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In this paper, we present a new model as an alternative to the classical 3D Richards model for the description of water flow in shallow aquifers. The new model is designed to achieve three goals. First, it provides a good approximation to the Richards model over a wide range of time scales. More specifically we show that both models characterize a flow with the same dominant components when the ratio of the horizontal length to the depth of the aquifer is small. Second, the new model accurately describes the velocity field. In particular, it is not based on the Dupuit hypothesis, which is often used in the context of shallow aquifers. This allows the new model to be well accurate even in the presence of wells and in aquifers with variable bedrock. Third, the new model can be viewed as a coupling of numerous 1D vertical Richards problems with a 2D elliptic one. In practice, this coupling is treated numerically using a Picard fixed point scheme, avoiding the need to solve any three-dimensional problem. This results in a significant reduction in computational cost compared to solving the 3D Richards problem directly. [ABSTRACT FROM AUTHOR]

We present and analyze a new derivation of the meso-level behavior of a discrete microscopic model of heat transfer. This construction is based on the principle of dynamic consistency. Our work reproduces and corrects, when needed, all the major previous expressions which provide modifications to the standard heat PDE. However, unlike earlier efforts, we do not allow the microscopic level parameters to have zero limiting values. We also give insight into the difficulties of constructing physically valid heat equations within the framework of the general mathematically inequivalent of difference and differential equations. [ABSTRACT FROM AUTHOR]

A simple HLLE-type scheme is proposed for all Mach number flows. In the proposed scheme, no extra wave structure is added in the HLLE scheme to resolve the shear wave while the contact wave is resolved by adding a wave structure similar to the HLLEM scheme. The resolution of the shear layers and the flow features at low Mach numbers is achieved by a velocity reconstruction method based on the face normal Mach number. Robustness against the numerical instabilities is achieved by scaling the velocity reconstruction function in the vicinity of shock with a multi-dimensional pressure sensor. The ability of the proposed scheme to resolve low Mach flow features is demonstrated through asymptotic analysis while the stability of the proposed scheme for strong shock is demonstrated through linear perturbation and matrix stability analyses. A set of numerical test cases are solved to show that the proposed scheme is free from numerical shock instability problems at high speeds and is capable of resolving the flow features at very low Mach numbers. • A new velocity reconstruction method for an all Mach number HLLE scheme. • Velocity reconstruction based on face normal Mach number and pressure function. • Velocity reconstruction resolves shear layers and low Mach flow features while presrving shock stability. • HLLEM type method for resolving contact wave while preserving shock stability. • Analytical and numerical demonstration of all Mach capability of the proposed scheme. [ABSTRACT FROM AUTHOR]

An asymptotic analysis is conducted of the effect surface vibrations may play in mitigating the resistance present in the system consisting of a fluid contained between a pair of parallel flat plates in relative motion. Previous numerical computations of this situation, which has applications in both biological and technological contexts, have suggested that small vibrations that move at a fast phase speed can lead to substantial reductions in the resistance. The effect seems to be a somewhat intricate function of the wavelength of the vibration modes. Our analysis uncovers the properties of the various phenomena that are at work and describes the interplay between them. The techniques employed here lend themselves to the extension to other situations including problems in which propulsion effects in fluid systems can be induced by vibrations. [ABSTRACT FROM AUTHOR]

This paper quantifies the asymptotic order of the largest singular value of a centered random matrix built from the path of a Block Markov Chain (BMC). In a BMC there are n labeled states, each state is associated to one of K clusters, and the probability of a jump depends only on the clusters of the origin and destination. Given a path X 0 , X 1 , ... , X T n started from equilibrium, we construct a random matrix N ˆ that records the number of transitions between each pair of states. We prove that if ω (n) = T n = o (n 2) , then ‖ N ˆ − E [ N ˆ ] ‖ = Ω P ( T n / n). We also prove that if T n = Ω (n ln n) , then ‖ N ˆ − E [ N ˆ ] ‖ = O P ( T n / n) as n → ∞ ; and if T n = ω (n) , a sparser regime, then ‖ N ˆ Γ − E [ N ˆ ] ‖ = O P ( T n / n). Here, N ˆ Γ is a regularization that zeroes out entries corresponding to jumps to and from most-often visited states. Together this establishes that the order is Θ P ( T n / n) for BMCs. [ABSTRACT FROM AUTHOR]

The sudden generation of wave disturbances in the ocean is associated with a change in the pressure field in the liquid layer. Consequently, compression-type of waves, known as acoustic-gravity waves, form and radiate at the speed of sound, carrying information on the event source, namely its magnitude and location. This information can be recorded by hydrophones hundreds of kilometres away from the event, and if analyzed at near-real time using an inverse model it could allow for an early warning system. The near-real time analysis requires an explicit form of the pressure field induced by the acoustic-gravity waves. In this work, we derive an explicit solution for the bottom pressure field using stationary phase approximation. Then, to demonstrate the effectiveness of the warning system, we apply an inverse model to calculate the generation time and location of the event. [ABSTRACT FROM AUTHOR]

