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We study the temperature evolution in the three-dimensional skin tissue exposed to an electromagnetic beam of millimeter wavelength. The skin absorption coefficient of the beam frequency determines how deep the electromagnetic energy penetrates into the skin tissue, which gives a sub-millimeter penetration depth for a 94 GHz wave. In contrast, in the lateral directions perpendicular to the depth, the beam size is usually much larger than the penetration depth. Based on this separation of length scales, we establish an asymptotic formulation in which each term has separable dependences on the depth coordinate and on the lateral coordinates. We solve it analytically to obtain a two-term asymptotic solution of the temperature distribution in the three-dimensional skin tissue. This closed-form analytical solution provides a practical and accurate way of predicting the temperature. When the beam size is moderately larger than the penetration depth (a ratio of 20), the effect of lateral heat conduction is well captured in the asymptotic solution with maximum error less than 0.0017 in the normalized temperature of magnitude well above 1. [ABSTRACT FROM AUTHOR]

In this work, we employ the multiple scale method to introduce a novel analytical solution for an extended four-degrees-of-freedom dynamical system modeled on a swinging Atwood machine. We provide a methodology for obtaining the asymptotic solution up to the second-order approximation for both the swinging and modified swinging Atwood machine, demonstrating its solvability through the multiple scale approach. Subsequently, we present a comparative analysis of time histories between numerical and analytical solutions. These analytical solutions are of particular significance in applied mechanics, given their practical applications in parametric dynamical models grounded in the pendulum concept. [ABSTRACT FROM AUTHOR]

The elasticity theory problem for a radially inhomogeneous cylinder of small thickness, whose elastic moduli are arbitrary continuous functions depending on the radius of the cylinder, is considered. It is assumed that the side surface of the cylinder is stress-free, and the boundary conditions that keep the cylinder in equilibrium are given at its seats. Asymptotic solutions are constructed by the asymptotic integration method. It is shown that the asymptotic solution consists of the sum of the penetrating solution, simple boundary effect and boundary layer solutions. The character of the stress-strain state corresponding to the penetrating solution, simple boundary effect and boundary layer solutions is determined. Asymptotic formulas are obtained for displacements and stresses, which allow to calculate the stress-strain state of a cylinder.The problem of torsion of a radially inhomogeneous cylinder is studied, with its lateral surface free from stress and the boundary conditions keeping it in equilibrium at its seats. By applying the asymptotic integration method, it is determined that the asymptotic solution for the torsion problem consists of the sum of the penetrating solution and boundary layer solutions.Numerical analysis is performed and the effect of material inhomogeneity on the stress-strain state of the cylinder is evaluated.

Fractional differential calculus is a mathematical tool that has found applications in the study of social and physical behaviors considered "anomalous". It is often used when traditional integer derivatives models fail to represent cases where the power law is observed accurately. Fractional calculus must reflect non-local, frequency- and history-dependent properties of power-law phenomena. This tool has various important applications, such as fractional mass conservation, electrochemical analysis, groundwater flow problems, and fractional spatiotemporal diffusion equations. It can also be used in cosmology to explain late-time cosmic acceleration without the need for dark energy. We review some models using fractional differential equations. We look at the Einstein–Hilbert action, which is based on a fractional derivative action, and add a scalar field, ϕ , to create a non-minimal interaction theory with the coupling, ξ R ϕ 2 , between gravity and the scalar field, where ξ is the interaction constant. By employing various mathematical approaches, we can offer precise schemes to find analytical and numerical approximations of the solutions. Moreover, we comprehensively study the modified cosmological equations and analyze the solution space using the theory of dynamical systems and asymptotic expansion methods. This enables us to provide a qualitative description of cosmologies with a scalar field based on fractional calculus formalism. [ABSTRACT FROM AUTHOR]

