Formal Asymptotic Expansion Research Articles

Correction to: Microstructure evolution and fluid transport in porous media: a formal asymptotic expansions approach

Correction to: Microstructure evolution and fluid transport in porous media: a formal asymptotic expansions approach

Microstructure evolution and fluid transport in porous media: a formal asymptotic expansions approach

Microstructure evolution and fluid transport in porous media: a formal asymptotic expansions approach

Upscaling and Effective Behavior for Two-Phase Porous-Medium Flow Using a Diffuse Interface Model

We investigate two-phase flow in porous media and derive a two-scale model, which incorporates pore-scale phase distribution and surface tension into the effective behavior at the larger Darcy scale. The free-boundary problem at the pore scale is modeled using a diffuse interface approach in the form of a coupled Allen–Cahn Navier–Stokes system with an additional momentum flux due to surface tension forces. Using periodic homogenization and formal asymptotic expansions, a two-scale model with cell problems for phase evolution and velocity contributions is derived. We investigate the computed effective parameters and their relation to the saturation for different fluid distributions, in comparison to commonly used relative permeability saturation curves. The two-scale model yields non-monotone relations for relative permeability and saturation. The strong dependence on local fluid distribution and effects captured by the cell problems highlights the importance of incorporating pore-scale information into the macro-scale equations.

Time-fractional Allen–Cahn equations versus powers of the mean curvature

We show by a formal asymptotic expansion that level sets of solutions of a time-fractional Allen–Cahn equation evolve by a geometric flow whose normal velocity is a positive power of the mean curvature.This connection is quite intriguing, since the original equation is nonlocal and the evolution of its solutions depends on all previous states, but the associated geometric flow is of purely local type, with no memory effect involved.

Nonlinear model reduction to temporally aperiodic spectral submanifolds.

We extend the theory of spectral submanifolds (SSMs) to general non-autonomous dynamical systems that are either weakly forced or slowly varying. Examples of such systems arise in structural dynamics, fluid-structure interactions, and control problems. The time-dependent SSMs we construct under these assumptions are normally hyperbolic and hence will persist for larger forcing and faster time dependence that are beyond the reach of our precise existence theory. For this reason, we also derive formal asymptotic expansions that, under explicitly verifiable nonresonance conditions, approximate SSMs and their aperiodic anchor trajectories accurately for stronger, faster, or even temporally discontinuous forcing. Reducing the dynamical system to these persisting SSMs provides a mathematically justified model- reduction technique for non-autonomous physical systems whose time dependence is moderate either in magnitude or speed. We illustrate the existence, persistence, and computation of temporally aperiodic SSMs in mechanical examples under chaotic forcing.

Compactons in Higher-Order Nesterenko's-Type Equations

Abstract A model describing propagation of waves in a prestressed granular media is considered. The model, having the form of evolutionary partial differential equation (PDE), is obtained from the system of ordinary differential equations (ODEs) describing dynamics of a chain of prestressed granules by means of formal asymptotic expansion. It is shown in our previous papers that in the lowest asymptotic approximation, in which both nonlinear effects and the presence of media structure are taken into account, the model equation possesses traveling wave (TW) solutions with compact support (compactons) manifesting soliton properties. In this paper, we study a higher-order evolutionary PDE obtained by taking into account previously discarded terms of the asymptotic expansion, as well as another PDE (called analogue), differing from the original one in the values of parameters and having compacton solutions expressed in analytical form. Numerical and analytical studies of both the higher-order model and its analogue allow to conclude that both models have compacton solutions exhibiting some properties of “true” solitons. This, in turn, testifies the stability of the previously used model with respect to the inclusion of the discarded terms of the asymptotic expansion.

A quasi-static model of a thermoelastic body reinforced by a thin thermoelastic inclusion

The problem of description of quasi-static behavior is studied for a planar thermoelastic body incorporating an inhomogeneity, which geometrically is a strip with a small cross-section. This problem contains a small positive parameter δ describing the thickness of the inhomogeneity, i.e., the size of the cross-section. Relying on the variational formulation of the problem, we investigate the behavior of solutions as δ tends to zero. As the result, by the version of the method of formal asymptotic expansions, we derive a closed limit model in which the inhomogeneity is thin (of zero width). After this, using the Galerkin method and the classical techniques of derivation of energy estimates, we prove existence, uniqueness, and stability of a weak solution to this model.

Double‐Scale Expansions for a Logarithmic Type Solution to a q‐Analog of a Singular Initial Value Problem

We examine a linear q−difference differential equation, which is singular in complex time t at the origin. Its coefficients are polynomial in time and bounded holomorphic on horizontal strips in one complex space variable. The equation under study represents a q−analog of a singular partial differential equation, recently investigated by the author, which comprises Fuchsian operators and entails a forcing term that combines polynomial and logarithmic type functions in time. A sectorial holomorphic solution to the equation is constructed as a double complete Laplace transform in both time t and its complex logarithm logt and Fourier inverse integral in space. For a particular choice of the forcing term, this solution turns out to solve some specific nonlinear q−difference differential equation with polynomial coefficients in some positive rational power of t. Asymptotic expansions of the solution relatively to time t are investigated. A Gevrey‐type expansion is exhibited in a logarithmic scale. Furthermore, a formal asymptotic expansion in power scale is displayed, revealing a new fine structure involving remainders with both Gevrey and q−Gevrey type growth.

