotic homogenisation method. We consider a family of heterogeneous problems with periodic, rapidly-oscillating and piece-wise coefficients that model a structure with two hierarchical levels of organisation. Here, we assume continuity contact conditions at the interfaces among the constituents and set a straightforward geometrical configuration in order to gain a better insight of the multiscale problem. The main goal is to provide a general overview of the procedure and validate the approach by means of a comparison between the solution of the original heterogeneous problem, the homogenised problem and the formal asymptotic solution. In addition, we show that the three-scale approach presents a clear improvement over the recursive two-scale one, and we illustrate the convergence of the solutions towards the solution of the homogenised problem when the asymptotic parameters approach.

Microfiltration of particles is modelled by the motion of particles embeddedin a Stokes flow near a porous membrane in which Darcy equations apply.Stokes flow also applies on the other side of the membrane. A pressure gradient isapplied across the membrane. Beavers and Joseph slip boundary condition applies along the membrane surfaces.This coupled Stokes-Darcy problem is solved by a perturbation method, considering that the particle size is muchlarger than the pores of the membrane. The formal asymptotic solution is developed in detail up to 3rd order.The method is applied to the example case of a spherical particle moving normal to a membrane. The solution,limited here to an impermeable slip surface (described from 3rd order expansion), uses as an intermediate step theboundary integral technique for Stokes flow near an impermeable surface with a no-slip boundary condition. Results of the perturbation solution are in good agreement with O'Neill and Bhatt analytical solution for this case.

The singularly perturbed systems for the nonlinear three-species food-chain reaction systems with time delay are considered. By using the method of the stretched variable, the formal asymptotic solution is obtained under suitable conditions.

In this paper, we consider the nonlinear boundary value problems for ordinary differential equations containing a small parameter by means of a modified method of multiple scales in weaker conditions than those used by other authors. The existence of solutions is proved, the asymptotic solutions are constructed, and the remainder terms are estimated by using a simpler method. The results in [1]—[6] are thus improved and extended.