We construct time-periodic asymptotic solutions for a one-dimensional system of nonlinear shallow water equations in a pool of variable depth D(x) with two shallow banks (which means that the function D(x) vanishes at the points defining the banks) or with one shallow bank and a vertical wall. Such solutions describe standing waves similar to well-known Faraday waves in pools with vertical walls. In particular, they approximately describe seiches in elongated bodies of water. The construction of such solutions consists of two stages. First, time-harmonic exact and asymptotic solutions of the linearized system generated by the eigenfunctions of the operator are determined, and then, using a recently developed approach based on simplification and modification of the Carrier–Greenspan transformation, solutions of nonlinear equations are reconstructed in parametric form. The resulting asymptotic solutions are compared with experimental results based on the parametric resonance excitation of waves in a bench experiment. [ABSTRACT FROM AUTHOR]

We continue the spectral analysis of Sturm-Liouville operators with ccmplex coefficients. By means of asymptotic integration the Titchmarsh-Weyl m-function is determined without the nesting circle analysis. With it the resolvent is constructed. The primary case is that of a dominant leading coefficient, but Euler type cases are also considered. This leads to resolvents that are compact and even Hilbert-Schmidt. [ABSTRACT FROM AUTHOR]

Purpose: In this paper, the analytic solution for a free rotatory motion under the influence of a motor of limited power is investigated; we aim to prove that the motion of the carrier body is close to rotation about a fixed axis depending upon the problem's parameters and the initial conditions. Method: Tensor calculus tools, asymptotic method, and kinematic equations of motion (EOM) are used. Results: At a large time, the asymptotic properties of solutions and a system of linear differential equations that describes the approximate gyrostat motion are obtained. Conclusion: The motion of the carrier body, which is close to the rotation around an axis, whose direction is fixed, depends upon the problem's parameters and the initial conditions. [ABSTRACT FROM AUTHOR]

Fractional differential calculus is a mathematical tool that has found applications in the study of social and physical behaviors considered “anomalous”. It is often used when traditional integer derivatives models fail to represent cases where the power law is observed accurately. Fractional calculus must reflect non-local, frequency- and history-dependent properties of power-law phenomena. This tool has various important applications, such as fractional mass conservation, electrochemical analysis, groundwater flow problems, and fractional spatiotemporal diffusion equations. It can also be used in cosmology to explain late-time cosmic acceleration without the need for dark energy. We review some models using fractional differential equations. We look at the Einstein–Hilbert action, which is based on a fractional derivative action, and add a scalar field, ϕ, to create a non-minimal interaction theory with the coupling, ξRϕ2, between gravity and the scalar field, where ξ is the interaction constant. By employing various mathematical approaches, we can offer precise schemes to find analytical and numerical approximations of the solutions. Moreover, we comprehensively study the modified cosmological equations and analyze the solution space using the theory of dynamical systems and asymptotic expansion methods. This enables us to provide a qualitative description of cosmologies with a scalar field based on fractional calculus formalism.

Asymptotic solutions of the boundary layer equations leading the flow of viscous fluid caused by stretching surface; outcomes are based on the asymptotic integration of second order linear differential equations. Also proved that the asymptotic formulae will exhibit non-oscillatory behaviour. These asymptotic solutions will always be bounded in nature. [ABSTRACT FROM AUTHOR]

In this paper, we use the Papkovich–Neuber potentials to derive the variant of general solution for the plain problems of strain gradient elasticity theory (SGET) in bounded domains. General solution contains complete sets of functions satisfying 2D Laplace equations and modified Helmholtz equations, including the polynomial and modified Bessel functions of integer and fractional orders with corresponding angular periodicity. It is shown that proposed form of general solution allows to derive the known SGET asymptotic solutions for the crack-tip fields. [ABSTRACT FROM AUTHOR]