Stages and main tasks of the century-long control theory and system identification development. Part III. The problem of complex systems identifying by inaccurate data

The article presents the main most noteworthy research results on the identification problem and methods that are widely used in solving identification problems. The most significant here is the problem of modeling especially complex systems, for which it is impossible, even in general, but in constructive form for identification, to write down a class of models in which the system under study has the most accurate description in which there is one of them adequate to the available data obtained from experiments. If the available knowledge on the object under study does not allow us to write down a suitable class of models in mathematical form, then asymptotic modeling should be used, as is done in computational mathematics. Instead of differential, integral and other equations, for example, algebraic equations approximating them are solved, the solutions of which approach the exact ones with increasing dimension and discretization step. Various forms of asymptotic expansions are used in identification, including rational approximation as applied to irrational and infinite-dimensional systems. However, in the presence of errors in the data, the use of asymptotic modeling leads to the fact that, with intentions to increase the accuracy of the resulting model due to its large dimension, in many cases the identification problems becomes incorrectly posed. Therefore, we have to look for a compromise or trade-off between bias and variance, which should determine the best-quality model. In addition to the modeling problem, the article discusses two concepts on the basis of which most existing methods for solving identification problems are based. Both of them give their own interpretation of the errors present in the data, namely, stochastic and nonstochastic. With the stochastic concept, errors are treated as a random process or sequence. The concept of estimation consistency is introduced, which makes it possible to use widely the theory of statistics in justifying methods for solving identification problems. The nonstochastic identification paradigm allows for arbitrary but bounded uncertainties in the data, i.e. belonging to some bounded sets. The largest number of different methods, as well as algorithms and software implementations, are made within the framework of the stochastic concept. Within the framework of nonstochastic identification, we mainly consider various modifications of the so-called subspace methods. In relation to complex systems, problems were noted that did not allow the development of sufficiently universal and effective methods for solving identification problems. A number of research areas have been pointed out that need further development in order to obtain more accurate models of complex systems.

ASYMPTOTIC SOLUTION OF FOKKER-PLANCK EQUATION BASED ON DARCY-BUCKINGHAM APPROACH IN UNSATURATED SOIL

The present study discusses one dimensional groundwater infiltration phenomenon in unsaturated soil. The mathematical formulation is developed using Dracy’s law, equation of continuity, and standard relations between diffusivity coefficient and hydraulic conductivity. We solved the obtained system using double power series method in the form of asymptotic expansion which represents moisture content in soil.

A Singularly Perturbed Dirichlet Problem for a Ring With Singular Boundaries

Uniform asymptotic expansions of small-parameter solutions of bisingular Dirichlet problems for a ring with any degree of accuracy are constructed by a generalized method of boundary functions and a small-parameter method. The investigated problems have two features: equations with a small parameter in the first derivatives and external solutions simultaneously have increasing features on the boundaries of the domain, i. e. limit equations have singularities at the same time on both boundaries of the ring. Formal asymptotic expansions are based on the maxima principle and the method of differential inequalities. The resulting asymptotic series are represented by the Puiseux series. The main term of the asymptotic expansions of the solutions has a negative fractional power with respect to the small parameter.

A new thin layer model for viscous flow between two nearby non‐static surfaces

AbstractWe propose a two‐dimensional flow model of a viscous fluid between two close moving surfaces. We show, using a formal asymptotic expansion of the solution, that its asymptotic behavior, when the distance between the two surfaces tends to zero, is the same as that of the the Navier‐Stokes equations.The leading term of the formal asymptotic expansions of the solutions to the new model and Navier‐Stokes equations are solution of the same limit problem, and the type of the limit problem depends on the boundary conditions. If slip velocity boundary conditions are imposed on the upper and lower bound surfaces, the limit is a solution of a lubrication model, but if the tractions and friction forces are known on both bound surfaces, the limit is a solution of a thin fluid layer model.The model proposed has been obtained to be a valuable tool for computing viscous fluid flow between two nearby moving surfaces, without the need to decide a priori whether the flow is typical of a lubrication or a thin fluid layer problem, and without the enormous computational effort that would be required to solve the Navier‐Stokes equations in such a thin domain.