• A constructive link between Willis' method and the two-scale asymptotic method. • Effective impedance tensor characterizing Willis' effective elastodynamic law. • Recursion formula for solving the hierarchical motion equations issuing from asymptotic analysis. Two-scale asymptotic method and Willis' method are jointly examined for the elastodynamic homogenization of periodic composites. The effective elastodynamic constitutive law given by Willis' method, non-local both in time and space, is first reformulated so as to make appear a compact expression for the effective impedance tensor characterizing it. The two-scale asymptotic method is then revisited by exploiting the fact that the results obtained by it constitute an approximation to the general ones delivered by Willis' method. The solutions for the hierarchical motion equations issuing from asymptotic analysis are shown to admit a compact recursive representation. A generic compact expression is derived for the effective impedance tensor associated with the n th-order approximation of asymptotic analysis. Remarkably, this expression turns out to be formally identical to the one of the effective impedance tensor obtained via Willis' method. As an example of illustration, the elastodynamic homogenization of a layered composite undergoing anti-plane shear is investigated in detail. [ABSTRACT FROM AUTHOR]

In this article, we study the impact of a change in the type of boundary conditions of an elliptic boundary value problem. In the context of the conductivity equation we consider a reference problem with mixed homogeneous Dirichlet and Neumann boundary conditions. Two different perturbed versions of this "background" situation are investigated, when (i) The homogeneous Neumann boundary condition is replaced by a homogeneous Dirichlet boundary condition on a "small" subset ω ε of the Neumann boundary; and when (ii) The homogeneous Dirichlet boundary condition is replaced by a homogeneous Neumann boundary condition on a "small" subset ω ε of the Dirichlet boundary. The relevant quantity that measures the "smallness" of the subset ω ε differs in the two cases: while it is the harmonic capacity of ω ε in the former case, we introduce a notion of "Neumann capacity" to handle the latter. In the first part of this work we derive representation formulas that catch the structure of the first non trivial term in the asymptotic expansion of the voltage potential, for a general ω ε , under the sole assumption that it is "small" in the appropriate sense. In the second part, we explicitly calculate the first non trivial term in the asymptotic expansion of the voltage potential, in the particular geometric situation where the subset ω ε is a vanishing surfacic ball. [ABSTRACT FROM AUTHOR]

The paper discusses the nonlinear behavior of a viscoelastic 3D thin film in frictional contact with a flat substrate. The frictional contact is modeled by Coulomb's law of friction at substrate. A function is included for damage caused by strain or excessive film stress to which the materials may be subjected over time. Assumptions and a functional framework are identified to allow for the existence and uniqueness of the weak solution to the problem. In contrast, preliminary estimates of the potential damage and stresses to the system during a specified operating period T are disclosed, the study concluded in the latter by an asymptotic analysis model of the problem explained with mathematical justifications and intended results. [ABSTRACT FROM AUTHOR]

In this work, we explore the rogue wave patterns in the coupled Fokas–Lenells equation by using the Darboux transformation. We demonstrate that when one of the internal parameters is large enough, the general high-order rogue wave solutions generated at a branch point of multiplicity three can be decomposed into some first-order outer rogue waves and a lower-order inner rogue wave. Remarkably, the positions and the orders of these outer and inner rogue waves are intimately related to Okamoto polynomial hierarchies. • A compact determinant formula of rogue wave solutions is constructed. • The rogue wave patterns are first derived with the aid of the Darboux transformation. • Some examples of rogue wave patterns are exhibited. [ABSTRACT FROM AUTHOR]

An asymptotic analysis on the dynamics of adiabatic and nonadiabatic spherically expanding laminar flame was performed for real gas described by the Noble-Abel equation of state. The effect of the finite molecular volume was considered and characterized by a covolume parameter. It was demonstrated that the non-ideal model leads to higher flame speed at quasi-steady state and to lower Markstein number compared to the perfect gas based solutions. The change of the value of these two parameters is proportional to the covolume parameter. For example, with a covolume of 0.1, the flame speed was increased by 10%, while the Markstein number could be reduced by up to 35%, depending on the value of the Lewis number. In addition, the finite molecular volume effect promotes the quenching of nonadiabatic flame in the vicinity of the critical state. New relations for extrapolation of the zero-stretch flame speed were established. It was shown that non-negligible modifications of the unstretched laminar flame speed and Markstein number are introduced by the finite molecular volume effect at high pressure. Strategies for high-pressure laminar flame speed extrapolation were discussed and the relative uncertainty related to the finite volume effect is of 20%–45% over the total uncertainty for pressures in the range 1–2.5 MPa, indicating that this effect cannot be neglected to determine the laminar flame speed under such pressure conditions. The impact of real gas effect was further illustrated with numerical flame simulations for a stoichiometric hydrogen–air mixture. Novelty and Significance Statement 1. First asymptotic analysis of adiabatic and nonadiabatic unsteady expanding spherical flame in a Noble-Abel gas 2. Covolume promotes higher flame speed but mitigates ignition 3. Non-negligible effect of the covolume even at a pressure of few MPa is demonstrated 4. First unsteady numerical simulations of spherical flame in a Noble-Abel gas [ABSTRACT FROM AUTHOR]

