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We study the Boussinesq approximation for the incompressible Euler equations using Lagrangian description. The conditions for the Lagrangian fluid map are derived in this setting, and a general method is presented to find exact fluid flows in both the two-dimensional and the three-dimensional case. There is a vast amount of solutions obtainable with this method and we can only showcase a handful of interesting examples here, including a Gerstner type solution to the two-dimensional Euler–Boussinesq equations. In two earlier papers we used the same method to find exact Lagrangian solutions to the homogeneous Euler equations, and this paper serves as an example of how these same ideas can be extended to provide solutions also to related, more involved models. [ABSTRACT FROM AUTHOR]

We consider a system of two coupled parallel queues with infinite waiting rooms. The time setting is discrete. In either queue, the service of a customer requires exactly one discrete time slot. Arrivals of new customers occur independently from slot to slot, but the numbers of arrivals into both queues within a slot may be mutually dependent. Their joint probability generating function (pgf) is indicated as A (z 1 , z 2) and characterizes the whole model. In general, determining the steady-state joint probability mass function (pmf) u (m , n) , m , n ≥ 0 or the corresponding joint pgf U (z 1 , z 2) of the numbers of customers present in both queues is a formidable task. Only for very specific choices of the arrival pgf A (z 1 , z 2) are explicit results known. In this paper, we identify a multi-parameter, generic class of arrival pgfs A (z 1 , z 2) , for which we can explicitly determine the system-content pgf U (z 1 , z 2) . We find that, for arrival pgfs of this class, U (z 1 , z 2) has a denominator that is a product, say r 1 (z 1) r 2 (z 2) , of two univariate functions. This property allows a straightforward inversion of U (z 1 , z 2) , resulting in a pmf u (m , n) which can be expressed as a finite linear combination of bivariate geometric terms. We observe that our generic model encompasses most of the previously known results as special cases. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the motion of the wind in the steady atmospheric Ekman layer with suitable boundary conditions in ellipsoidal coordinates. Firstly, we consider the system with the piecewise eddy viscosity coefficients. When the eddy viscosity of the upper layer is constant and the eddy viscosity of the bottom layer is related to the height, we transform the system to a first-order Riccati equation with a suitable initial value. Then, we also construct the solution to the system with the piecewise-constant eddy viscosity. Secondly,we derive the explicit solution to the system with a small perturbation of a constant. Finally, by choosing an exponential variation of the eddy viscosity, we get the explicit solution to the system. [ABSTRACT FROM AUTHOR]

Solutions of a group of conjugate time-varying matrix equations are discussed in this paper. Through mathematical derivation, the solutions to this group of equations are equivalent to the solutions to a class of conjugate time-invariant matrix equations. Further, the related conditions of solvability are obtained and the general explicit solutions are represented by using quasicontrollability and quasiobservability matrices. A detailed algorithm is presented to make the calculation process clear, and the effectiveness of the algorithm is verified by a concrete example. The proposed algorithm can provide complete solutions to the considered equation in explicit parametric form and its main computation includes solving an ordinary linear algebraic equation and some matrix multiplication operations. [ABSTRACT FROM AUTHOR]

In this paper, we present a new general system of equations describing the steady motion of atmosphere with uniform density in ellipsoidal coordinates, which is derived from the general governing equations for viscous fluids. We first show that this new system can be reduced to the classic Ekman equations. Secondly, we obtain the explicit solution of the Ekman equations in ellipsoidal coordinates. Thirdly, for the viscosity related to the height, we obtain the solution of the classical problem with zero acceleration at the bottom of Ekman layer. Finally, the uniqueness and dynamical properties of solution are demonstrated. [ABSTRACT FROM AUTHOR]

This article studies a class of integrable semi-discrete equations with one continuous and two discrete independent variables. At present, in the literature there are nine integrable equations of the form u n + 1 , x j = f (u n , x j , u n j + 1 , u n j , u n + 1 j , u n + 1 j − 1) up to point transformations. An efficient method based on some relation that generalizes the notion of the local conservation law is proposed for searching for Miura-type transformations relating to semi-discrete equations in 3D. The efficiency of the method is illustrated with the equations from the list. For one of the equations, which is little studied, the continuum limit is calculated. For this equation, the problem of finite-field reductions in the form of Darboux integrable systems of equations of a hyperbolic type is discussed. For reductions of small orders, N = 1 and N = 2 , complete sets of characteristic integrals are presented. Note that the existence of characteristic integrals makes it possible to construct particular solutions to the original lattice. For the case N = 1 , an explicit solution was found in this paper. A new semi-discrete equation is found that lies beyond the considered class. For this equation, the Lax pair is presented. [ABSTRACT FROM AUTHOR]