Асимптотика собственных значений оператора Лапласа с двумя разбегающимися возмущениями на плоскости (случай произвольной кратности)

The Laplacian with a pair of distant perturbations is studied in two-dimensional space. Perturbations are understood as real finite continuous potentials. The discrete spectrum of the perturbed Laplacian is studied when the distance between the potentials increases. The presence of its eigenvalues and eigen-functions that correspond to them is considered for various cases of multiplicities of the limiting eigenvalue. The first case of the considered multiplicity is the double limiting eigenvalue. By this we mean the simple and isolated Laplacian eigenvalue with the first potential, as well as the simple and isolated Laplacian eigenvalue with the second potential. The second case under consideration is the case of arbitrary multiplicity of the limiting eigenvalue. By this we mean the Laplacian eigenvalue with the first potential of arbitrary multiplicity and the Laplacian eigenvalue with the second potential also of arbitrary multiplicity. In both cases under consideration (of multiplicity two and arbitrary), the first terms of formal asymptotic expansions of the eigenvalues and eigenfunctions of the perturbed Laplacian are constructed. A complex exponential-power structure of the constructed asymptotic is demonstrated. Also, in both cases under consideration, symmetry with respect to zero of the first corrections of the asymptotics of the eigenvalues of the perturbed Laplacian is shown

Asymptotics of the Solution to the Cauchy Problem for a Singularly Perturbed Operator Differential Transport Equation with Weak Diffusion

Formal asymptotic expansions of the solution to the Cauchy problem for a singularly perturbed operator differential transport equation with weak diffusion and small nonlinearity are constructed in the critical case. Under certain conditions imposed on the data of the problem, an asymptotic expansion of the solution is constructed in the form of series in powers of a small parameter with coefficients depending on stretched variables. Problems for determining all terms of the asymptotic expansion are obtained. It is shown that the leading term of the solution asymptotics is determined by solving Cauchy problems for a parabolic Burgers-type equation and, under certain conditions, for a Korteweg–de Vries–Burgers type equation. The remainder terms are estimated with respect to the residual.

De Moivre and Bell polynomials

We survey a family of polynomials that are very useful in all kinds of power series manipulations, and appearing more frequently in the literature. Applications to formal power series, generating functions and asymptotic expansions are described, and we discuss the related work of De Moivre, Arbogast and Bell.

A phase‐field porous media fracture model based on homogenization theory

AbstractA novel regularized fracture model for crack propagation in porous media is proposed. Our model is obtained through formal asymptotic expansions. We start with a regularized quasi‐static fracture model posed in a periodically perforated domain obtained by periodic extension of a rescaled unit cell with a hole. This setup allows us to write two separated minimality conditions for the primary (displacement) and secondary (or phase field) variables plus a balance of energy relation. Then we apply the usual asymptotic expansion matching to deduce limit relations when the rescaling parameter of the unit cells vanishes. By introducing cell problems solutions and a homogenized tensor, we can recast the obtained relations into a novel model for crack propagation in porous media. The proposed model can be interpreted as a regularized quasi‐static fracture model for porous media. This model yields two separated (homogenized) minimality conditions for the primary and secondary variables and a balance of a homogenized energy relation.

Multiscale analysis of stationary thermoelastic vibrations of a composite material.

The problem of description of stationary vibrations is studied for a planar thermoelastic body incorporating thin inclusions. This problem contains two small positive parameters [Formula: see text] and [Formula: see text], which describe the thickness of an individual inclusion and the distance between two neighbouring inclusions, respectively. Relying on the variational formulation of the problem, by means of the modern methods of asymptotic analysis, we investigate the behaviour of solutions as [Formula: see text] and [Formula: see text] tend to zero. As the result, we construct two models corresponding to the limit cases. At first, as [Formula: see text], by the version of the method of formal asymptotic expansions we derive a limit model in which inclusions are thin (of zero width). Then, from this limit model, as [Formula: see text], we derive a homogenized model, which describes effective behaviour on the macroscopic scale, i.e. on the scale where there is no need to take into account each individual inclusion. The limiting passage as [Formula: see text] is based on the use of the two-scale convergence theory. This article is part of the theme issue 'Non-smooth variational problems and applications'.

On the approximation of vorticity fronts by the Burgers–Hilbert equation

This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers–Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers–Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived.

On the Multiscale Landau–Lifshitz–Gilbert Equation: Two-Scale Convergence and Stability Analysis

Permalloy is a nickel-iron magnetic alloy, which typically has a face-centered cubic phase but may form an irregular polycrystalline structure. Its magnetization dynamics is modeled by the multiscale Landau–Lifshitz–Gilbert (LLG) equation with locally periodic material coefficients. We consider homogenization of the multiscale LLG equation in this work, and the novelty lies in three aspects. First, we derive the homogenized LLG equation using the formal asymptotic expansion and prove the rigorous convergence using the notion of two-scale convergence. Second, we establish a stability result of the homogenized LLG equation under small disturbances of material coefficients. Third, a modified Gauss–Seidel projection method is implemented to verify the convergence between the multiscale and homogenized LLG equations and the stability result.

Construction of boundary conditions for hyperbolic relaxation approximations II: Jin-Xin relaxation model

This is our second work in the series about constructing boundary conditions for hyperbolic relaxation approximations. The present work is concerned with the one-dimensional linearized Jin-Xin relaxation model, a convenient approximation of hyperbolic conservation laws, with non-characteristic boundaries. Assume that proper boundary conditions are given for the conservation laws. We construct boundary conditions for the relaxation model with the expectation that the resultant initial-boundary-value problems are approximations to the given conservation laws with the boundary conditions. The constructed boundary conditions are highly non-unique. Their satisfaction of the generalized Kreiss condition is analyzed. The compatibility with initial data is studied. Furthermore, by resorting to a formal asymptotic expansion, we prove the effectiveness of the approximations.