Showing papers on "Method of matched asymptotic expansions published in 2017"
TL;DR: In this paper, asymptotic properties of a class of fourth-order delay differential equations are studied. But the results in this paper do not generalize some previous results, but also improve the earlier ones.
Abstract: The aim of this work is to study asymptotic properties of a class of fourth-order delay differential equations. Our results in this paper not only generalize some previous results, but also improve the earlier ones. Examples are considered to elucidate the main results.
59 citations
TL;DR: In this article, the asymptotic form of fixed-point solutions in functional truncations, in particular the f(R) approximation, has been studied both physically and mathematically.
Abstract: As already hinted at in Sect. 1.4.3, in order to understand fixed point solutions of the RG equation both physically and mathematically it is necessary to study their asymptotic behaviour. In this chapter we explain how to find the asymptotic form of fixed-point solutions in functional truncations, in particular the f(R) approximation.
53 citations
TL;DR: In this paper, it was shown that if the leading-order spin and quadrupole moment vanish, then through second order in its mass it moves on a geodesic of a smooth, locally causal vacuum metric defined in its local neighbourhood.
Abstract: When a small, uncharged, compact object is immersed in an external background spacetime, at zeroth order in its mass it moves as a test particle in the background. At linear order, its own gravitational field alters the geometry around it, and it moves instead as a test particle in a certain effective metric satisfying the linearized vacuum Einstein equation. In the letter [Phys. Rev. Lett. 109, 051101 (2012)], using a method of matched asymptotic expansions, I showed that the same statement holds true at second order: if the object's leading-order spin and quadrupole moment vanish, then through second order in its mass it moves on a geodesic of a certain smooth, locally causal vacuum metric defined in its local neighbourhood. Here I present the complete details of the derivation of that result. In addition, I extend the result, which had previously been derived in gauges smoothly related to Lorenz, to a class of highly regular gauges that should be optimal for numerical self-force computations.
50 citations
TL;DR: This paper discusses the analysis of a cross-diffusion PDE system for a mixture of hard spheres derived in Bruna and Chapman from a stochastic system of interacting Brownian particles using the method of matched asymptotic expansions.
Abstract: In this paper, we discuss the analysis of a cross-diffusion PDE system for a mixture of hard spheres, which was derived in Bruna and Chapman (J Chem Phys 137:204116-1–204116-16, 2012a) from a stochastic system of interacting Brownian particles using the method of matched asymptotic expansions. The resulting cross-diffusion system is valid in the limit of small volume fraction of particles. While the system has a gradient flow structure in the symmetric case of all particles having the same size and diffusivity, this is not valid in general. We discuss local stability and global existence for the symmetric case using the gradient flow structure and entropy variable techniques. For the general case, we introduce the concept of an asymptotic gradient flow structure and show how it can be used to study the behavior close to equilibrium. Finally, we illustrate the behavior of the model with various numerical simulations.
29 citations
TL;DR: A numerical method comprising a standard finite difference scheme on a rectangular piecewise uniform fitted mesh of Nx × Nt elements condensing in the boundary layers is suggested and it is proved to be parameter-uniform.
21 citations
TL;DR: A high-order uniformly convergent method to solve singularly perturbed delay parabolic convection diffusion problems exhibiting a regular boundary layer is introduced and it is shown that the method is -uniformly convergent of second-order accurate in time and in the spatial direction.
Abstract: In this article, we aim to introduce a high-order uniformly convergent method to solve singularly perturbed delay parabolic convection diffusion problems exhibiting a regular boundary layer. The domain is discretized by a uniform mesh in the time direction and a piecewise-uniform Shishkin mesh for the spatial direction. We use the Crank–Nicolson method for the time derivative and we develop a fourth-order compact difference method to solve the set of ordinary differential equations at each time level. The stability analysis and the truncation error are discussed. Parameter-uniform error estimates are derived and it is shown that the method is e-uniformly convergent of second-order accurate in time, and in the spatial direction it is of second-order outside region of boundary layer, and of almost fourth-order inside the layer region. Numerical examples are presented to verify the theoretical results and to confirm the efficiency and high accuracy of the proposed method.
19 citations
TL;DR: A new numerical methodology that is based on a spectral method that uses an expansion in Hermite functions to approximate the solution of an appropriate Poisson equation, which is used in order to calculate the coefficients of the homogenized equation.
Abstract: This paper presents a new method for the solution of multiscale stochastic differential equations at the diffusive time scale. In contrast to averaging-based methods, e.g., the heterogeneous multiscale method (HMM) or the equation-free method, which rely on Monte Carlo simulations, in this paper we introduce a new numerical methodology that is based on a spectral method. In particular, we use an expansion in Hermite functions to approximate the solution of an appropriate Poisson equation, which is used in order to calculate the coefficients of the homogenized equation. Spectral convergence is proved under suitable assumptions. Numerical experiments corroborate the theory and illustrate the performance of the method. A comparison with the HMM and an application to singularly perturbed stochastic PDEs are also presented.
