Showing papers on "Method of matched asymptotic expansions published in 2002"

Kolmogorov two-thirds law by matched asymptotic expansion

Abstract: The Kolmogorov two-thirds law is derived for large Reynolds number isotropic turbulence by the method of matched asymptotic expansions. Inner and outer variables are derived from the Karman–Howarth equation by using the von Karman self-preservation hypothesis. Matching the resulting large Reynolds number asymptotic expansions yields the Kolmogorov law. The Kolmogorov similarity hypotheses are not assumed; only the Navier–Stokes equation is employed and the assumption that dissipation is finite. This indicates that the Kolmogorov results are a direct consequence of the Navier–Stokes equations.

The Dynamics of Multispike Solutions to the One-Dimensional Gierer--Meinhardt Model

Abstract: The dynamical behavior of spike-type solutions to a simplified form of the Gierer--Meinhardt activator-inhibitor model in a one-dimensional domain is studied asymptotically and numerically in the limit of small activator diffusivity $\varepsilon$. In the limit $\varepsilon \to 0$, a quasi-equilibrium solution for the activator concentration that has n localized peaks, or spikes, is constructed asymptotically using the method of matched asymptotic expansions. For an initial condition of this form, a differential-algebraicsystem of equations describing the evolution of the spike locations is derived. The equilibrium solutions for this system are discussed. The spikes are shown to evolve on a slow time scale $\tau=\varepsilon^2 t$ towards a stable equilibrium, provided that the inhibitor diffusivity D is below some threshold and that a certain stability criterion on the quasi-equilibrium solution is satisfied throughout the slow dynamics. If this stability condition is not satisfied initially or else is no l...

Well-posedness and Asymptotic Behaviour of Non-autonomous Linear Evolution Equations

Abstract: There is a striking difference between autonomous and non-autonomous linear evolution equations. Autonomous problems are well understood in the framework of strongly continuous operator semigroups and their generalizations. The Hille-Yosida type theorems settle the question of well-posedness to a great extend, many perturbation and approximation results have been established, and for a large class of problems the asymptotic behaviour can be studied on the basis of spectral theory and transform methods. In these and many other areas semigroup theory has reached a considerable degree of maturity, and its applications thrive in plenty of fields.

Perturbation Methods for Differential Equations

Abstract: Preface Asymptotic Series and Expansions Regular Perturbation Methods The Method of Strained Coordinates/Parameters Method of Averaging The Method of Matched Asymptotic Expansions Method of Multiple Scales Miscellaneous Perturbation Methods References Answers to Selected Problems Index Permissions

Asymptotic expansions of lattice Green's functions

Abstract: We derive asymptotic expansions for the Green functions associated with coercive difference equations on general lattices. These expansions lead to rigorous methods for approximating the lattice Green functions by polyharmonic rational functions.

Two‐Point Taylor Expansions of Analytic Functions

Abstract: Taylor expansions of analytic functions are considered with respect to two points Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals The method is also used for obtaining Laurent expansions in two points

The inertial lift on a small particle in a weak-shear parabolic flow

Abstract: The lateral migration of a small spherical particle translating within a vertical channel flow with large channel Reynolds numbers Rc is investigated. The weak-shear case is studied when the ratio of the slip velocity to maximum velocity of the channel flow, Vs, is finite, while two other dimensionless groups, particle Reynolds number Rs and Rc−1, are asymptotically small. The disturbance flow at large distances from the sphere is governed by Oseen-like equations. The ratio of Oseen length to channel width is e=ls/l=(Rc|Vs|)−1≪1, i.e., the Oseen region is only a small part of the channel while the major portion of the disturbance flow is inviscid. A solution of the governing equations is constructed in terms of two-dimensional Fourier transform of the disturbance field in a plane parallel to the channel walls. The ordinary differential equation for Fourier transform of lateral velocity Γz(k,z) is solved using the method of matched asymptotic expansions based on e. Several domains in (k,z) space are distin...

A quasi-steady approach to the instability of time-dependent flows in pipes

Abstract: Asymptotic solutions for unsteady one-dimensional axisymmetric laminar flow in a pipe subject to rapid deceleration and/or acceleration are derived and their stability investigated using linear and weakly nonlinear analysis. In particular, base flow solutions for unsteady one-dimensional axisymmetric laminar flow in a pipe are derived by the method of matched asymptotic expansions. The solutions are valid for short times and can be successfully applied to the case of an arbitrary (but unidirectional) axisymmetric initial velocity distribution. Excellent agreement between asymptotic and analytical solutions for the case of an instantaneous pipe blockage is found for small time intervals. Linear stability of the base flow solutions obtained from the asymptotic expansions to a three-dimensional perturbation is investigated and the results are used to re-interpret the experimental results of Das & Arakeri (1998). Comparison of the neutral stability curves computed with and without the planar channel assumption shows that this assumption is accurate when the ratio of the unsteady boundary layer thickness to radius (i.e. δ1/R) is small but becomes unacceptable when this ratio exceeds 0.3. Both the current analysis and the experiments show that the flow instability is non-axisymmetric for δ1/R = 0.55 and 0.85. In addition, when δ1/R = 0.18 and 0.39, the neutral stability curves for n = 0 and n = 1 are found to be close to one another at all times but the most unstable mode in these two cases is the axisymmetric mode. The accuracy of the quasi-steady assumption, employed both in this research and in that of Das & Arakeri (1998), is supported by the fact that the results obtained under this assumption show satisfactory agreement with the experimental features such as type of instability and spacing between vortices. In addition, the computations show that the ratio of the rate of growth of perturbations to the rate of change of the base flow is much larger than 1 for all cases considered, providing further support for the quasi-steady assumption. The neutral stability curves obtained from linear stability analysis suggest that a weakly nonlinear approach can be used in order to study further development of instability. Weakly nonlinear analysis shows that the amplitude of the most unstable mode is governed by the complex Ginzburg–Landau equation which reduces to the Landau equation if the amplitude is a function of time only. The coefficients of the Landau equation are calculated for two cases of the experimental data given by Das & Arakeri (1998). It is shown that the real part of the Landau constant is positive in both cases. Therefore, finite-amplitude equilibrium is possible. These results are in qualitative agreement with experimental data of Das & Arakeri (1998).