• Phase-change problems with different overspecified boundary conditions. • Thermal coefficient formulae through an inverse Stefan problem. • Relationship between problems with Robin-Neumann and Dirichlet-Neumann conditions. • Convergence analysis when the heat transfer coefficient (h) is large. • Absolute error with an upper bound of order o (1 / h). The phase-change processes are found in a wide variety of dynamic systems, for example in the study of snow avalanches. When a thermal property of the material is unknown, we can add a boundary condition to formulate an Inverse Stefan Problem, and determine this property. In this paper we study a heat conduction phase-change problem with Robin and Neumann boundary condition at a fixed face. This overspecified condition allows to simultaneously determine two unknown thermal coefficients through a moving boundary problem or a free boundary problem. Formulae for different cases where obtained by Ceretani and Tarzia (2015) [6]. The formulation with these type of boundary conditions is a more realistic one than the heat conduction phase-change problems with Dirichlet and Neumann boundary condition at the fixed face, considered by Tarzia, (1982-1983). Therefore we propose to study the relationship between the problems with Robin-Neumann conditions, and the problems with Dirichlet-Neumann conditions. The main result of this work is the convergence analysis of these problems, when the heat transfer coefficient h of the Robin condition is very large. We present for each case of the free and moving boundary problems, an upper bound for the error of the two unknown parameters, obtaining in every case a bound of order o (1 h). Finally we show a numerical example of the convergence, for a phase change material commonly used in heating or cooling processes. [ABSTRACT FROM AUTHOR]

The equilibrium equation governing the plane strain static problem for a thin elastic layer is studied using asymptotic analysis for a material with generalized Young's modulus weakly dependent on the mass density. Within the context of the adopted scaling, the effect of a physical nonlinearity arises at the second order along with shear deformation and other linear phenomena. Refined equations for plate bending and transverse compression are derived. Numerical examples illustrating the role of the corrections due to the considered constitutive behavior are presented. • Density dependent Young's modulus in the equations of linear elasticity. • Analysis of the associated small physical nonlinearity. • Higher order asymptotic analysis of a plane strain problem for thin plate extension and transverse compression. • Discussion of refined plate equations. [ABSTRACT FROM AUTHOR]

In this paper we study the following nonlinear Choquard equation − Δ u + u = ln 1 | x | ∗ F (u) f (u) , in R 2 , where f ∈ C 1 (R , R) and F is the primitive of the nonlinearity f vanishing at zero. We use an asymptotic approximation approach to establish the existence of positive solutions to the above problem in the standard Sobolev space H 1 (R 2). We give a new proof and at the same time extend part of the results established in (Cassani-Tarsi, Calc.Var.PDE, 2021) [11]. [ABSTRACT FROM AUTHOR]

Gerdjikov–Ivanov equation describes the propagation of the weakly nonlinear and dispersive Alfvén waves along the ambient electromagnetic field in a partially ionized plasma in the presence of the Hall, ambipolar, and Ohmic effects. With respect to the transverse components of the electromagnetic field to the lowest order, N solitons in the determinant form are derived and analyzed via the asymptotic analysis on the basis of an existing binary DT and an existing Lax pair, where N is a positive integer. Without loss of generality, expressions of the N asymptotic solitons in the determinant form are obtained, from which we can obtain the amplitudes, positions, velocities and phase shifts of the N asymptotic solitons. Those results reveal that the interaction among the N solitons is elastic. When N = 3 , we graphically illustrate the elastic interaction among the three solitons. This work may provide a physical mechanism for the generations and interactions of the Alfvén solitons in the magnetized plasmas. [ABSTRACT FROM AUTHOR]

Interaction of two solitons with different structures is exhibited. By adjusting the corresponding parameters, distinct solitons and their interaction can be achieved. It can also simulate a process of energy concentration of solitons at different times. [Display omitted] • We investigated a two-dimensional Gross-Pitaevskii equation with time-varying trapping potential in the Bose-Einstein condensation. • The Hirota bilinear method is established to solve the two-dimensional Gross-Pitaevskii equation and its parabolic soliton, line-soliton and dromion-like structure can be exhibited via some appropriate parameters chosen. Their interaction structures are discussed. • The interaction of two-soliton solutions is investigated through asymptotic analysis. The Gross-Pitaevskii equation is a class of the nonlinear Schrödinger equation, whose exact solution, especially soliton solution, is proposed for understanding and studying Bose-Einstein condensate and some nonlinear phenomena occurring in the intersection field of Bose-Einstein condensate with some other fields. It is an important subject to investigate their exact solutions. We give multi-soliton of a two-dimensional Gross-Pitaevskii system which contains the time-varying trapping potential with a few interactions of multi-soliton. Through analytical and graphical analysis, we obtain one-, two- and three-soliton which are affected by the strength of atomic interaction. The asymptotic expression of two-soliton embodies the properties of solitons. We can give some interactions of solitons of different structures including parabolic soliton, line-soliton and dromion-like structure. By constructing an appropriate Hirota bilinear form, the multi-soliton solution of the system is obtained. The soliton elastic interaction is analyzed via asymptotic analysis. The results in this paper theoretically provide the analytical bright soliton solution in the two-dimensional Bose-Einstein condensation model and their interesting interaction. To our best knowledge, the discussion and results in this work are new and important in different fields. The study enriches the existing nonlinear phenomena of the Gross-Pitaevskii model in Bose-Einstein condensation, and prove that the Hirota bilinear method and asymptotic analysis method are powerful and effective techniques in physical sciences and engineering for analyzing nonlinear mathematical-physical equations and their solutions. These provide a valuable basis and reference for the controllability of bright soliton phenomenon in experiments for high-dimensional Bose-Einstein condensation. [ABSTRACT FROM AUTHOR]

