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A flow of Eyring–Powell type constitutes a remarkable area of analysis to model non-Newtonian processes in fluids. The associated diffusion term comes from the general kinetic theory of liquids and permits to account for a wider diffusivity, which is applicable for qualitatively low to higher shear stresses. The goal of the present article is to introduce a generalization of an Eyring–Powell fluid by the introduction of a porous reaction term (of Darcy–Forchheimer type) and a perturbation with a higher order operator. In particular, we consider that our model is an extension of a classical Eyring–Powell fluid in the same manner as introduced for other equations (see the extended Fisher–Kolmogorov model). The obtained equation is novel and requires analysis about existence, regularity and uniqueness of solutions. Stationary solutions are explored under the definition of a Hamiltonian. In addition, profiles of solutions are obtained with an exponential scaling that ends in a Hamilton–Jacobi equation. Eventually, some numerical assessments are introduced to validate the hypothesis done, and to discuss about the accuracy of the analytical approach followed. 2022-23

In the paper, we investigate the mixed type transmission problem arising in the model of fluid–solid acoustic interaction when a piezoceramic elastic body (Ω+) is embedded in an unbounded fluid domain (Ω−). The corresponding physical process is described by the boundary-transmission problem for second-order partial differential equations. In particular, in the bounded domain Ω+, we have a 4×4 dimensional matrix strongly elliptic second-order partial differential equation, while in the unbounded complement domain Ω−, we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations based on the Wiener–Hopf factorization method, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. We derive asymptotic expansion of solutions, and on the basis of asymptotic analysis, we establish optimal Holder smoothness results for solutions. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we derive an asymptotic expansion for the semi-infinite sum of Dirac-δ functions centered at discrete equidistant points defined by the set Na={x∈R:∃n∈N∧x=na,∀a>0}. The method relies on the Laplace transform of the semi-infinite sum of Dirac-δ functions. The derived series distribution takes the form of the Euler-Maclaurin summation when the distributions are defined for complex or real-valued continuous functions over the interval [0,∞). For n=1, the series expansion contributes with a term equal to δ(x)/2, which survives in the limit when a→0+. This term represents a correction term, which is in general omitted in calculations of the density of states of quantum confined systems by finite-size effects.

We consider the Lame system for an elastic medium consisting of an inclusion embedded in a homogeneous background medium. Based on the field expansion method and layer potential techniques, we rigorously derived the asymptotic expansion of the perturbed displacement field because of small perturbations in the interface of the inclusion. We extend these techniques to determine a relationship between traction-displacement measurements and the shape of the object and derive an asymptotic expansion for the perturbation in the elastic moment tensors because of the presence of small changes in the interface of the inclusion. Copyright © 2016 John Wiley & Sons, Ltd.

We consider a version of the stationary phase method in one dimension of A. Erdelyi, allowing the phase to have stationary points of non-integer order and the amplitude to have integrable singularities. After having completed the original proof and improved the error estimate in the case of regular amplitude, we consider a modification of the method by replacing the smooth cut-off function employed in the source by a characteristic function, leading to more precise remainder estimates. We exploit this refinement to study the time-asymptotic behaviour of the solution of the free Schrodinger equation on the line, where the Fourier transform of the initial data is compactly supported and has a singularity. We obtain asymptotic expansions with respect to time in certain space-time cones as well as uniform and optimal estimates in curved regions, which are asymptotically larger than any space-time cone. These results show the influence of the frequency band and of the singularity on the propagation and on the decay of the wave packets. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we address a simplified version of a problem arising from volcanology. Specifically, as a reduced form of the boundary value problem for the Lame system, we consider a Neumann problem for harmonic functions in the half-space with a cavity C. Zero normal derivative is assumed at the boundary of the half-space; differently, at ∂C, the normal derivative of the function is required to be given by an external datum g, corresponding to a pressure term exerted on the medium at ∂C. Under the assumption that the (pressurized) cavity is small with respect to the distance from the boundary of the half-space, we establish an asymptotic formula for the solution of the problem. Main ingredients are integral equation formulations of the harmonic solution of the Neumann problem and a spectral analysis of the integral operators involved in the problem. In the special case of a datum g, which describes a constant pressure at ∂C, we recover a simplified representation based on a polarization tensor. Copyright © 2016 John Wiley & Sons, Ltd.