An accurate fracture analysis of multi-material V-notched Reissner plates under bending or twisting is performed by a new coupling approach of isogeometric analysis (IGA) and eigenfunction expansion method. In this approach, the overall plate is divided into two regions: a singular region near the notch tip (near field) and a regular region far away from the notch tip (far field). The overall model is discretized by a T-spline-based IGA. Asymptotic solutions of singular stress fields are applied to the near field, and therefore, the large number of fundamental unknowns of control points is reduced into a small set of undetermined coefficients. The unknowns in the far field remain unchanged. Consequently, the computational cost is significantly reduced, and explicit expressions of singular stress components in the vicinity of notch tip are obtained without post processing. Comparisons are presented to demonstrate the accuracy and convergence of the present approach. Effects of key influencing factors on the singularity order and singular items of stresses are investigated also. [ABSTRACT FROM AUTHOR]

We investigate a susceptible-infected-susceptible (SIS) epidemic model based on the Caputo–Fabrizio operator. After performing an asymptotic analysis of the system, we study a related finite horizon optimal control problem with state constraints. We prove that the corresponding value function satisfies in the viscosity sense a dynamic programming equation. We then turn to the asymptotic behavior of the value function, proving its convergence to the solution of a stationary problem, as the planning horizon tends to infinity. Finally, we present some numerical simulations providing a qualitative description of the optimal dynamics and the value functions involved. [ABSTRACT FROM AUTHOR]

This paper aims to investigate a boundary layer flow with heat and mass transfer of a special third grade fluid over a stretchable surface in a parallel free stream. Using the similarity variables generated by the Lie group analysis, the governing nonlinear partial differential equations (PDEs) are transformed into a system of nonlinear ordinary differential equations (ODEs). The transformed equations are then solved numerically using the shooting technique. In addition, an attempt is made to carry the asymptotic solution behavior for large stretching and suction parameters, and the asymptotic results of skin friction coefficient are then compared with the direct numerical solutions, which shows a good agreement. It is observed that the similarity equations exhibit dual solutions in a certain range of shrinking strength. These two solution branches show different behavior on the skin friction coefficient, local Nusselt number, Sherwood number, velocity, and temperature profiles. Thus, emphasis has been given to carrying out a stability analysis to determine the physically reliable solution. The stability analysis shows that the upper branch solution is stable. It is observed that the suction parameter increases the range of dual solutions and the magnitude of the critical point from where the dual solutions bifurcate; however, the non-Newtonian parameter shows the opposite behavior. The momentum, thermal and concentration boundary layer thicknesses in the upper branch solution are lower than the lower branch solution. [ABSTRACT FROM AUTHOR]

The paper deals with the asymptotic expressions for the solutions along with their first derivatives, the distribution of the eigenvalues and the normalized eigenvector for large eigenvalues corresponding to a system of second order differential equations having turning points at both ends of the interval, under certain suitable boundary conditions. [ABSTRACT FROM AUTHOR]

Hyperthermia, an alternative medical approach aiming at locally increasing the temperature of a tumor, can cause the "death" of cancer cells or the sensitization of them to chemotherapeutic drugs and radiation. In contrast with the conventional treatments, hyperthermia provokes no injury to normal tissues. In particular, magnetic hyperthermia can utilize iron oxide nanoparticles, which can be administered intravenously to heat tumors under an alternating magnetic field. Currently, there is no theoretical model in the relative literature for the effective thermal conductivity of blood and magnetic nanoparticles. The scope of the present study is twofold: (a) development of a theoretical relationship, based on experimental findings and blood structure and (b) study of the laminar natural convection in a simplified rectangular porous enclosure, by using the asymptotic expansions method for deriving ordinary differential equations of the mass, momentum and energy balances, as a first approach of investigating heat transfer and providing theoretical guidelines. In short, the thermal conductivity of the resulting bio-nanofluid tends to increase by both increasing the concentration of the nanoparticles and the temperature. Furthermore, the heat transfer is enhanced for more intense internal heating (large Rayleigh numbers) and more permeable media (large Darcy numbers), while larger nanoparticle concentrations tend to suppress the flow. [ABSTRACT FROM AUTHOR]

Several classes of asymptotic solutions of the discrete Painlevé equation of second type (dPII) for large values of the independent variable are found. The cases of complex and real solutions are considered. as well as special solutions related to symmetric group representations. [ABSTRACT FROM AUTHOR]