• The prostate tumor growth model. • An asymptotic analysis. • A generalized moving least squares technique. • Experimental data. This paper focuses on presenting an asymptotic analysis and employing simulations of a model describing the growth of local prostate tumor (known as a moving interface problem) in two-dimensional spaces. We first demonstrate that the proposed mathematical model produces the diffuse interfaces with the hyperbolic tangent profile using an asymptotic analysis. Next, we apply a meshless method, namely the generalized moving least squares approximation for discretizing the model in the space variables. The main benefits of the developed meshless method are: First, it does not need a background mesh (or triangulation) for constructing the approximation. Second, the obtained numerical results suggest that the proposed meshless method does not need to be combined with any adaptive technique for capturing the evolving interface between the tumor and the neighboring healthy tissue with proper accuracy. A semi-implicit backward differentiation formula of order 1 is used to deal with the time variable. The resulting fully discrete scheme obtained here is a linear system of algebraic equations per time step solved by an iterative scheme based on a Krylov subspace, namely the generalized minimal residual method with zero–fill incomplete lower–upper preconditioner. Finally, some simulation results based on estimated and experimental data taken from the literature are presented to show the process of prostate tumor growth in two-dimensional tissues, i.e., a square, a circle and non-convex domains using both uniform and quasi-uniform points. [ABSTRACT FROM AUTHOR]

• An asymptotic analysis is performed for modeling radial forging process. • It can predict the material flow inhomogeneity and deviation of neutral points profile. • A process feasibility criterion is developed in terms of the die angle, friction and back-push stress for the 1st time. • A breakthrough method is proposed that can effectively find the location of the neutral plane far easier and faster. In this paper, an analytical approach based on the asymptotic analysis is proposed to develop an axisymmetric model for the radial forging process of rods. This model is able to predict the deformation inhomogeneity in the radial direction and approves that the profile of the neutral plane is not a planar cross section of the workpiece. For the first time, a relationship has been developed between the die angle, friction coefficient and back-push stress that can be used as a criterion for examining the feasibility of the radial forging process. A breakthrough method has also been proposed that can effectively find the location of the neutral plane far easier and faster than the traditional approach. Finite element simulation of the process has been performed to validate the deformation pattern obtained by the asymptotic model. Models are also verified against available experimental data. The findings of this study are especially useful for online design, analysis and control of the radial forging process. [ABSTRACT FROM AUTHOR]

The Stefan problem (whether in its classical form or not) has assumed that the interface moves instantaneously with time as a prescribed initial condition. This assumption is valid in the problems subjected to an isothermal boundary. Yet, it may not represent the physics for a convective boundary since a certain amount of time is needed to cool or melt the convective surface to its fusion depending on the heat transfer coefficient (or Biot number in a dimensionless way). In this study, we formulate a two-phase Stefan problem in an annulus for outward solidification subjected to a convective or Robin boundary condition while not assuming that the interface moves right away at time t = 0. A comprehensive asymptotic analysis is performed by expanding around a small Stefan number; four spatial and five temporal scales are characterized based on the Stefan problems involving a convective surface. The method of property averaging is also employed at the scale where the equilibrium freezing occurs. The developed asymptotic solution is verified with numerical data generated by the enthalpy method at various Biot and Stefan numbers. The results demonstrate the significance of abandoning the ansatz (i.e., the interface does not move immediately with time), especially towards small Biot numbers. Further, it is found that the presented asymptotic solution considerably extends the valid range of the Stefan number when compared with the conventional asymptotic technique. • Two-phase Stefan problem with a Robin boundary, yet no instant motion of the interface. • Systematic asymptotic analysis, including four spatial and five temporal scales. • Property-averaging method is employed to enlarge the validity of the asymptotic solution. • The results are verified with numerical data for various Biot and Stefan numbers. • The framework can be extended to other Stefan problems with a convective boundary. [ABSTRACT FROM AUTHOR]

