The Interaction of a Strong Bore with Small Disturbances in Shallow Water

This paper gives an extension of previous work [2] on weakly nonlinear shallow water waves over a variable bottom to include the effects of strong bores and small surface disturbances. We first consider the interaction of a strong bore with quiescent water over an isolated bottom disturbance to highlight some of the modifications that are introduced in our results for both noncritical and transcritical Froude numbers. We also exhibit the secular effect on the bore trajectory of a bottom disturbance that has a nonzero average. In a second example, we consider the interaction of a strong bore with a small amplitude periodic surface disturbance upstream. We show that downstream of the bore, the wave length of this disturbance increases, whereas its amplitude increases (decreases) depending on whether the bore speed is larger (smaller) than a critical value. We also use this example to illustrate the derivation of the solution and bore trajectory to second order accuracy. All our asymptotic results, obtained in the form of multiple scale expansions, are compared with numerical solutions for a number of illustrative cases.     

The compression of a thin strip of material in a superplasticity state

The viscoplastic flow of a thin strip of material in a superplasticity state between rigid, converging parallel planes (an analogue of Prandtl's problem) is investigated. An analytical quadrature solution of the problem is constructed, asymptotically precise in the same sense as Prandtl's solution. Special cases are considered where the solution (including an approximate solution) is written out fully. The effects of superplasticity are determined.     

Zaremba's problem for elliptic equations in the neighborhood of a singular point or in the infinity

In the present paper the behavior of solutions of the mixed Zaremba's problem in the neighborhood of a boundary point and at infinity is studied. In part I of this paper[4] the concept of Wiener's generalized solution of Zaremba's problem was introduced and the so called Growth Lemma for the class of domains, satisfying isoperimetric condition, was proven. In part II regularity criterion for joining points of Neumann's and Dirichlet's boundary conditions is formulated. Generalized solution in unlimited domains as a limit of Zaremba's problem's solutions in a sequence of limited domains is introduced and a regularity condition allowed to obtain an analogue of Phragmen-Lindeloeff theorem for the solutions of Zaremba's problem. Main results of the present paper are formulated in terms of divergence of Wiener's type series.     

The Evolution of Internal Undular Bores over a Slope in the Presence of Rotation

The large‐amplitude internal waves commonly observed in the coastal ocean often take the form of unsteady undular bores. Hence, here, we examine the long‐time combined effect of variable topography and background rotation on the propagation of internal undular bores, using the framework of a variable‐coefficient Ostrovsky equation. Because the leading waves in an internal undular bore are close to solitary waves, we first examine the evolution of a single solitary wave. Then, we consider an internal undular bore, for which two methods of generation are used. One method is the matured undular bore developed from an initial shock box in the Korteweg–de Vries equation, that is the Ostrovsky equation with the rotational term omitted, and the other method is a modulated cnoidal wave solution of the same Korteweg–de Vries equation. It transpires that in the long‐time model simulations, the rotational effect disintegrates the nonlinear waves into inertia‐gravity waves, and then there emerge complicated interactions between these inertia‐gravity waves and the modulated periodic waves of the undular bore, especially at the rear part of the undular bore. However, near the front of the undular bore, nonlinear effects further modulate these waves, with the eventual emergence of nonlinear envelope wave packets.     

Solving a multiobjective traveling salesman problem by dynamic programming

In this paper Klötzler's method of multiobjective dynamic programming is applied to the solution of a two-dimensional traveling salesman problem. In this way Bellman's and Held/Karp's dynamic programming approach to one-dimensional traveling salesman problems is extended to the multiobjective case.     

Radial deformation of a nonhomogeneous spherically isotropic elastic sphere with a concentric spherical inclusion

The elasticity of a spherically isotropic medium bounded by two concentric spherical surfaces subjected to normal pressures is discussed. The material of the structure is spherically isotropic and, in addition, is continuously inhomogeneous with mechanical properties varying exponentially as the square of the radius. An exact solution of the problem in terms of Whittaker functions is presented. The St. Venant’s solution in the case of homogeneous material and Lamé’s solution in the case of homogeneous isotropic material are derived from the general solution. The problem of a solid sphere of the same medium under the external pressure is also solved as a particular case of the above problem. Finally, the displacements and stresses of a composite sphere consisting of a solid spherical body made of homogeneous material and a nonhomogeneous concentric spherical shell covering the inclusion, both of them being spherically isotropic, are obtained when the sphere is under uniform compression.     

The diffraction in a class of unbounded domains connected through a hole

In this paper, the unique solvability, Fredholm property, and the principle of limiting absorption are proved for a boundary value problem for the system of Maxwell's equations in a semi‐infinite rectangular cylinder coupled with a layer by an aperture of arbitrary shape. Conditions at infinity are taken in the form of the Sveshnikov–Werner partial radiation conditions. The method of solution employs Green's functions of the partial domains and reduction to vector pseudodifferential equations considered in appropriate vectorial Sobolev spaces. Singularities of Green's functions are separated both in the domain and on its boundary. The smoothness of solutions is established. Copyright © 2003 John Wiley & Sons, Ltd.     

The loss of stability of a two-step rod when it hits a rigid barrier

The critical precollision velocity of a two-step rod of finite length when it collides with a rigid obstacle, leading to a loss of its stability, is calculated by an analytical solution of the wave equation using d’Alembert's method. The critical force and velocity are calculated using Euler's formula for a static load.     