Based on a devised gradient-holonomic integrability testing algorithm, we analyze a class of dark type nonlinear dynamical systems on spatially one-dimensional functional manifolds possessing hidden symmetry properties and allowing their linearization on the associated cotangent spaces. We described main spectral properties of nonlinear Lax type integrable dynamical systems on periodic functional manifolds particular within the classical Floquet theory, as well as we presented the determining functional relationships between the conserved quantities and related geometric Poisson and recursion structures on functional manifolds. For evolution flows on functional manifolds, parametrically depending on additional functional variables, naturally related with the classical Bellman-Pontriagin optimal control problem theory, we studied a wide class of nonlinear dynamical systems of dark type on spatially one-dimensional functional manifolds, which are both of diffusion and dispersion classes and can have interesting applications in modern physics, optics, mechanics, hydrodynamics and biology sciences. We prove that all of these dynamical systems possess rich hidden symmetry properties, are Lax type linearizable and possess finite or infinite hierarchies of suitably ordered conserved quantities. [ABSTRACT FROM AUTHOR]

We describe a method for calculating the crack tip stress intensity factors for the problem of one or two cracks at the apex of an arbitrary angle wedge. The kernels for a dislocation in an arbitrary angle wedge described in part 1 of this paper are used extensively. Consideration is given to variations of crack length, crack angle and wedge angle. [ABSTRACT FROM AUTHOR]

We consider the state of stress created by the presence of an edge dislocation at an arbitrary position, in a wedge of arbitrary internal angle. A method for determining the state of stress in the wedge is demonstrarted and verified against finite element method simulations. Furthermore, a Mellin transform is employed to ensure that the free surfaces of the wedge remain traction free along their length. [ABSTRACT FROM AUTHOR]

We comprehensively analyse the dynamics for the gravitational field equations for the Chiral-Quintom theory in a Friedman–Lemaître–Robertson–Walker cosmology with an additional matter source. We consider a new set of dimensionless variables and write the field equations in the equivalent form of an algebraic-differential system. Specifically, we consider two families of quintom models where the two scalar fields interact in the kinetic sector. We mathematically focus on the dynamical effect of spatial curvature. Physically, we find two periods of inflation related to the Universe's early and late-time acceleration phases. [ABSTRACT FROM AUTHOR]

Humans have been arguing about the benefits of dictatorial versus democratic regimes for millennia. Despite drastic differences between the dictatorships in the world, one of the key common features is the Dictator's Dilemma as defined by Wintrobe [1] : a dictator will never know the true state of affairs in his country and is perpetually presented distorted information, thus having difficulties in making the right governing decisions. The dictator's dilemma is essential to most autocratic regimes and is one of the key features in the literature on the subject. Yet, no quantitative theory of how the distortion of information develops from the initial state has been developed up to date. I present a model of the appearance and evolution of such information distortion, with subsequent degradation of control by the dictator. The model is based on the following fundamental and general premises: a) the dictator governs aiming to follow the desired trajectory of development based only on the information from the advisors; b) the deception from the advisors cannot decrease in time; and c) the deception change depends on the difficulties the country encounters. The model shows effective control in the short term (a few months to a year), followed by instability leading to the country's gradual deterioration of the state over many years. I derive some universal parameters applicable to all dictators and show that advisors' deception increases parallel with the decline of the control. In contrast, the dictator thinks the government is doing a reasonable, but not perfect, job. Finally, I present a match of our model to the historical data of grain production in the Soviet Union in 1928–1940. • The Dictator Dilemma prevents the dictator from knowing accurate information; • A model of the corruption of information in authoritarian societies is developed; • The model is based on the corruption of information from advisors; • The model shows effective control in the short term, followed by deterioration; • The instability of the model is shown for general cases of corruption dynamics; • The instability is caused by the random forcing terms and structure of equations; • The model is extended for the case of advisory board and general forcing; • This model is applied to the historical data of grain harvest in the Soviet Union. [ABSTRACT FROM AUTHOR]