• Williams' crack tip solution including crack face pressure is analyzed. • T-stress results including the pressure term are reported. • Confusion about T-stress determination is cleared. Williams' series expansion crack tip solution in linear elasticity is modified to include a uniform crack face pressure. It is shown that the crack face pressure p only enters the constant or the T -stress term. Practical ways to calculate the T -stress from near crack tip stresses are outlined. The analytical results are consistent with numerical results such as from ABAQUS. [ABSTRACT FROM AUTHOR]

We study the large-order and infinite-order solitons of the coupled nonlinear Schrödinger equation with the Riemann–Hilbert method. By using the Riemann–Hilbert representation of the high-order Darboux dressing matrix, the large-order and infinite-order solitons can be analyzed directly without using inverse scattering transform. We firstly disclose the asymptotics for large-order solitons, which are divided into four different regions—the genus one region, the genus zero region, the exponential decay and the algebraic decay region. We verify the consistency between asymptotic solutions and exact solutions by the Darboux dressing method numerically. Moreover, we consider the property and dynamics for infinite-order solitons—a special limitation for the larger order solitons. It is shown that the genus one region and exponential decay region will disappear for the infinite-order solitons. • The large-order asymptotics of 3 × 3 Lax pair is analyzed, including four different asymptotic regions. • By tedious calculation, the leading order asymptotics in the oscillatory region is written as the Riemann-Theta function. • We give two types of asymptotics in the infinite-order case, one is the large χ asymptotics and the other is the large τ asymptotics. [ABSTRACT FROM AUTHOR]

A two-order asymptotic solution for sharp V-notch tip mechanics field in elasto-viscoplastic solids with power-law form under mode I loading is developed. The stress exponents and angular distributions for various notch angles and viscoplastic exponents for the second terms are reported. The stress exponent of the first order is singular and is non-singular except for low creep exponent with small notch opening angle. Compared with finite element results from several V-notched specimens, it is found that the two-order solution can have a better estimate for those cases with small notch opening angles, e.g., α = 30° under n = 5, than that of one-order solution regardless of viscoplastic extent, specimen type and loading level. One-order term solution is demonstrated to be sufficient to characterize the full field of sharp V-notch specimen for those conditions that elastic term is considered into higher order term, e.g., α = 60° under n = 5, in all specimens under various viscoplasticity extents. [ABSTRACT FROM AUTHOR]

New expressions are derived for the Green function (GF) for diffraction and radiation of waves by a two-dimensional (2D) body in finite water depth. The finite depth GF is expressed as a sum of singularities, the infinite depth GF and smoothly-varying integrals that are convergent for all parameter values. The infinite depth component is given explicitly, making it very fast to compute. Explicit expressions are derived for the limiting cases of zero and infinite frequency, for both finite and infinite water depth. The low frequency limit of the 2D GF is inconsistent with the zero-frequency 2D GF, with the real part tending to infinity when the water depth is infinite and the imaginary part tending to infinity in finite water depth. The inconsistencies with the zero frequency GF differ between the 2D and 3D cases. These inconsistencies lead to differences between the low-frequency behaviour of the added mass and damping of an oscillating body and the values at zero frequency. It is shown that these differences can be inferred directly from the behaviour of the GF at low frequencies. • Finite depth Green function (GF) expressed as infinite depth GF plus perturbations. • Explicit expression derived for infinite depth Green function. • Explicit expressions derived for zero and infinite frequency GF in finite depth. • Inconsistencies in low frequency added mass and damping related to behaviour of GF. [ABSTRACT FROM AUTHOR]

Accurate asymptotic solutions are presented for axisymmetric deformation of thin layers constrained by either two rigid plates or two rigid spheres. Those solutions are developed using Saint-Venant's principle and the layer thinness as the only assumptions. The solutions are valid in the entire range of Poisson's ratios, and allow one to distinguish among compressible, nearly incompressible, and incompressible layers. That classification involves both material and geometric parameters. [ABSTRACT FROM AUTHOR]

This paper studies the game of guessing riffle-shuffled cards with complete feedback. A deck of n cards labeled 1 to n is riffle-shuffled once and placed on a table. A player tries to guess the cards from top and is given complete feedback after each guess. The goal is to find the guessing strategy with maximum reward (expected number of correct guesses). We give the optimal strategy for this game and prove that the maximum expected reward is n ∕ 2 + 2 ∕ π ⋅ n + O (1) , partially solving an open problem of Bayer and Diaconis (Bayer and Diaconis, 1992). [ABSTRACT FROM AUTHOR]

Two asymptotic solutions are presented for linear elastic thin, not necessarily circular, cylindrical layers fully constrained by two rigid plates. Other than being small, the plate displacements and rotations are not restricted, and therefore, in general, a constrained layer is subjected to combined stretching, bending, shearing, and twisting. The first solution is restricted to layers formed by compressible solids, whose Poisson's ratio is not too close to one half. This solution is a superposition of a polynomial displacement field, valid in the bulk of the layer, and a corrective displacement field, which decays exponentially fast away from the cylindrical surface and becomes negligible at distances comparable to the layer thickness. The second solution is not restricted in terms Poisson's ratio but it is correct only to a leading order. This solution unifies leading-order solutions for layers formed by compressible, nearly incompressible, and incompressible solids. The unification involves a parameter characterizing the competition between compressibility and thinness. • General solutions are presented for thin layers constrained by rigid plates. • The solution in the bulk is elementary for compressible solids. • A more complicated solution is valid for any Poisson's ratio. [ABSTRACT FROM AUTHOR]