This paper is concerned to study the Dirichlet type quasi-static boundary value problem of coupled theory of elasticity for porous circular ring. The obtained solution is represented as absolutely and uniformly convergent series. [ABSTRACT FROM AUTHOR]

Positioning an object with biased distance measurements is exactly solvable if exact knowledge of the reference locations and noise-free range measurements are assumed. By examining the positioning algebra, this paper obtains explicit necessary and sufficient conditions for the positioning problem to have a unique or twin solutions. The intersection of negative hyperconic halves is shown to be an exact geometric interpretation of the positioning solutions. The placement of non-coplanar references largely ensures but does not guarantee unique positioning. Given a set of references, object region for non-unique positioning is identifiable by using the conditions derived in this work. The placement of five references to form an asymmetric hexahedron is postulated to be sufficient for unique positioning in a three-dimensional environment. Illustrative examples explain these findings. [ABSTRACT FROM AUTHOR]

In this paper, we provide a systematic way of finding explicit solutions for a class of continuous fragmentation equations with growth or decay in the state space and derive new explicit solutions in the cases of constant and linear growth/decay coefficients. [ABSTRACT FROM AUTHOR]

In this paper, the extended double (2+1)-dimensional sine-Gorden equation is studied. First of all, using the symmetry method, the corresponding vector fields, Lie algebra and infinitesimal generators are derived. Then, from infinitesimal generators, the symmetry reductions are presented. In addition, these reduced equations are converted into the corresponding partial differential equations, which including classical double (1+1)-dimensional sine-Gorden equation. Moreover, based on the Lie symmetry method again, these reduced equations are investigated. Meanwhile, based on traveling wave transformation, some explicit solutions of the extended double (2+1)-dimensional sine-Gorden equation are obtained. Consequently, a conservation law is derived via conservation law multiplier method. Finally, especially with the help of the fractional complex transform, some solutions of double time fractional (2+1)-dimensional sine-Gorden equation are also derived. These results might explain complex nonlinear phenomenon. [ABSTRACT FROM AUTHOR]

In this paper, we study the logarithmic-KP equation. The analysis depends mainly on the Lie symmetry method. The corresponding vector fields and symmetry reductions are derived. Furthermore, the conservation laws of the equation are constructed. [ABSTRACT FROM AUTHOR]

In this paper, we study the standard problem of the wind in the steady atmospheric Ekman layer with classical boundary conditions. We consider the system with varying eddy viscosity coefficients that are small perturbation of a constant. We derive the explicit solution by using a different argument in the previous works. For two layers, the eddy viscosity is constant in the upper layer, while is only continuous with height in the lower layer, we transform the system to a first order Riccati equation with a suitable initial value and derive the solution for piecewise-constant eddy viscosity. [ABSTRACT FROM AUTHOR]

This paper presents an efficient real-time implementation of embedded model predictive control, adopted in the context of active vibration control with the objective of minimizing the tip deflection of lightly damped cantilever beams. In particular, we focus on memory and time-efficient explicit solutions of the associated constrained optimal control problems that are easily implementable on low-end embedded hardware. To this end, we exploit the concept of convex lifting and show how it can be used to devise low-complexity, regionless piecewise affine controllers without any loss of optimality and performance. The efficiency of this constructive procedure is quantified via an extensive complexity analysis, evidenced by a successful practical deployment and optimal vibration control performance using a family of 32-bit ARM Cortex-M-based microcontroller platforms. [ABSTRACT FROM AUTHOR]

The present paper deals with the three-dimensional linear theory of elastic materials with inner structure whose particles, in addition to the classical displacement and temperature fields, possess microtemperatures. In the half-space solutions of some basic boundary value problems are constructed in quadratures. [ABSTRACT FROM AUTHOR]

This paper is concerned to study quasi-static boundary value problems of coupled linear theory of elasticity for porous circle and for plane with a circular hole. The Dirichlet type boundary value problem for a circle and the Neumann boundary value problem for a plane with a circular hole are solved explicitly. All the formulas are presented in explicit ready-to-use form. The solutions are represented by means of absolutely and uniformly convergent series. [ABSTRACT FROM AUTHOR]