We consider the problem of distributing two conducting materials with a prescribed volume ratio in a given domain so as to minimize the first eigenvalue of an elliptic operator with Dirichlet conditions. The gap between the two conductivities is assumed to be small (low contrast regime). For any geometrical configuration of the mixture, we provide a complete asymptotic expansion of the first eigenvalue. We then consider a relaxation approach to minimize the second-order approximation with respect to the mixture. We present numerical simulations in dimensions two and three to illustrate optimal distributions and the advantage of using a second-order method. Copyright © 2016 John Wiley & Sons, Ltd.

The topological derivative concept has been successfully applied in many relevant physics and engineering problems. In particular, the topological asymptotic analysis has been fully developed for a wide range of problems modeled by partial differential equations. In this paper, the topological asymptotic analysis of the energy shape functional associated with a diffusive/convective steady-state heat equation is developed. The topological derivative with respect to the nucleation of a circular inclusion is derived in its closed form with help of a non-standard adjoint state. Finally, we provide the estimates for the remainders of the topological asymptotic expansion and perform a complete mathematical justification for the derived formulas. The obtained result is new and can be applied in the context of topology design of heat sinks, for instance. Copyright © 2015 John Wiley & Sons, Ltd.

We present two efficient iterative schemes for solving the self-consistent field equations of flexible–semiflexible diblock copolymers. One is a semi-implicit scheme developed by employing asymptotic expansion, and the other is a hybrid scheme combining the robustness of the steepest descent method with the efficiency of the conjugate gradient method. In our position-one-dimensional and position-two-dimensional numerical experiments, we demonstrate that these schemes are much more efficient than the steepest descent method. Copyright © 2013 John Wiley & Sons, Ltd.

An asymptotic expansion of the contrasting structure-like solution of the generalized Kolmogorov–Petrovskii–Piskunov equation is presented. A generalized maximum principle for the pseudoparabolic equations is developed. This, together with the generalized differential inequalities method, allows to prove the consistence and convergence of the asymptotic series method. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider the three-dimensional Riquier-type and Dirichlet-type screen boundary value problems for the polymetaharmonic equation with real wave numbers k1 and k2. We investigate these problems by means of the potential method and the theory of pseudodifferential equations, prove the existence and uniqueness of solutions in Sobolev–Slobodetski spaces, and on the basis of asymptotic analysis, we establish the best Holder smoothness results for solutions. Copyright © 2012 John Wiley & Sons, Ltd.

In this paper, we provide an asymptotic expansion for the mean integrated squared error (MISE) of nonlinear wavelet estimator of survival density for a censorship model when the data exhibit some kind of dependence. It is assumed that the observations form a stationary and α-mixing sequence. This asymptotic MISE expansion, when the density is only piecewise smooth, is same. However, for the kernel estimators, the MISE expansion fails if the additional smoothness assumption is absent. Also, we establish the asymptotic normality of the nonlinear wavelet estimator. Copyright © 2011 John Wiley & Sons, Ltd.

In this paper, we find a new asymptotic expansion valid in enzymatic reactions, where the total amount of enzyme exceeds greatly the total amount of substrate. In such a case, it is well known that the Michaelis–Menten approximation is no longer valid; therefore our asymptotic expansion, which improves known results, is a new tool to approximate in a closed form the concentrations of the reactants in the presence of an enzyme excess. Copyright © 2011 John Wiley & Sons, Ltd.

In a previous paper, we have studied the asymptotic behavior of a viscous fluid satisfying Navier's law on a periodic rugous boundary of period e and amplitude δe, with δe/e tending to zero. In the critical size, δe∼e3/2, in order to obtain a strong approximation of the velocity and the pressure it is necessary to consider a boundary layer term in the corresponding ansatz. The purpose of this paper is to estimate the approximation given by this ansatz. Copyright © 2011 John Wiley & Sons, Ltd.