An asymptotic numerical method for singularly perturbed third-order ordinary differential equations of convection-diffusion type

Abstract: Singularly perturbed two-point boundary value problems (SPBVPs) for third-order ordinary differential equations (ODEs) with a small parameter multiplying the highest derivative are considered. A numerical method is suggested in this paper to solve such problems. In this method, the given BVP is transformed into a weakly coupled system of two ODEs subject to suitable initial and boundary conditions. Then, the computational method, presented in this paper, is applied to this system. In this method, we reduce the weakly coupled system into a decoupled system. Then, to solve this decoupled system numerically, we apply a ‘boundary value technique (BVT)’, in which the domain of definition of the differential equation is divided into two nonoverlapping subintervals called inner and outer regions. Then, we solve the decoupled system over these regions as two point boundary value problems. An exponentially fitted finite difference scheme is used in the inner region and a classical finite difference scheme, in the outer region. The boundary conditions at the transition point are obtained using the zero-order asymptotic expansion approximation of the solution of the problem. This computational method is distinguished by the facts that the decoupling reduces the computational time very much and it is well suited for parallel computing. This method can be extended to a system of two ordinary differential equations, of which, one is of first order and the other is of second order. Numerical examples are given to illustrate the method.

On combined asymptotic expansions in singular perturbations

Abstract: A structured and synthetic presentation of Vasil'eva's combined expansions is proposed. These expansions take into account the limit layer and the slow motion of solutions of a singularly perturbed dierential equation. An asymptotic formula is established which gives the distance between two exponentially close solutions. An \input-output" relation around a canard solution is carried out in the case of turning points. We also study the distance between two canard values of dierential equations with given parameter. We apply our study to the Liouville equation and to the splitting of energy levels in the one-dimensional steady Schrodinger equation in the double well symmetric case. The structured nature of our approach allows us to give eective symbolic algorithms.

A second-order scheme for singularly perturbed differential equations with discontinuous source term

Abstract: Abstract A Galerkin finite element method that uses piecewise linear functions on Shishkin- and Bakhvalov–Shishkin-type of meshes is applied to a linear reaction-diffusion equation with discontinuous source term. The method is shown to be convergent, uniformly in the perturbation parameter, of order N –2 ln2 N for the Shishkin-type mesh and N –2 for the Bakhvalov–Shishkin-type mesh, where N is the mesh size number. Numerical experiments support our theoretical results.

Three-dimensional motion of a vortex filament and its relation to the localized induction hierarchy

Abstract: Three-dimensional motion of a slender vortex tube, embedded in an inviscid incompressible fluid, is investigated under the localized induction approximation for the Euler equations. Using the method of matched asymptotic expansions in a small parameter e, the ratio of core radius to curvature radius, the velocity of a vortex filament is derived to O(e3), whereby the influence of elliptical deformation of the core due to the self-induced strain is taken into account. It is found that there is an integrable line in the core whose evolution obeys a summation of the first and third terms of the localized induction hierarchy.

An asymptotic membrane-like theory for long wave motion in a pre-stressed elastic plate

Abstract: An asymptotically consistent two–dimensional theory is developed to help elucidate dynamic response in finitely deformed layers. The layers are composed of incompressible elastic material, with the theory appropriate for long–wave motion associated with the fundamental mode and derived in respect of the most general appropriate strain energy function. Leading–order and refined higher–order equations for the mid–surface deflection are derived. In the case of zero normal initial static stress and in–plane tension, the leading–order equation reduces to the classical membrane equation, with its refined counterpart also being obtained. The theory is applied to a one–dimensional edge loading problem for a semi–infinite plate. In doing so, the leading– and higher–order governing equations are used as inner and outer asymptotic expansions, the latter valid within the vicinity of the associated quasi–front. A solution is derived by using the method of matched asymptotic expansions.

Interior Layers for Second-Order Singular Equations

Abstract: Our aim in this article is to study singularly perturbed problems which display boundary layers in the interior of the domain. These interior boundary layers which supplement the usual boundary layers at the boundary, are generated by discontinuities in the data. Second-order linear elliptic one-dimensional and multi-dimensional problems are considered in this article.