Wave motions play an important role in fluid dynamics and engineering issues. In this paper, a systematic study to the generalized KdV–Burgers–Kuramoto equation is presented resort to symmetry method. First, based on the Lie point symmetries of the generalized KdV–Burgers–Kuramoto equation, we get invariants and invariant solutions. In particular, we obtain series solutions of generalized KdV–Burgers–Kuramoto equation. Meanwhile, we find that this equation just exists in Lie point symmetries. Then, we present a conservation law, and derive reciprocal Bäcklund transformations of conservation law for the first time. Furthermore, Bäcklund transformation of KdV–Burgers–Kuramoto equation associated with truncated Painlevé expansion is studied for the first time. Subsequently, mCK method is employed to study KdV–Burgers–Kuramoto equation. Lastly, we investigate symmetries of generalized time fractional KdV–Burgers–Kuramoto equation, and derive a conservation law. The obtained results illustrate that symmetry method is a very powerful method to deal with nonlinear partial differential equations. The results provide theoretical support for explaining complex nonlinear phenomena. [ABSTRACT FROM AUTHOR]

In this paper we apply the method of stochastic characteristics to a Lighthill–Whitham–Richards model. The stochastic perturbation can be seen as errors in measurement of the traffic density. For concrete examples we solve the equation perturbed by a standard Brownian motion and the geometric Brownian motion without drift. [ABSTRACT FROM AUTHOR]

An advantage of a high-order fully-actuated (HOFA) system is that there exists a controller such that a constant linear closed-loop system with an arbitrarily assignable eigenstructure can be obtained. In this paper, a generalised form of the conventional first-order strict-feedback systems (SFSs) is firstly proposed, and a recursive solution is proposed to convert equivalently the generalised SFS into a HOFA model. Then the second- and high-order SFSs are defined and their equivalent HOFA models are also derived. It is further shown that, under certain common conditions, the recursive solutions for converting the generalised SFSs into HOFA models can be rearranged into direct analytical explicit solutions. Such a high-order system approach is more direct and simpler than the first-order system approach since it avoids the process of converting firstly these second- and high-order SFSs into first-order ones for control, and can finally produce a constant linear closed-loop system. Particularly, it is more effective than the well-known method of backstepping since, for the generalised complicated SFSs with more subsystems, the method of backstepping may simply be not applicable due to more serious 'differential explosion' problem. Two examples are worked out to demonstrate the effect of the approach. [ABSTRACT FROM AUTHOR]

In this paper, we study the classical problem of the wind in the steady atmospheric Ekman layer with classical boundary conditions and the eddy viscosity is an arbitrary height-dependent function with a finite limit value. We present existence and uniqueness and smooth results to justify computing first order approximation of solutions. Using a different argument that in previous works, we construct the Green's function to derive the solution by a perturbation approach. [ABSTRACT FROM AUTHOR]

We obtain for the two-phase Lame-Clapeyron-Stefan problem for a semi-infinite material an equivalence between the temperature and convective boundary conditions at the fixed face in the case that an inequality for the convective transfer coefficient is satisfied. Moreover, an inequality for the coefficient which characterizes the solid-liquid interface of the classical Neumann solution is also obtained. This inequality must be satisfied for data of any phase-change material, and as a consequence the result given in our previous paper is also recovered when a heat flux condition was imposed at the fixed face. [ABSTRACT FROM AUTHOR]

In this paper, Lie symmetry method is applied to investigate the invariance properties of the time fractional Kolmogorov–Petrovskii–Piskunov equation with Riemann–Liouville derivative. In view of point symmetry, the vector fields for the governing equation are well constructed. And then the equation can be reduced to a fractional ODE with the help of Erdélyi–Kober operator. Moreover, a kind of explicit power series solutions of the equation is derived by virtue of the power series theory. Lastly, based on Ibragimov's method, two kinds of conservation laws for the equation are established. [ABSTRACT FROM AUTHOR]

In this paper, we consider an explicit solution of system of Sylvester matrix equations of the form A1V1 − E1V1F1 = B1W1 and A2V2 − E2V2F2 = B2W2 with F1 and F2 being arbitrary matrices, where V1,W1,V2 and W2 are the matrices to be determined. First, the definitions, of the matrix polynomial of block matrix, Sylvester sum, and Kronecker product of block matrices are defined. Some definitions, lemmas, and theorems that are needed to propose our method are stated and proved. Numerical test problems are solved to illustrate the suggested technique. [ABSTRACT FROM AUTHOR]