In this paper, a three-dimensional, nonlinear model is proposed to incorporate the effect of submarine volcanic eruption on equatorial oceanic flows. The model inspired by the Constantin and Johnson model, considers an appropriate density function to model the density of the oceanic system with a volcano erupting at a point on the sea-bed, synchronizing with the temperature variations. The formulated model contains term related to Coriolis effect and is consistent with β -plane approximation. Exact results for the model have been obtained using the asymptotic system of equations. [ABSTRACT FROM AUTHOR]

We present detailed analytical and experimental results for a system of two comparator-based relaxation oscillators that are coupled via threshold modulation. Starting with the op-amp circuit equations, we derive parameter conditions that lead to n:1 phase-locked solutions, thereby obtaining predictions for the shapes of Arnold tongues for this system. These tongues are compared to experimental data and good agreement is found. We follow up with a stability analysis of phase-locked solutions. Specifically, we use two-timescale asymptotic analysis, and we check some results obtained therein against experimentally captured dynamics. [ABSTRACT FROM AUTHOR]

The article is aimed to provide an effective digital solution in support of the infill microstructural design within objectives generated from normal computer aided design (CAD) systems. The key here is to deploy the microstructure and to analyse the performance of the resulting multiscale objective in the parameter domain, from which a CAD objective is mapped, usually by means of Non-Uniform Rational B-Splines (NURBS) functions. In the regularly shaped domain defined in the parameter space, another set of mapping functions are defined for the description of spatially-varying microstructure (in the parameter space), and the actual multiscale CAD objective gets represented through composition of these mapping functions with the NURBS-based functions for CAD geometry generation. When the introduced mapping functions are also NURBS-based, the proposed representation strategy becomes more favoured by CAD systems, in the sense that the CAD geometry and its infill can be tuned simultaneously but independently with a same set of NURBS basis. Moreover, our computer-aided engineering (CAE) module is also installed over the parameter domain, based on a machine-learning-based homogenisation formulation for compliance and strength analysis. Thus the finite element meshes stay unchanged although the accommodating CAD geometry keeps varying. With the aforementioned treatments, a consecutive CAD-consistent scheme is proposed. Our numerical results show that each time as the control points are moved (by a CAD designer), one may choose to wait several seconds for two-dimensional cases and/or a few minutes for three-dimensional cases, to get a CAD geometry filled with microstructure bearing optimised compliance. [ABSTRACT FROM AUTHOR]

We consider positive solutions of a fractional Lane–Emden-type problem in a bounded domain with Dirichlet conditions. We show that uniqueness and nondegeneracy hold for the asymptotically linear problem in general domains. Furthermore, we also prove that all the known uniqueness and nondegeneracy results in the local case extend to the nonlocal regime when the fractional parameter s is sufficiently close to 1. [ABSTRACT FROM AUTHOR]

This paper focuses on the asymptotic behaviors of the solutions to the second-kind Fredholm integral equations with a highly oscillatory and weakly singular kernel. We employ van der Corput lemmas, resolvents of the Volterra integral equations, and the contraction mapping principle to analyse these behaviors. Numerical results are provided to show how the singularity and oscillatory parameters affect the solution. [ABSTRACT FROM AUTHOR]

In this paper, a simplified multi-layered plate model is proposed, which can be used to study the growth-induced deformations of multi-layered hyperelastic plates. First, we introduce a full-form multi-layered plate theory of growth, which has a relatively complicated plate equation system. To simplify the plate equations, systematic asymptotic analyses are conducted with some specified magnitudes of growth functions and surface tractions. It is found that the terms containing a high-order stress tensor always have higher asymptotic orders. By dropping these higher-order terms from the original plate equations, a simplified multi-layered plate model can be derived. Despite its simplicity, this new plate model is valid in a wide range of growth and loading conditions. To be prepared for practical applications, the associated weak forms of the simplified plate equations are derived and implemented in a finite element software. Some typical examples are further studied to show the efficiency of the plate model. By virtue of the plate model, the growth deformations of multi-layered plate samples are simulated, which show very good consistency with the results of 3D volume model. Especially, the distributions of residual stresses on the interfaces of different layers can be predicted accurately, which has an important meaning for the design of soft devices with multi-layered plate structures. [Display omitted] • A simplified multi-layered plate model of growth is developed. • Asymptotic analyses with different growth and surface traction conditions are conducted. • The weak form of the plate equation system is derived and implemented in a FE software. • Some typical examples are studied to demonstrate the efficiency of the plate model. • The multi-layered plate model shows advantages in the aspect of numerical calculations. [ABSTRACT FROM AUTHOR]