In this study, by assuming a small bottom slope, asymptotic solutions were developed to discuss natural convection within rooted emergent vegetation in response to different heating and cooling mechanisms. Based upon the maximum water depth in comparison to the penetration depth of solar radiation, two scenarios in shallow and deep waters were examined. The temperature structures showed that isotherms in shallows are near vertical but become stable stratified layers (horizontal isotherms) in deep regions. In shallow regions, horizontal velocity profiles perform classic cubic shapes, while the horizontal velocity in deep regions is constant near the surface, and a local upslope flow occurs near the bottom. In shallow water, viscous effects are dominant to shape the velocity profiles, whereas vegetation drag becomes more important in deep regions. By using turbulent parameters, horizontal exchange flowrates and velocities predicted by the asymptotic solutions show good agreements with the existing measurements. [ABSTRACT FROM AUTHOR]

We consider the evolution of the solution of a class of scalar nonlinear hyperbolic reaction–diffusion equations which incorporate a relaxation time and with a reaction function given by a monostable cubic polynomial. An initial-value problem is studied when the prescribed starting data are given by a simple step function. It is established that the large-time structure of the solution is governed by the evolution of a propagating wave-front. The character of this front can be one of three forms, either reaction–diffusion, reaction–relaxation or reaction–relaxation–diffusion, which is relevant and depends on the particular values of the problem parameters that describe the underlying reaction polynomial. [ABSTRACT FROM AUTHOR]

Summary: In this work, we have comprehensively investigated the impact of non‐linearity (α − μ) and shadowing (inverse gamma) collectively over the wireless channel. To this effect, an approximate closed‐form probability density function (PDF) expression is derived over α − μ/inverse gamma fading channel by employing Gauss–Laguerre quadrature polynomial. The accuracy of the proposed approximate solution is validated through the Kullback–Leibler divergence test. Subsequently, the proposed solution is utilized in the development of various fundamental statistics like commutative distribution function (CDF), moment generating function (MGF) and nth moment. This contribution also includes the derivation of coding gain and diversity gain to study average symbol error probability (SEP) with MRC and EGC diversity. It is observed that diversity gain is independent of shadowing and as the channel condition improves, the coding gain decreases. Finally, the closed‐form solutions of spectral efficiency under different adaptive transmission policies are also developed specifically, optimal power and rate adaptation (OPRA) and effective capacity along with simpler low‐ and high‐power expressions. Monte‐Carlo simulations are utilized to validate the proposed analytical and numerical results. [ABSTRACT FROM AUTHOR]

Anisotropic fiber-reinforced composites are used in lightweight construction, which is of great industrial relevance. During mold filling of fiber suspensions, the microstructural evolution of the local fiber arrangement and orientation distribution is determined by the local velocity gradient. Based on the Folgar–Tucker equation, which describes the evolution of the second-order fiber orientation tensor in terms of the velocity gradient, the present study addresses selected states of deformation rates that can locally occur in complex flow fields. For such homogeneous flows, exact solutions for the asymptotic fiber orientation states are derived and discussed based on the quadratic closure. In contrast to the existing literature, the derived exact solutions take into account the fiber-fiber interaction. The analysis of the asymptotic solutions relying upon the common quadratic closure shows disadvantages with respect to the predicted material symmetry, namely, the anisotropy is overestimated for strong fiber-fiber interaction. This motivates us to suggest a novel normalized fully symmetric quadratic closure. Two versions of this new closure are derived regarding the prediction of anisotropic properties and the fiber orientation evolution. The fiber orientation states determined with the new closure approach show an improved prediction of anisotropy in both effective viscous and elastic composite behaviors. In addition, the symmetrized quadratic closure has a simple structure that reduces the effort in numerical implementation compared to more elaborated closure schemes. [ABSTRACT FROM AUTHOR]