One-dimensional unsteady-state advection-dispersion models are useful in practice for predicting time-varying concentrations in streams for phenomena such as chemical spills or combined sewer overflows. The challenge in modeling such phenomena, which demonstrate sharp concentration gradients, is to ensure that the numerical dispersion, an artificial erroneous smearing of concentrations, not overshadow the true dispersion, which, while of significance due to sharp gradients, is ordinarily small compared to advection as a transport mechanism. This paper revisits stability criteria and numerical dispersion properties of explicit Lagrangian and Eulerian solutions of the one-dimensional advection-dispersion equation. It is shown that the Leith method of correcting for numerical dispersion (which was developed prior to the recent increase in popularity of Lagrangian methods) has similar stability restrictions on the step sizes as the latter when the zero-overshoot criterion for stability is used. For uncorrected Eulerian formulations, the stable regions for Dx and Dt depend on the approximation of the advection term. For the forward and central difference approximations, the region is bounded, while for the backward difference formulation, Dx is unbounded. For Lagrangian formulations, the stable region has no upper bound, but has a lower bound on Dx. Correcting numerical dispersion in Eulerian formulations not only results in substantially better accuracy but changes the stable regions for Dx and Dt. The stable regions for corrected Eulerian formulations indeed, the formulations themselves — are independent of the type of approximation of the first derivative, and are unbounded, although they become increasingly narrow as the true dispersion decreases. As the true dispersion approaches zero, the stable region approaches a line which corresponds to a Courant number of one. Thus, in the limit where the dispersion approaches zero, corrected Eulerian formulations, like Lagrangian formulations, must use a Courant number of one, but corrected Eulerian formulations do not have the lower bound on step size. [ABSTRACT FROM AUTHOR]

The Colebrook-White equation is often used for calculation of the friction factor in turbulent regimes; it has succeeded in attracting a great deal of attention from researchers. The Colebrook–White equation is a complex equation where the computation of the friction factor is not direct, and there is a need for trial-error methods or graphical solutions; on the other hand, several researchers have attempted to alter the Colebrook-White equation by explicit formulas with the hope of achieving zero-percent (0%) maximum deviation, among them Dejan Brkić and Pavel Praks. The goal of this paper is to discuss the results proposed by the authors in their paper:" Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function" and to propose more accurate formulas. [ABSTRACT FROM AUTHOR]

Despite numerous studies of epidemiological systems, the role of seasonality in the recurrent epidemics is not entirely understood. During certain periods of the year incidence rates of a number of endemic infectious diseases may fluctuate dramatically. This influences the dynamics of mathematical models describing the spread of infection and often leads to chaotic oscillations. In this paper, we are concerned with a generalization of a classical Susceptible–Infected–Recovered epidemic model which accounts for seasonal effects. Combining numerical and analytic techniques, we gain new insights into the complex dynamics of a recurrent disease influenced by the seasonality. Computation of the Lyapunov spectrum allows us to identify different chaotic regimes, determine the fractal dimension and estimate the predictability of the appearance of attractors in the system. Applying the homotopy analysis method, we obtain series solutions to the original nonautonomous SIR model with a high level of accuracy and use these approximations to analyze the dynamics of the system. The efficiency of the method is guaranteed by the optimal choice of an auxiliary control parameter which ensures the rapid convergence of the series to the exact solution of the forced SIR epidemic model. [ABSTRACT FROM AUTHOR]

This paper studies the properties of the optimal portfolio-consumption strategies in a finite horizon robust utility maximization framework with different borrowing and lending rates. In particular, we allow for constraints on both investment and consumption strategies, and model uncertainty on both drift and volatility. With the help of explicit solutions, we quantify the impacts of uncertain market parameters, portfolio-consumption constraints and borrowing costs on the optimal strategies and their time monotone properties. [ABSTRACT FROM AUTHOR]

In this paper we consider a one-dimensional one-phase Stefan problem corresponding to the solidification process of a semi-infinite material with a convective boundary condition at the fixed face. The exact solution of this problem, available recently in the literature, enable us to test the accuracy of the approximate solutions obtained by applying the classical technique of the heat balance integral method and the refined integral method, assuming a quadratic temperature profile in space. We develop variations of these methods which turn out to be optimal in some cases. Throughout this paper, a dimensionless analysis is carried out by using the parameters: Stefan number (Ste) and the generalized Biot number (Bi). In addition it is studied the case when Bi goes to infinity, recovering the approximate solutions when a Dirichlet condition is imposed at the fixed face. Some numerical simulations are provided in order to estimate the errors committed by each approach for the corresponding free boundary and temperature profiles. [ABSTRACT FROM AUTHOR]