Within the framework of the Kirchhoff-Love theory, a thin homogeneous layer (called adhesive) of small width between two plates (called adherents) is considered. It is assumed that elastic properties of the adhesive layer depend on its width which is a small parameter of the problem. Our goal is to perform an asymptotic analysis as the parameter goes to zero. It is shown that depending on the softness or stiffness of the adhesive, there are seven distinct types of interface conditions. In all cases, we establish weak convergence of the solutions of the initial problem to the solutions of limiting ones in appropriate Sobolev spaces. The asymptotic analysis is based on variational properties of solutions of corresponding equilibrium problems. [ABSTRACT FROM AUTHOR]

We consider weak positive solutions to the critical p -Laplace equation with Hardy potential in R N − Δ p u − γ | x | p u p − 1 = u p ⁎ − 1 where 1 < p < N , 0 ⩽ γ < (N − p p) p and p ⁎ = N p N − p. The main result is to show that all the solutions in D 1 , p (R N) are radial and radially decreasing about the origin. [ABSTRACT FROM AUTHOR]

The self-organisation behaviour of dislocation upon loading is still a mechanistically mysterious phenomenon. The present article aims to present a computationally tractable dislocation dynamics model, which can be used for helping analyse the mechanism behind dislocation pattern formation. With an integrated use of asymptotic analysis and machine learning tools, the discrete dislocation dynamics on two parallel slip planes is self-consistently reformulated at a coarse-grained level. For the present case, asymptotic analysis helps in 1) reformulating the original globally discrete problem by a continuum model underpinned by a database resolving the almost singular short-range elastic interaction of discrete dislocations; 2) identifying the proper input and output quantities for implementing machine learning tools; 3) digging out a hidden but explicit interrelation between mean-field quantities to reduce the dimensionality of the data space. Machine learning tools serve for 1) inferring the inherently implicit interrelationships between continuum quantities; 2) capturing a low-dimensional manifold in the data space corresponding to the local flow stress. The non-monotonically increasing profile of the flow stress - density relationship, as revealed by machine learning, is found to play a key role in the onset of dislocation patterning behaviour seen in the simulation results. A scaling law relating the applied stress to pattern wavelength is also derived, providing a rationale to the widely used empirical similitude relation. The experimentally observed swing in similitude coefficient value is attributed to the randomness in slip plane distributions. In a methodological viewpoint, the treatment of using asymptotic analysis to help design the curriculum for implementing machine learning tools, offers a paradigm for self-consistently upscaling more complicated multiscale systems. [ABSTRACT FROM AUTHOR]

• A new class of flow models for unconfined aquifers is proposed. • The models couple the fast and slow components of the flows. • The 3D-Richards equation is properly approached in a wide range of time scales. • The validity of the models is specified using a non-conventional asymptotical analysis. • The performances and the tractability of the models are illustrated by numerical simulations. In this paper, we present a new class of efficient models for water flow in shallow unconfined aquifers, providing an alternative to the classical but less tractable 3d-Richards model. Its derivation is guided by two objectives: to obtain a model that has low computational cost and yields relevant results on every time scale. Thus, we keep track of two types of flow that occur in such a context and are dominant when the ratio of thickness to longitudinal length is small: the first is dominant on a small time scale and is described by a vertical 1d-Richards problem; the second corresponds to a large time scale, when the evolution of the hydraulic head becomes independent of the vertical variable. These two types of flow are appropriately modeled by a one-dimensional and two-dimensional system of PDE boundary value problems, respectively. They are coupled at an artificial level below which the Dupuit hypothesis holds true (i.e. , the vertical flow is instantaneous) so that the global model is mass conservative. Tuning the artificial level, which can even depend on an unknown of the problem, we obtain the new class of models. Using asymptotic expansions, we prove that the 3d-Richards model and each model in the class behave identically on every considered time scale (short, intermediate, and large) in thin aquifers. The results are illustrated by numerical simulations, and it is demonstrated that they fit well with those obtained by the original 3d-Richards model even in non-thin aquifers. [ABSTRACT FROM AUTHOR]

In this article, we perform an asymptotic analysis of a nonlocal reaction-diffusion equation, with a fractional laplacian as the diffusion term and with a nonlocal reaction term. Such equation models the evolutionary dynamics of a phenotypically structured population. We perform a rescaling considering large time and small effect of mutations, but still with algebraic law. With such rescaling, we expect that the phenotypic density will concentrate as a Dirac mass which evolves in time. To study such concentration phenomenon, we extend an approach based on Hamilton-Jacobi equations with constraint, that has been developed to study models from evolutionary biology, to the case of fat-tailed mutation kernels. However, unlike previous works within this approach, the WKB transformation of the solution does not converge to a viscosity solution of a Hamilton-Jacobi equation but to a viscosity supersolution of such equation which is minimal in a certain class of supersolutions. Such property allows to derive the concentration of the population density as an evolving Dirac mass, under monotony conditions on the growth rate, similarly to the case with thin-tailed mutation kernels [ABSTRACT FROM AUTHOR]