The paper deals with an application of the direct scheme method, consisting of immediately substituting a postulated asymptotic solution into a problem condition and determining a series of control problems for finding asymptotics terms, for asymptotics construction of a solution of a weakly nonlinearly perturbed linear-quadratic optimal control problem with three-tempo state variables. For the first time, explicit formulas for linear-quadratic optimal control problems, from which all terms of the asymptotic expansion are found, are justified, and the estimates of the proximity between the asymptotic and exact solutions are proved for the control, state trajectory, and minimized functional. Non-increasing of the minimized functional, if a next approximation to the optimal control is used, following from the proposed algorithm of the asymptotics construction, is also established.

Hyperthermia, an alternative medical approach aiming at locally increasing the temperature of a tumor, can cause the “death” of cancer cells or the sensitization of them to chemotherapeutic drugs and radiation. In contrast with the conventional treatments, hyperthermia provokes no injury to normal tissues. In particular, magnetic hyperthermia can utilize iron oxide nanoparticles, which can be administered intravenously to heat tumors under an alternating magnetic field. Currently, there is no theoretical model in the relative literature for the effective thermal conductivity of blood and magnetic nanoparticles. The scope of the present study is twofold: (a) development of a theoretical relationship, based on experimental findings and blood structure and (b) study of the laminar natural convection in a simplified rectangular porous enclosure, by using the asymptotic expansions method for deriving ordinary differential equations of the mass, momentum and energy balances, as a first approach of investigating heat transfer and providing theoretical guidelines. In short, the thermal conductivity of the resulting bio-nanofluid tends to increase by both increasing the concentration of the nanoparticles and the temperature. Furthermore, the heat transfer is enhanced for more intense internal heating (large Rayleigh numbers) and more permeable media (large Darcy numbers), while larger nanoparticle concentrations tend to suppress the flow.

In this note, we extend the problem treated in (Lok, Math Modelling Anal 24:617–634 (2019)) to the case of permeable surface which is shrinking in mutually orthogonal directions. Both numerical and asymptotic solutions are obtained for two important governing parameters, γ the shrinking rate and S characterizing the fluid transfer through the boundary. In this problem, a restriction on S is required for a solution to exist. This contrasts with the problem in (Lok, Math Modelling Anal 24:617–634 (2019)) where no restriction on S is needed. Numerical solutions show that for a fixed value of S, two critical points γ c are observed for S > 2 . Conversely, two critical points S c are found for a given value of γ when S > 2 . A discussion on the nonexistence of solution for S = 2 is given and asymptotic solutions for S large and (S - 2) small are also presented. [ABSTRACT FROM AUTHOR]

Based on a devised gradient-holonomic integrability testing algorithm, we analyze a class of dark type nonlinear dynamical systems on spatially one-dimensional functional manifolds possessing hidden symmetry properties and allowing their linearization on the associated cotangent spaces. We described main spectral properties of nonlinear Lax type integrable dynamical systems on periodic functional manifolds particular within the classical Floquet theory, as well as we presented the determining functional relationships between the conserved quantities and related geometric Poisson and recursion structures on functional manifolds. For evolution flows on functional manifolds, parametrically depending on additional functional variables, naturally related with the classical Bellman-Pontriagin optimal control problem theory, we studied a wide class of nonlinear dynamical systems of dark type on spatially one-dimensional functional manifolds, which are both of diffusion and dispersion classes and can have interesting applications in modern physics, optics, mechanics, hydrodynamics and biology sciences. We prove that all of these dynamical systems possess rich hidden symmetry properties, are Lax type linearizable and possess finite or infinite hierarchies of suitably ordered conserved quantities.