18 citations
TL;DR: In this paper, the authors study the effective slip length for unidirectional flow over a superhydrophobic mattress of bubbles in the small-solid-fraction limit and elucidate the singularity of the slip length as relative to the periodicity.
Abstract: We study the effective slip length for unidirectional flow over a superhydrophobic mattress of bubbles in the small-solid-fraction limit . Using scaling arguments and utilising an ideal-flow analogy we elucidate the singularity of the slip length as : relative to the periodicity it scales as for protrusion angles and as for . We continue with a detailed asymptotic analysis using the method of matched asymptotic expansions, where ‘inner’ solutions valid close to the solid segments are matched with ‘outer’ solutions valid on the scale of the periodicity, where the bubbles protruding from the solid grooves appear to touch. The analysis yields asymptotic expansions for the effective slip length in each of the protrusion-angle regimes. These expansions overlap for intermediate protrusion angles, which allows us to form a uniformly valid approximation for arbitrary protrusion angles . We thereby explicitly describe the transition with increasing protrusion angle from a logarithmic to an algebraic small-solid-fraction slip-length singularity.
18 citations
TL;DR: In this article, a coarse-grained Bloch-wave dispersion problem is solved by a generalized Fourier series, whose singular asymptotics in the vicinities of scatterers yield the dispersion relation governing modes that are strongly perturbed from plane-wave solutions existing in the absence of the scatterer; there are also empty-lattice waves that are only weakly perturbed.
Abstract: We study waves governed by the planar Helmholtz equation, propagating in an infinite lattice of subwavelength Dirichlet scatterers, the periodicity being comparable to the wavelength. Applying the method of matched asymptotic expansions, the scatterers are effectively replaced by asymptotic point constraints. The resulting coarse-grained Bloch-wave dispersion problem is solved by a generalized Fourier series, whose singular asymptotics in the vicinities of scatterers yield the dispersion relation governing modes that are strongly perturbed from plane-wave solutions existing in the absence of the scatterers; there are also empty-lattice waves that are only weakly perturbed. Characterizing the latter is useful in interpreting and potentially designing the dispersion diagrams of such lattices. The method presented, which simplifies and expands on Krynkin and McIver [Waves Random Complex, 19 (2009), pp. 347--365], could be applied in the future to study more sophisticated designs entailing resonant subwavelen...
16 citations
TL;DR: In this article, Cauchy's integral formula is employed to compute the coefficient functions to a high order of accuracy for linear second-order differential equations having a large real parameter and turning point in the complex plane.
Abstract: Linear second-order differential equations having a large real parameter and turning point in the complex plane are considered. Classical asymptotic expansions for solutions involve the Airy function and its derivative, along with two infinite series, the coefficients of which are usually difficult to compute. By considering the series as asymptotic expansions for two explicitly defined analytic functions, Cauchy’s integral formula is employed to compute the coefficient functions to a high order of accuracy. The method employs a certain exponential form of Liouville–Green expansions for solutions of the differential equation, as well as for the Airy function. We illustrate the use of the method with the high accuracy computation of Airy-type expansions of Bessel functions of complex argument.
16 citations
TL;DR: In this article, it was shown that the Euler equation describing the motion of an ideal fluid in a given order is well-posed in a class of functions allowing spatial asymptotic expansions.
Abstract: In this paper we prove that the Euler equation describing the motion of an ideal fluid in $${\mathbb{R}^d}$$
is well-posed in a class of functions allowing spatial asymptotic expansions as $${|x|\to\infty}$$
of any a priori given order. These asymptotic expansions can involve log terms and lead to a family of conservation laws. Typically, the solutions of the Euler equation with rapidly decaying initial data develop non-trivial spatial asymptotic expansions of the type considered here.
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TL;DR: In this paper, the authors discuss the asymptotic behavior of the solutions of the classical heat equation posed in the whole Euclidean space, and present several methods of proof: first, the scaling method, then several versions of the representation method, and finally, the Boltzmann entropy method, coming from kinetic equations.
Abstract: In this expository work we discuss the asymptotic behaviour of the solutions of the classical heat equation posed in the whole Euclidean space.
After an introductory review of the main facts on the existence and properties of solutions, we proceed with the proofs of convergence to the Gaussian fundamental solution, a result that holds for all integrable solutions, and represents in the PDE setting the Central Limit Theorem of probability. We present several methods of proof: first, the scaling method. Then several versions of the representation method. This is followed by the functional analysis approach that leads to the famous related equations, Fokker-Planck and Ornstein-Uhlenbeck. The analysis of this connection is also given in rather complete form here. Finally, we present the Boltzmann entropy method, coming from kinetic equations.