Asymptotic behavior of solutions of a class of second order quasilinear ordinary differential equations

Abstract: We study asymptotic behavior of solutions of a class of second order quasilinear ordinary differential equations. All solutions are classified into six types by means of their asymptotic behavior. Necessary and/or sufficient conditions are given for such equations to possess a solution of each of the six types.

Singularly perturbed volterra integro-differential equations

Abstract: Several investigations have been made on singularly perturbed integral equations. This paper aims at presenting an algorithm for the construction of asymptotic solutions and then provide a proof asymptotic correctness to singularly perturbed systems of Volterra integro-differential equations. Mathematics Subject Classification (2000) : 45D05, 45F05, 45J05, 41A60. Key words: Singular perturbation; Volterra integro-differential equations Quaestiones Mathematicae 25 (2002), 229-248

Order Reduction Is Invalid for Singularly Perturbed Control Problems with a Vector Fast Variable

Abstract: The order reduction method for singularly perturbed optimal control systems consists of employing the system obtained while setting the small parameter to zero. In many situations the differential-algebraic system thus obtained indeed provides an appropriate approximation to the singularly perturbed optimal control problem under consideration. In this paper we show, however, that the set of singularly perturbed optimal control systems for which the order reduction approach is invalid is dense (in the L∞ norm) in the class of systems which we consider. This is established under the assumption that the fast variable in the singularly perturbed system is not a scalar.

Boundary Value Technique for Finding Numerical Solution to Boundary Value Problems for Third Order Singularly Perturbed Ordinary Differential Equations

Abstract: A class of singularly perturbed two point boundary value problems (BVPs) for third order ordinary differential equations is considered. The BVP is reduced to a weakly coupled system of one first order Ordinary Differential Equation (ODE) with a suitable initial condition and one second order singularly perturbed ODE subject to boundary conditions. In order to solve this system, a computational method is suggested in this paper. This method combines an exponentially fitted finite difference scheme and a classical finite difference scheme. The proposed method is distinguished by the fact that, first we divide the domain of definition of the differential equation into three subintervals called inner and outer regions. Then we solve the boundary value problem over these regions as two point boundary value problems. The terminal boundary conditions of the inner regions are obtained using zero order asymptotic expansion approximation of the solution of the problem. The present method can be extended to system of two equations, of which, one is a first order ODE and the other is a singularly perturbed second order ODE. Examples are presented to illustrate the method.

Singularly Perturbed Vector and Scalar Nonlinear Schrödinger Equations with Persistent Homoclinic Orbits

Abstract: Singularly perturbed vector nonlinear Schrodinger equations (PVNLS) are investigated. Emphasis is placed upon the relation with their restriction: the singularly perturbed scalar nonlinear Schrodinger equation (PNLS) studied in [1]. It turns out that the persistent homoclinic orbit for the PNLS [1] is the only one for the PVNLS, asymptotic to the same saddle.

Asymptotic Properties of Input-Output Operators Norm Associated with Singularly Perturbed Systems with Multiplicative White Noise

Abstract: In this paper, we study the problem of the asymptotic property of the norm of input-output operators related to a class of singularly perturbed stochastic linear systems. The system is under perturbation of multiplicative white noise. By using reduction order and boundary layer techniques, it is shown that the norm of the operator of the perturbed system is less than a given number $\gamma$ when the small perturbation $\varepsilon$ tends to zero if both the related norms of the reduced subsystem and the boundary layer subsystem are less than $\gamma$. Furthermore, a stabilizing robust controller is designed, which is independent of perturbation $\varepsilon$.

A Class Of Nonlinear Nonlocal Singularly Perturbed Problems For Reaction Diffusion Equations

Abstract: A class of nonlinear nonlocal singularly perturbed problems for reaction diffusion equations are considered. Under suitable conditions, using theory of differential inequalities the asymptotic behaviors of solution for the initial boundary value problems are studied.

Singularly perturbed Volterra integral equations with weakly singular kernels

Abstract: We consider finding asymptotic solutions of the singularly perturbed linear Volterra integral equations with weakly singular kernels. An interesting aspect of these problems is that the discontinuity of the kernel causes layer solutions to decay algebraically rather than exponentially within the initial (boundary) layer. To analyse this phenomenon, the paper demonstrates the similarity that these solutions have to a special function called the Mittag-Leffler function.

Optimal control of discrete-time singularly perturbed systems

Abstract: In this paper, we present a new algorithm for solving the optimal control of discrete-time singularly perturbed systems. The main idea of this algorithm is based on two steps. First, the Hamiltonian difference equation is reduced to the backward recursive form rather than the forward recursive form. Second, the bilinear transformation is applied to transform the derived non-symmetric discrete-time Riccati equations into continuous-time equations. In order to improve the efficiency of this scheme, two matrix permutations are introduced into this algorithm by taking into account the previous work of Gajic and Shen (1991). Therefore, substantial numerical advantages are gained; namely, computation and memory requirements. The F-8 aircraft model is used to illustrate the efficiency of the proposed method.