The main purpose of this article is to study the qualitative analysis and explicit solutions of perturbed Chen–Lee–Liu equation with refractive index. By utilizing wave transformation, two-dimensional dynamics in a plane are obtained. Using the qualitative theory of planar dynamical systems, a series of planar phase portraits are presented. By analyzing the bifurcations of these phase portraits and the characteristics of equilibrium points, some explicit solutions such as Jacobian function solutions and hyperbolic function solutions are constructed. These can further understand the dynamic behavior of perturbed Chen–Lee–Liu equation and their wave propagation. • The main purpose of this paper is to study the qualitative analysis and traveling wave solution of the perturbed Chen–Lee–Liu equation with refractive index. • Some specific parameters are given to draw the plane phase portrait of the dynamic system of the perturbed Chen–Lee–Liu equation. [ABSTRACT FROM AUTHOR]

We introduce the traveling salesman problem with forbidden neighborhoods (TSPFN). This is an extension of the Euclidean TSP in the plane where direct connections between points that are too close are forbidden. The TSPFN is motivated by an application in laser beam melting. In the production of a workpiece in several layers using this method one hopes to reduce the internal stresses of the workpiece by excluding the heating of positions that are too close. The points in this application are often arranged in some regular (grid) structure. In this paper we study optimal solutions of TSPFN instances where the points in the Euclidean plane are the points of a regular grid. Indeed, we explicitly determine the optimal values for the TSPFN and its associated path version on rectangular regular grids for different minimal distances of the points visited consecutively. For establishing lower bounds on the optimal values we use combinatorial counting arguments depending on the parities of the grid dimensions. Furthermore we provide construction schemes for optimal TSPFN tours for the considered cases. [ABSTRACT FROM AUTHOR]

In this paper, we presented a noncommutative (NC) generalization of nonlinear Schrödinger equation (NLSE) in 2 + 1 dimensions. A matrix Darboux transformation (MDT) is used to generate multiple soliton solutions for NC-NLSE and commutative NLSE in 2 + 1 dimensions. We expressed multiple soliton solutions in terms of quasideterminants and as ratios of ordinary determinants for NC and commutative NLSE in 2 + 1 dimensions, respectively. The quasideterminant formula for K-times repeated MDT enables us to compute single, double and triple soliton solutions for NC and commutative (2 + 1)-dimensional NLSE. Some interesting localized solutions are obtained for the NC and commutative NLSE in 2 + 1 dimensions. [ABSTRACT FROM AUTHOR]

The boundary value problems of the coupled linear theory of elasticity are solved for isotropic one-porous solids of specific shape. Special representations of a general solution of a system of differential equations are constructed by means of elementary functions. With the help of these representations, the solutions to the problems are presented explicitly, in the form of absolutely and uniformly convergent series. The question of the uniqueness of regular solutions to the problems under consideration is investigated. [ABSTRACT FROM AUTHOR]

We present some explicit solutions with radial character in the setting of Eulerian framework to the three-dimensional nonlinear water waves in arctic regions. In particular, we focus on the motion of water waves with a moving bed and the pressure varying on free surface. [ABSTRACT FROM AUTHOR]

In this paper, by means of similarity transfomations, we obtain explicit solutions to the cubic-quintic nonlinear Schrödinger equation with varying coefficients, which involve four free functions of space. Four types of free functions are chosen to exhibit the corresponding nonlinear wave propagations. [ABSTRACT FROM AUTHOR]

This paper considers multiclass queueing network systems with fixed routing. With the service control as one player and the arrival and service rates as the other, the problem of network regulation can be formulated as a differential game. Representations of the value function are developed by studying the geometric properties of the associated Hamiltonians and are expressed in terms of related simpler halfspace problems. Also, a method of constructing the optimal feedback controls through the representation and the projected Isaacs equations is provided. The controls so constructed give both a guaranteed level of performance and robust stability over a range of rate perturbations. [ABSTRACT FROM AUTHOR]

In this paper we present an analysis of the mean flow velocities, and related mass transport, which are induced by certain equatorially-trapped water waves. In particular, we examine a recently-derived exact and explicit solution to the geophysical governing equations in the $${\beta}$$ -plane approximation at the equator which incorporates a constant underlying current. [ABSTRACT FROM AUTHOR]