• Analysed in detail a two-step nonadiabatic competitive exothermic reaction scheme. • Identified two flame regimes and showed the difference between the adiabatic and nonadiabatic combustion waves. • Investigated the effect of heat loss on the behaviour of combustion waves. • Determined the extinction conditions analytically and numerically. We study the properties of travelling combustion waves in a diffusional thermal model with a two-step competitive exothermic reaction mechanism considering the system in one spatial dimension under nonadiabatic conditions. Based on the notion of the crossover temperature, the model is first examined analytically to predict the behaviour of travelling combustion waves in the limit of large activation energies. It is then studied numerically over a wide range of parameter values, such as those describing the ratios of the enthalpies, pre-exponential factors and activation energies. It is demonstrated that the nonadiabatic flame speed as a function of these parameters is a single-valued monotonic function with two flame regimes identified, each of which represents the region of the parameter values when one reaction dominates the other, while the flame speed as a function of the activation energy of the reaction R 2 (the exothermicity parameter) is either a c- or m-shaped. The extinction conditions for these two flame regimes are investigated analytically. [ABSTRACT FROM AUTHOR]

The characterization of the stress field for notch contained structures and components at elevated temperature is a preliminary to assess their safety and integrity reasonably. In this paper, the characteristics of the sharp V-notch tip field in power-law creeping materials are presented from the asymptotic viewpoint. Through asymptotic analysis and solution, the stress exponents and the angular distribution functions for various notch angles and creep exponents are presented. The asymptotic structure of the stress field for the sharp V-notch tip field is presented by proposing the amplitude factor K (t) which is dependent on the creep time, specimen geometry, notch angle as well as loading condition. The amplitude factor K (t) can bridge the fracture parameters between notch and crack in power-law creeping solids. A reasonable and accurate estimation method for the evaluation of the K (t) factor is also proposed. With the determination of K (t), the notch tip stress fields of single edged sharp V-notch specimens in power-law creeping materials with various notch angles and different depths are presented, compared and analyzed. It has been verified that the stress components for sharp V-notch specimens can be estimated accurately and reasonably with the theory and method proposed in this paper. The differences of the asymptotic solutions for sharp V-notch tip in power-law creeping solids and power-law hardening materials are also discussed and clarified. [ABSTRACT FROM AUTHOR]

The paper is concerned with the derivation of a general imperfect interface law in a linear multiphysics framework for a composite, constituted by two solids, separated by a thin adhesive layer. The analysis is performed by means of the asymptotic expansions technique. After defining a small parameter ε, which will tend to zero, associated with the thickness and the constitutive coefficients of the intermediate layer, we characterize three different limit models and their associated limit problems: the soft interface model, in which the constitutive coefficients depend linearly on ε; the hard interface model, in which the constitutive properties are independent of ε; the rigid interface model, in which they depend on 1 ε. The asymptotic expansion method is reviewed by taking into account the effect of higher order terms and by defining a general multiphysics interface law which comprises the above aforementioned models. [ABSTRACT FROM AUTHOR]

Abstract We consider a family of linearly elastic shells all sharing the same middle surface and in unilateral contact with a rigid foundation on the lower face. The shells are elliptic and their lateral face is clamped. Under these conditions, when the thickness tends to zero, the solution of the three-dimensional problem converges to the solution of a two-dimensional obstacle problem for an elastic membrane shell. In this paper we provide error estimates for this convergence. The proof uses a corrector method. [ABSTRACT FROM AUTHOR]

Fault detection and fault isolation performance of a model based diagnosis system mainly depends on the level of model uncertainty and the time allowed for detection. The longer time for detection that can be accepted, the more certain detection can be achieved and the main objective of this paper is to show how the window length relates to a diagnosis performance measure. A key result is an explicit expression for asymptotic performance with respect to window length and it is shown that there exists a linear asymptote as the window length tends to infinity. The gradient of the asymptote is a system property that can be used in the evaluation of diagnosis performance when designing a system. A key property of the approach is that the model of the system is analyzed directly, which makes the approach independent of detection filter design. [ABSTRACT FROM AUTHOR]

Bifurcations in the equilibrium shape of a thick elastic layer attached to a circular rigid cylinder rotating about its axis are investigated analytically and numerically. The centrifugal force breaks the symmetry of the system leading to deformations invariant along the axis as the result of an instability. The instability threshold depends at linear order on the relative thickness of the compliant layer, and on a dimensionless control parameter based on the elastic modulus, the angular velocity and the outer radius. A weakly non linear analysis, carried out for layers following the Mooney–Rivlin constitutive law, points out the discontinuous (sub-critical) features of the bifurcation, except for relative thickness laying in a very narrow range in which the bifurcation is super-critical. Numerical simulations in the fully post-buckled regime yield the absolute instability threshold, and the order in the rotational symmetry of the developed equilibrium shape. [ABSTRACT FROM AUTHOR]