The different methods are interesting because of the possible extension to prove the asymptotic behaviour or stabilization analysis for more general equations, linear or nonlinear. It all depends a lot on the particular features, and only one or some of the methods work in each case.Other settings of the Heat Equation are briefly discussed in Section 9 and a longer mention of results for different equations is done in Section 10.
TL;DR: In this article, the singularly perturbed boundary value problem for a linear second order delay differential equation is solved by an exponentially fitted finite difference scheme. But the stability of the scheme is investigated.
TL;DR: In this paper, a family of new finite difference (NFD) methods for solving the convection-diffusion equation with singularly perturbed parameters are considered, which can achieve the predicted convergence orders on uniform mesh regardless of the perturbed parameter.
Abstract: In this paper, a family of new finite difference (NFD) methods for solving the convection-diffusion equation with singularly perturbed parameters are considered. By taking account of infinite terms in the Taylor's expansions and using the triangle function theorem, we construct a series of NFD schemes for the one-dimensional problems firstly and derive the error estimates as well. Then, applying the ADI technique, the idea is extended to two dimensional equations. Besides no numerical oscillation, there are mainly three advantages for the proposed methods: one is that the schemes can achieve the predicted convergence orders on uniform mesh regardless of the perturbed parameter for 1D equations; Secondly, no matter which convergence order the scheme is, the generated linear systems have diagonal structures; Thirdly, the methods are easily expanded to the special mesh technique such as Shishkin mesh. Some numerical experiments are shown to verify the prediction.
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TL;DR: In this paper, a system of interacting Brownian particles subject to short-range repulsive potentials is considered, and a nonlinear diffusion equation is derived systematically in the dilute limit using the method of matched asymptotic expansions.
Abstract: A system of interacting Brownian particles subject to short-range repulsive potentials is considered. A continuum description in the form of a nonlinear diffusion equation is derived systematically in the dilute limit using the method of matched asymptotic expansions. Numerical simulations are performed to compare the results of the model with those of the commonly used mean-field and Kirkwood-superposition approximations, as well as with Monte Carlo simulation of the stochastic particle system, for various interaction potentials. Our approach works best for very repulsive short-range potentials, while the mean-field approximation is suitable for long-range interactions. The Kirkwood superposition approximation provides an accurate description for both short- and long-range potentials, but is considerably more computationally intensive.
TL;DR: In this article, a modified version of the barycentric interpolation collocation method is proposed to overcome this disadvantage and two numerical examples are provided to show the effectiveness of the present method.
Abstract: The barycentric interpolation collocation method is discussed in this paper, which is not valid for singularly perturbed delay partial differential equations. A modified version is proposed to overcome this disadvantage. Two numerical examples are provided to show the effectiveness of the present method.
TL;DR: In this paper, a quantitative asymptotic, stability estimate for solutions to nonlinear evolution equations has been derived by measuring the distance in time of a solution to a parabolic problem.
Abstract: We study a quantitative asymptotic, stability estimates for solutions to nonlinear evolution equations. More precisely, we measure the distance in time of a solution to a parabolic problem from a solution to a stationary one.
TL;DR: In this paper, the oscillatory and asymptotic behavior results for a class of third-order nonlinear neutral dynamic equations on time scales are presented, and the results can be extended to more general nonlinear NDEs of the type considered here.
Abstract: The oscillatory and asymptotic behavior results for a class of third-order nonlinear neutral dynamic equations on time scales are presented. The results obtained can be extended to more general third-order neutral dynamic equations of the type considered here. Examples are provided to illustrate the applicability of the results.
TL;DR: In this article, the Lomov regularization method is generalized to integro-partial differential equations and an algorithm for constructing regularized asymptotics is developed for the case in which the upper limit of the integral operator coincides with the differentiation variable.
Abstract: The Lomov regularization method [1] is generalized to integro-partial differential equations. It turns out that the regularization procedure essentially depends on the type of integral operator. The case in which the upper limit of the integral is not the differentiation variable is the most difficult one. It is not considered in the present paper. Only the case in which the upper limit of the integral operator coincides with the differentiation variable is studied. For such problems, an algorithm for constructing regularized asymptotics is developed.
TL;DR: In this paper, a fitted finite difference method on Shishkin mesh is suggested to solve a class of third order singularly perturbed boundary value problems for ordinary delay differential equations of convection-diffusion type.
Abstract: In this paper, a fitted finite difference method on Shishkin mesh is suggested to solve a class of third order singularly perturbed boundary value problems for ordinary delay differential equations of convection-diffusion type. Numerical solution converges uniformly to the exact solution. The order of convergence of the numerical method is almost first order. Numerical results are provided to illustrate the theoretical results.