In this paper we consider the problem of two-dimensional (2D) steady flow of a viscous incompressible fluid at low Reynolds number past a porous body of arbitrary shape whose boundary is a closed Lipschitz curve. Assuming that the flow inside the porous body is governed by the Brinkman equation, we consider indirect layer potential representations corresponding to Brinkman, Stokes and Oseen systems. We show that the boundary integral equations of the Oseen–Brinkman coupling turn out to be a regular perturbation of those of the Stokes–Brinkman coupling. This allows us to prove that the difference between the first terms of the matched asymptotic expansions of the Stokes system and of the Oseen system is of the order |ln Re|−3 uniformly in any compact region of ℝ2. Since it is not known whether the coupled Brinkman–Navier–Stokes 2D interface problem has a solution and whether it can be approximated (if exists) by the Brinkman–Oseen transmission problem, these estimates can be considered as a first step in the complete analysis of 2D low Reynolds number flow past a porous body. Copyright © 2010 John Wiley & Sons, Ltd.

The motion of a mass point in the gravitational field of two bodies traveling along arbitrary orbits about their center of mass is considered. The mass ratio of the two bodies is equal to e≪1. When the mass point passes close to the smaller mass, the character of its trajectory changes abruptly, and the trajectory asymptotics as e0 is complex. The uniform asymptotic expansion of the entire trajectory up to any power of e is constructed and justified. In particular, an algorithm is presented for finding the limiting turning angle of the trajectory after the mass point passes a neighbourhood of the smaller mass. Copyright © 2010 John Wiley & Sons, Ltd.

Motivated by the well-posedness theory in fluid mechanics, we investigate the question of uniform convergence of solutions to the following heat equation perturbed with a complex-valued potential and a small positive parameter: ∂ t u e ― Δu e + i e V(X)u e = 0 in ℝ + t × ℝ d x . We introduce a class of potentials allowing a full representation of u e by the so-called Feynman-Kac integral in the Wiener space. We show that this ansatz provides an asymptotic expansion of u e as e tends to zero as well as a uniform control of ∥∇u e (t, -)∥ L 2 (ℝ d ) for small times. The proof is carried out by the stationary phase method for the Wiener measure introduced by Ben Arous in (Stochastics 1988; 25:125―153) and essentially applied in the context of the semi-classical approximation of solutions to the Schrodinger equation ( J. Funct. Anal. 1996; 137:243-280). At last, we prove as an application a uniform local existence result for inhomogeneous rotating fluid problem in the frame of anisotropic Sobolev spaces.

We study the asymptotic behavior of the eigenelements of the Dirichlet problem for the Laplacian in a two-dimensional bounded domain with thin shoots, depending on a small parameter e. Under the assumption that the width of the shoots goes to zero, as e tends to zero, we construct the limit (homogenized) problem and prove the convergence of the eigenvalues and eigenfunctions to the eigenvalues and eigenfunctions of the limit problem, respectively. Under the additional assumption that the shoots, in a fixed vicinity of the basis, are straight and periodic, and their width and the distance between the neighboring shoots are of order e, we construct the two-term asymptotics of the eigenvalues of the problem, as e0. Copyright © 2009 John Wiley & Sons, Ltd.

In this paper we use the method of matched asymptotic expansions in order to study the two-dimensional steady flow of a viscous incompressible fluid at low Reynolds number past a porous body of arbitrary shape. One assumes that the flow inside the porous body is described by the Brinkman model, i.e. by the continuity and Brinkman equations, and that the velocity and boundary traction fields are continuous across the interface between the fluid and porous media. By considering some indirect boundary integral representations, the inner problems are reduced to uniquely solvable systems of Fredholm integral equations of the second kind in some Sobolev or Holder spaces, while the outer problems are solved by using the singularity method. It is shown that the force exerted by the exterior flow on the porous body admits an asymptotic expansion with respect to low Reynolds number, whose terms depend on the solutions of the abovementioned system of boundary integral equations. In addition, the case of small permeability of the porous body is also treated. Copyright © 2008 John Wiley & Sons, Ltd.