The concept of the renormalization group (RG) emerged from the renormalization of quantum field variables, which is typically used to deal with the issue of divergences to infinity in quantum field theory. Meanwhile, in the study of phase transitions and critical phenomena, it was found that the self–similarity of systems near critical points can be described using RG methods. Furthermore, since self–similarity is often a defining feature of a complex system, the RG method is also devoted to characterizing complexity. In addition, the RG approach has also proven to be a useful tool to analyze the asymptotic behavior of solutions in the singular perturbation theory. In this review paper, we discuss the origin, development, and application of the RG method in a variety of fields from the physical, social and life sciences, in singular perturbation theory, and reveal the need to connect the RG and the fractional calculus (FC). The FC is another basic mathematical approach for describing complexity. RG and FC entail a potentially new world view, which we present as a way of thinking that differs from the classical Newtonian view. In this new framework, we discuss the essential properties of complex systems from different points of view, as well as, presenting recommendations for future research based on this new way of thinking. [ABSTRACT FROM AUTHOR]

In this study, by assuming a small bottom slope, asymptotic solutions were developed to discuss natural convection within rooted emergent vegetation in response to different heating and cooling mechanisms. Based upon the maximum water depth in comparison to the penetration depth of solar radiation, two scenarios in shallow and deep waters were examined. The temperature structures showed that isotherms in shallows are near vertical but become stable stratified layers (horizontal isotherms) in deep regions. In shallow regions, horizontal velocity profiles perform classic cubic shapes, while the horizontal velocity in deep regions is constant near the surface, and a local upslope flow occurs near the bottom. In shallow water, viscous effects are dominant to shape the velocity profiles, whereas vegetation drag becomes more important in deep regions. By using turbulent parameters, horizontal exchange flowrates and velocities predicted by the asymptotic solutions show good agreements with the existing measurements.

The Korteweg–de Vries equation with variable coefficients and a small parameter of the first level at the higher-order derivative is considered. The concept of a stepwise asymptotic solution is introduced. On the basis of the nonlinear WKB method, an algorithm for creating such solutions is developed and substantiated. The order on small parameter of the asymptotic accuracy of the constructed approximate solution satisfying the original equation is established. [ABSTRACT FROM AUTHOR]

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We consider a standard Navier–Stokes system on the n-D torus T n = R n / Z n , valid when the flow is not very compressible and the temperature does not vary too much. We construct a sequence of approximate solutions that tend to satisfy the equations in a weak sense for arbitrary physical initial conditions. By weak compactness, we obtain Radon measure solutions in density and momentum when the velocity of the flow is finite, as numerically observed in all tests, and the absence of void regions in the flow. We notice that the method also applies without viscosity and complements results on other systems of evolution equations such as the systems of isothermal and isentropic gas flows considered in Colombeau (Z Angew Math Phys 66(5):2575–2599, 2015). [ABSTRACT FROM AUTHOR]

Starting with this paper, we intend to develop a program aiming at construction of boundary conditions (BCs) for hyperbolic relaxation systems. Physically, such BCs are not always available. The construction is based on the assumption that the relaxation systems and well-posed BCs for the corresponding equilibrium systems are given. This paper focuses on the linearized Suliciu model. We obtain strictly dissipative and compatible BCs for the linearized model with different non-characteristic boundaries. Moreover, the effectiveness of the constructed BCs is shown by resorting to formal asymptotic solutions and energy estimates. [ABSTRACT FROM AUTHOR]

In a recent paper, Al-Housseiny and Stone (J Fluid Mech 706:597–606, 2012) considered the dynamics of a stretching surface and how this interacts with the boundary-layer flow it generates. These authors discussed the cases c = - 3 for an elastic sheet and c = - 1 for the viscous fluid, c being representative for the stretching velocity of the sheet. The aim of the present paper is to extend the analysis of Al-Housseiny and Stone (2012) to the general values of c, to allow for both a stretching and a shrinking sheet and for the surface to be permeable through the parameter S, where S > 0 for the fluid withdrawal and S < 0 for fluid injection. Both the cases S = 0 (impermeable surface) and S ≠ 0 (permeable surface) are considered for both stretching surfaces and shrinking surfaces. In all these cases, asymptotic solutions are presented for large values c and S (both withdrawal and injection). [ABSTRACT FROM AUTHOR]