Dynamics of a two-layer beam with layer slip

A possible type of structural damping in mechanical systems when there is a distinct dependence of the oscillation decrement on the amplitude is investigated, using the example of the solution of a model problem on the oscillations of a two-layer beam. It is assumed that layer slip only occurs along the beam axis and that the layers together in the transverse direction. The interaction between the layers is of an elastic-friction form. The equations of motion of the beam in Timoshenko's form are numerically integrated using Godunov's difference scheme.     

Supply chain model for a deteriorating product with time-varying demand and production rate

The paper develops a two-echelon supply chain model with a single-buyer and a single-vendor. The buyer sells a seasonal product over a short selling period and its inventory is subject to deterioration at a constant rate over time. The vendor's production rate is dependent on the buyer's demand rate, which is a linear function of time. Also, the vendor's production process is not perfectly reliable; it may shift from an in-control state to an out-of-control state at any time during a production run and produce some defective (non-conforming) items. Assuming that the vendor follows a lot-for-lot policy for replenishment made to the buyer, the average total cost of the supply chain is derived and an algorithm for finding the optimal solution is developed. The numerical study shows that the supply chain coordination policy is more beneficial than those policies obtained separately from the buyer's and the vendor's perspectives.     

Likely Limit Sets of a Class of $p$-Order Feigenbaum's Maps

A continuous map from a closed interval into itself is called a $p$-order Feigenbaum's map if it is a solution of the Feigenbaum's equation $f^p (λx) = λf(x)$. In this paper, we estimate Hausdorff dimensions of likely limit sets of some $p$-order Feigenbaum's maps. As an application, it is proved that for any $0 < t < 1$, there always exists a $p$-order Feigenbaum's map which has a likely limit set with Hausdorff dimension $t$. This generalizes some known results in the special case of $p = 2$.     

Wave Breaking and the Generation of Undular Bores in an Integrable Shallow Water System

The generation of an undular bore in the vicinity of a wave-breaking point is considered for the integrable Kaup–Boussinesq (KB) shallow water system. In the framework of the Whitham modulation theory, an analytic solution of the Gurevich–Pitaevskii type of problem for a generic "cubic" breaking regime is obtained using a generalized hodograph transform, and a further reduction to a linear Euler–Poisson equation. The motion of the undular bore edges is investigated in detail.     

An approximation solution of a nonlinear equation with Riemann–Liouville's fractional derivatives by He's variational iteration method

In this article, an application of He's variational iteration method is proposed to approximate the solution of a nonlinear fractional differential equation with Riemann–Liouville's fractional derivatives. Also, the results are compared with those obtained by Adomian's decomposition method and truncated series method. The results reveal that the method is very effective and simple.     

Steady flow in a deformable porous medium

A nonlinear model for a steady flow in a deformable porous medium is considered. The flow is governed by the poroelasticity system consisting of an elasticity equation for the displacement of the porous medium and Darcy's equation for the pressure in the fluid. This poroelasticity system is nonlinear when the permeability in Darcy's equation is assumed to depend on the dilatation of the porous medium. Existence and uniqueness of a weak solution of this poroelasticity system is established under rather weak assumptions on the regularity of the data. Convergence of a finite element approximation is proved and verified through numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

On a model for electromagnetic processes inside and outside a ferromagnetic body

One‐dimensional Maxwell's equations are considered in a ferromagnetic body surrounded by vacuum. Existence and uniqueness of solution for the resulting system of partial differential equations with hysteresis on the whole real line is proved under suitable constitutive hypotheses. Copyright © 2008 John Wiley & Sons, Ltd.     

The a Posteriori Error Estimator of SDG Method for Variable Coefficients Time-Harmonic Maxwell's Equations

In this paper, we study the a posteriori error estimator of SDG method for variable coefficients time-harmonic Maxwell's equations. We propose two a posteriori error estimators, one is the recovery-type estimator, and the other is the residual-type estimator. We first propose the curl-recovery method for the staggered discontinuous Galerkin method (SDGM), and based on the super-convergence result of the postprocessed solution, an asymptotically exact error estimator is constructed. The residual-type a posteriori error estimator is also proposed, and it's reliability and effectiveness are proved for variable coefficients time-harmonic Maxwell's equations. The efficiency and robustness of the proposed estimators is demonstrated by the numerical experiments.     

Transcritical Flow Over a Hole

Transcritical flow over a localized obstacle generates upstream and downstream nonlinear wavetrains. In the weakly nonlinear long-wave regime, this flow has been modeled with the forced Korteweg-de Vries (fKdV) equation, where numerical simulations and asymptotic solutions have demonstrated that the upstream and downstream nonlinear wavetrains have the structure of unsteady undular bores, connected by a locally steady solution over the obstacle. Further, it has been shown that when the obstacle is replaced by a step of semi-infinite length, it is found that a positive step generates only an upstream-propagating undular bore, and a negative step generates only a downstream-propagating undular bore. This result suggests that for flow over a hole, that is a step down followed by a step up, the two wavetrains generated will interact over the hole. In this paper, this situation is explored by numerical simulations of the fKdV equation.