In this paper we propose an efficient method to reconstruct a small inclusion buried inside a body using the perturbation of modal parameters measured on the boundary of the body. We design a reconstruction algorithm based on the asymptotic expansions of the eigenvalue perturbations obtained by Ammari and Moskow (Math. Meth. Appl. Sci. 2003; 26:67–75). We then implement this algorithm and demonstrate its viability and limitations. Copyright © 2008 John Wiley & Sons, Ltd.

we consider the problem of determining the boundary perturbations of an object from far-field electric or acoustic measurements. Assuming that the unknown object boundary is a small perturbation of a circle, we develop a linearized relation between the far-field data that result from fixed Dirichlet boundary conditions, entering as parameters, and the shape of the object, entering as variables. This relation is used to find the Fourier coefficients of the perturbation of the shape and makes use of an expansion of the Dirichlet-to-Neumann operator.

Following a suggestion from A. Thiaville and J. Miltat, whose work and experiments are about ferromagnetic thin layers and nanowires, we study in this paper the behaviour of the Landau–Lifschitz equation in a straight ferromagnetic wire. As the diameter of the domain and the exchange coefficient in the equation simultaneously tend to zero, we perform an asymptotic expansion to precise the solution for well-prepared initial conditions and are led to consider 2D exterior problems. Copyright © 2008 John Wiley & Sons, Ltd.

This paper presents a slender body theory for the dynamics of a curved inertial viscous Newtonian fiber. Neglecting surface tension and temperature dependence, the fiber flow is modeled as a three-dimensional free boundary value problem in terms of instationary incompressible Navier–Stokes equations. From regular asymptotic expansions in powers of the slenderness parameter, leading-order balance laws for mass (cross-section) and momentum are derived that combine the unrestricted motion of the fiber centerline with the inner viscous transport. The physically reasonable form of the one-dimensional fiber model results thereby from the introduction of the intrinsic velocity that characterizes the convective terms. For the numerical investigation of the viscous, gravitational and rotational effects on the fiber dynamics, a finite volume approach on a staggered grid with implicit upwind flux discretization is applied. Copyright © 2007 John Wiley & Sons, Ltd.

The stabilization of a symmetric tree-shaped network of Euler–Bernoulli beams described by a system of partial differential equations is considered. The boundary controllers are designed based on passivity principle. The eigenfrequencies are analysed in detail and the asymptotic expansion of eigenvalues are presented. It is shown that there is a set of generalized eigenfunctions for the closed-loop system, which forms a Riesz basis with parentheses for the energy state space. This concludes the spectrum-determined growth condition and the exponential stability of the closed-loop system. Copyright © 2007 John Wiley & Sons, Ltd.

The paper is devoted to the mathematical modelling of an extrusion process. Usually, an extruder has a very complicated geometry. This generates a lot of difficulties for computations of three-dimensional flows. In the present paper, we develop and justify the asymptotic domain decomposition strategy in order to parallelize the computational process and reduce the memory. The error estimates are proved for the Stokes steady-state equation in the two-dimensional and three-dimensional cases. Then, the asymptotic domain decomposition procedure is applied for numerical testing and computations of the non-Newtonian fluid simulating a real process of the polymer extrusion. Copyright © 2007 John Wiley & Sons, Ltd.

In this paper, we study the global behaviour of an underground waste disposal in order to have an accurate upscaled model suitable for the computations involved in safety assessment processes. We start from a detailed model describing the transport of pollutant leaking from a high number of units. Using the method of homogenization, going to the limit, we obtain first a macroscopic model where the sources are now appearing globally. Then we compute a first-order matched asymptotic expansion and we give the error estimates for this approximation. Copyright © 2004 John Wiley & Sons, Ltd.

A conformal invariant asymptotic expansion approach is used to solve the (3+1)-dimensional non-linear Schrodinger (NLS) equation. Some new (3+1)-dimensional integrable models under the condition that they are conformal invariant and possess Painleve property are obtained. These Painleve integrable models can be used to solve the (3+1)-dimensional NLS equation approximately. In some special cases, the approximate solutions become exact.