In the present paper on the basis of the linear theory of thermoelasticity of homogeneous isotropic bodies with microtemperatures the zeroth order approximation of hierarchical models of elastic prismatic shells with microtemperatures in the case of constant thickness (but, in general, with bent face surfaces) is considered. The existence and uniqueness of solutions of basic boundary value problems when the projections of the bodies under consideration are bounded and unbounded domains with closed contours are established. The ways of solving boundary value problems in explicit forms and of their numerical solution are indicated. [ABSTRACT FROM AUTHOR]

In this paper, we study the atmospheric Ekman flows of the Northern Hemisphere related to latitude and height in ellipsoidal coordinates. Firstly, by using a suitable transformation, we convert the atmospheric Ekman equation into a second order elliptic partial differential equation. Further, we prove it is equivalent to a second order homogeneous ordinary differential equation with variable coefficients. Finally, we construct the explicit solution and give the laws of motion of the atmospheric Ekman flows. [ABSTRACT FROM AUTHOR]

In this paper, we consider the geometrical optics-or WKB-method for geophysical equatorial barotropic incompressible flows. The stability analysis of such flows is reduced to the stability analysis of an ordinary differential equation system along their trajectories. The analysis of this system in the case of equatorially trapped waves propagating eastward in stratified water shows that those waves for which the steepness parameter of the wave profile is higher than a specific value are unstable. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the symmetry analysis and conservation laws of the time fractional Clannish Random Walker's Parabolic (CRWP) equation. The vector fields and similarity reduction of the time fractional CRWP equation are obtained. In addition, based on the power series theory, a simple and effective approach for constructing explicit power series solutions is proposed. Finally, by use of the new conservation theorem, the conservation laws of the time fractional CRWP equation are constructed. [ABSTRACT FROM AUTHOR]

In this paper, we study the classical problem of the wind in the steady atmospheric Ekman layer with the constant eddy viscosity. Different from the previous work, we modify the boundary conditions and derive the explicit solution by using the notation of matrix cosine and matrix sine. For the arbitrary height-dependent eddy viscosity, we get the solution of the classical problem with zero velocity and acceleration at the bottom of the layer. In addition, uniqueness is shown and dynamical properties of solution are characterized. [ABSTRACT FROM AUTHOR]

Abstract Solutions of a group of conjugate time-varying matrix equations are discussed in this paper. Through mathematical derivation, the solutions to this group of equations are equivalent to the solutions to a class of conjugate time-invariant matrix equations. Further, the related conditions of solvability are obtained and the general explicit solutions are represented by using quasicontrollability and quasiobservability matrices. A detailed algorithm is presented to make the calculation process clear, and the effectiveness of the algorithm is verified by a concrete example. The proposed algorithm can provide complete solutions to the considered equation in explicit parametric form and its main computation includes solving an ordinary linear algebraic equation and some matrix multiplication operations.

We consider a system of two coupled parallel queues with infinite waiting rooms. The time setting is discrete. In either queue, the service of a customer requires exactly one discrete time slot. Arrivals of new customers occur independently from slot to slot, but the numbers of arrivals into both queues within a slot may be mutually dependent. Their joint probability generating function (pgf) is indicated as A(z1,z2) and characterizes the whole model. In general, determining the steady-state joint probability mass function (pmf) u(m,n),m,n≥0 or the corresponding joint pgf U(z1,z2) of the numbers of customers present in both queues is a formidable task. Only for very specific choices of the arrival pgf A(z1,z2) are explicit results known. In this paper, we identify a multi-parameter, generic class of arrival pgfs A(z1,z2), for which we can explicitly determine the system-content pgf U(z1,z2). We find that, for arrival pgfs of this class, U(z1,z2) has a denominator that is a product, say r1(z1)r2(z2), of two univariate functions. This property allows a straightforward inversion of U(z1,z2), resulting in a pmf u(m,n) which can be expressed as a finite linear combination of bivariate geometric terms. We observe that our generic model encompasses most of the previously known results as special cases.

In this work, the interaction between a moving anti-plane crack and a screw dislocation in the magneto-electro-elastic material is studied. The exact solutions are derived by the Muskhelishvili theory and the mechanical-electric-magnetic fields are presented. Comparing with the recent researching status, the present work has three prominent features. Firstly, both the remote loads with eight possible combinations and the coupling of the mechanical-electric-magnetic fields are considered. Secondly, the crack face is assumed to have the general boundary condition which is not the single permeable or impermeable case. Finally, the general solution is derived with the consideration of the crack moving velocity and the screw dislocation. Results presented in this paper would have potential applications on the analysis and design of magneto-electro-elastic structures, especially for the influence of microdefects on the initiation and propagation of cracks. [ABSTRACT FROM AUTHOR]