TL;DR: This approach works best for very repulsive short-range potentials, while the mean-field approximation is suitable for long-range interactions and the Kirkwood-superposition approximation provides an accurate description for both short- and long- range potentials but is considerably more computationally intensive.
Abstract: A system of interacting Brownian particles subject to short-range repulsive potentials is considered. A continuum description in the form of a nonlinear diffusion equation is derived systematically in the dilute limit using the method of matched asymptotic expansions. Numerical simulations are performed to compare the results of the model with those of the commonly used mean-field and Kirkwood-superposition approximations, as well as with Monte Carlo simulation of the stochastic particle system, for various interaction potentials. Our approach works best for very repulsive short-range potentials, while the mean-field approximation is suitable for long-range interactions. The Kirkwood-superposition approximation provides an accurate description for both short- and long-range potentials but is considerably more computationally intensive.
TL;DR: In this article, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to the singularly perturbed ODE.
Abstract: In this work, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, we employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show efficiency of this numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with the results that are obtained by other reported methods.
TL;DR: In this article, a thorough asymptotic analysis of nonoscillatory solutions of the q-difference equation D q ( r ( t ) D q y (t ) + p( t ) y ( q t ) = 0 considered on the lattice { q k : k ∈ N 0 }, q > 1 ǫ, q > 0,
TL;DR: In this article, the authors studied the asymptotic behavior of solutions of a class of non-autonomous delay differential equations and showed that every solution of the equations is bounded and tends to a constant as t → + ∞.
01 Dec 2017
TL;DR: In this paper, the authors introduce and describe a numerical scheme for the approximate solutions of the one-dimensional singularly perturbed boundary-value problems based on Haar wavelets and its main characteristic is that, it converts the given problem into a system of algebraic equations that can be solved easily with any of the usual methods.
Abstract: The basic aim of this study is to introduce and describe a numerical scheme for the approximate solutions of the one-dimensional singularly perturbed boundary-value problems. The method is based on Haar wavelets and its main characteristic is that, it converts the given problem into a system of algebraic equations that can be solved easily with any of the usual methods. Another distinguishing feature of the this method is that unlike several other numerical methods, it does not require conversion of a boundary value problem into initial-value problem and hence eliminates the possibility of unstable solutions. To show the accuracy and the efficiency of the method, several benchmark problems are implemented and the comparisons are given with other methods existing in the recent literature. The results of numerical tests confirm that the Haar wavelet collocation method is superior to other existing ones and is highly accurate.
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05 Feb 2017
TL;DR: In this paper, a numerical method for singularly perturbed delay differential equations with layer or oscillatory behavior for which a small shift (δ) is in the reaction term is presented.
Abstract: In this paper, we presented numerical method for solving singularly perturbed delay differential equations with layer or oscillatory behaviour for which a small shift (δ) is in the reaction term. First, the given singularly perturbed delay reaction-diffusion equation is converted into an asymptotically equivalent singularly perturbed two point boundary value problem and then solved by using fourth order finite difference method. The stability and convergence of the method has been investigated. The numerical results have been tabulated and further to examine the effect of delay on the boundary layer and oscillatory behavior of the solution, graphs have been given for different values of δ. Both theoretical and numerical rate of convergence have been established and are observed to be in agreement for the present method. Briefly, the present method improves the findings of some existing numerical methods in the literature.
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22 Jun 2017TL;DR: In this paper, the fixed point index theory is applied to the Banach space to obtain fixed points of the integral operator in order to obtain solutions that satisfy some certain kind of asymptotic behavior.
Abstract: In this work we will consider integral equations defined on the whole real line and look for solutions which satisfy some certain kind of asymptotic behavior. To do that, we will define a suitable Banach space which, to the best of our knowledge, has never been used before. In order to obtain fixed points of the integral operator, we will consider the fixed point index theory and apply it to this new Banach space.
TL;DR: It is shown that the algorithm gives almost fourth uniform numerical approximations for the exact solution, and that for small values of the perturbation parameter just one iteration is required to achieve the almost fourth-order accuracy.
Abstract: We consider a system of singularly perturbed semilinear reaction–diffusion equations. To solve this system numerically we develop an overlapping Schwarz domain decomposition algorithm, where we use the asymptotic behaviour of the exact solution for domain partitioning as well as to construct the iterative algorithm. The algorithm is analysed by defining some auxiliary problems, that allows to prove the uniform convergence of the method in two steps, splitting the discretization error and the iteration error. It is shown that the algorithm gives almost fourth uniform numerical approximations for the exact solution. More importantly, it is shown that for small values of the perturbation parameter just one iteration is required to achieve the almost fourth-order accuracy. Numerical results support our theoretical findings.
TL;DR: In this paper, a septic B-spline method was developed for solving a self-adjoint singularly perturbed two-point boundary value problem, which is applied directly to the numerical solution of the problems without reducing its order.