APPLICATION OF MATHEMATICS IN TECHNICAL AND NATURAL SCIENCES: 3rd International Conference—AMiTaNS'11
Preface: Application of Mathematics in Technical and Natural Sciences, 3rd International Conference—AMiTaNS'111404(2011); http://dx.doi.org/10.1063/1.3659899View Description Hide Description
1404(2011); http://dx.doi.org/10.1063/1.3659900View Description Hide Description
This is a further note on the (Guass‐Maxwell) force‐flux construct proposed previously (Goddard, J. D., A note on Eringen's moment balances, Int. J. Eng. Sci., in the press, 2011). Motivated in part by its promise as a homogenization technique for constructing micromorphic continua, the present work is focused rather on some additional representations and on novel applications, such as the derivation of dissipative thermodynamic potentials from force‐flux relations.
1404(2011); http://dx.doi.org/10.1063/1.3659901View Description Hide Description
Basic equations of continuum mechanics reflect deep symmetry properties of Euclidean space and medium, which moves in it. As a corollary, these equations admit a wide Lie group that allows us to apply group‐theoretical methods for continuum motion study. In this paper, both gravitational heat convection and thermocapillary convection equations are considered from the group‐theoretical point of view. It turned out that classical Ostroumov and Birikh's solutions have a group nature. We present a number of new solutions, which describe a non‐isothermal viscous liquid motion in channels, tubes and layers under action of gravity, centrifugal and thermocapillary forces.
1404(2011); http://dx.doi.org/10.1063/1.3659902View Description Hide Description
We study the propagation and interaction of solitons modeled by the equation governing the flexural deformation of an elastic rod on elastic foundation (EREFE). The model equation is only of fourth order, but because of the presence of the linear restoring force, it exhibits some of the phenomenology of the 6GBE, especially when the nonmonotonic shapes are concerned. To do that, we apply the Christov spectral method in Initially, we solve the problem in the moving frame and consider the obtained solutions as initial condition for the time dependant problem and study the propagation and interaction of solitons.
1404(2011); http://dx.doi.org/10.1063/1.3659903View Description Hide Description
The Boussinesq model of shallow water flow is considered, which contains nonlinearity and fourth‐order dispersion. Boussinesq equation has been one of the main soliton models in 1D. To find its 2D solutions, a perturbation series with respect to the small parameter is developed in the present work, where c is the phase speed of the localized wave. Within the order a hierarchy is derived consisting of fourth‐order ordinary differential equations (ODEs). The Bessel operators involved are reformulated to facilitate the creation of difference schemes for the ODEs from the hierarchy. The numerical scheme uses a special approximation for the behavioral condition in the singularity point (the origin). The results of this work show that at infinity the 2D wave shape decays algebraically, rather than exponentially as in the 1D cases. The new result can be instrumental for understanding the interaction of 2D Boussinesq solitons, and for creating more efficient numerical algorithms explicitly acknowledging the asymptotic behavior of the solution.
1404(2011); http://dx.doi.org/10.1063/1.3659904View Description Hide Description
The N‐soliton interactions of two classes of vector NLS (VNLS) equations are analyzed. It is shown that the N‐soliton interactions of the VNLS related to symmetric spaces of BD.I‐type, just like in the scalar case, reduce to shifts of the center of masses and of the phases, while the polarization vectors remain unchanged. The same approach applied to the standard VNLS (or the Manakov model) shows nontrivial effects of the interaction on the polarization vectors.
Theoretical and Numerical Aspects for Global Existence and Blow Up for the Solutions to Boussinesq Paradigm Equation1404(2011); http://dx.doi.org/10.1063/1.3659905View Description Hide Description
In this paper we prove that the global existence and the blow up of the weak solutions to Boussinesq Paradigm Equation (BPE) depend not only on the initial energy but also on the profiles of the initial data.
The constant d of the critical initial energy which guaranties the above properties of the solution is found explicitly by means of the exact constant of the Sobolev embedding theorem. We demonstrate numerically in the one dimensional case that this constant d is the best possible one for the global existence and a lack of global existence.
In this way we can find the intervals for the velocities c of the solitary wave solutions to BPE in which the solution to BPE with initial data close to the solitons exists globally in time. Thus for different parameters of BPE we give numerically some ranges of the stability of the solitary waves.
1404(2011); http://dx.doi.org/10.1063/1.3659906View Description Hide Description
The system of coupled nonlinear Schrödinger's equations (CNLSE) is solved numerically by means of a conservative difference scheme. Values of the cross‐modulation parameter, are chosen to induce collisions for which the initial profiles are eventually reversed, focusing on two specific effects‐delay and polarization. The resulting waveforms are fitted to generally polarized soliton profiles and studied via internal parameters including phase velocity, pulse width, and polarization angle to gain insight into the dynamics of the resulting collisions.
Traveling Wave Solutions of the Gardner Equation and Motion of Plane Curves Governed by the mKdV Flow1404(2011); http://dx.doi.org/10.1063/1.3659907View Description Hide Description
The Gardner equation is well‐known in the mathematical literature since the late sixties of 20th century. Initially, it appeared in the context of the construction of local conservation laws admitted by the KdV equation. Later on, the Gardner equation was generalized and found to be applicable in various branches of physics (solid‐state and plasma physics, fluid dynamics and quantum field theory). In this paper, we examine the travelling wave solutions of the Gardner equation and derive the full set of solutions to the corresponding reduced equation in terms of Weierstrass and Jacobi elliptic functions. Then, we use the travelling wave solutions of the focusing mKdV equation and obtain in explicit analytic form exact solutions of a special type of plane curve flow, known as the mKdV flow.
1404(2011); http://dx.doi.org/10.1063/1.3659908View Description Hide Description
In the current work, we develop a numerical method suitable for treating the problem of nonlinear two‐dimensional flows in rectangular domains. For the spatial approximation we employ the Fourier‐Galerkin approach. More specifically, our basis functions are products of trigonometric and Beam functions. This choice means that the solutions automatically satisfy the boundary and periodic conditions in the x and y directions respectively.
The accuracy of the method is assessed by applying it to model problems which admit exact analytical solutions. The numerical and analytic solutions are found to be in good agreement. The convergence rate of the spectral coefficients is found to be fifth‐order algebraic in the x‐direction and y‐direction, confirming the efficiency and speed of our technique.
Spectral Formulation for the Solution of Full‐Wave Scattering from a Conducting Wedge Tipped with a Corrugated Cylinder1404(2011); http://dx.doi.org/10.1063/1.3659909View Description Hide Description
A spectral mode‐matching technique is formulated to solve for the full‐wave scattering of a corrugated cylinder‐tipped wedge in the presence of an impressed electric or magnetic line source. Asymptotic approximations of large‐order Bessel or Henkel functions for a fixed argument were introduced in order to overcome numerical difficulties in their regular series expansions. The corrugations on the conducting cylinder have the shape of annular sectors. The primary objective of this work is to investigate the impact of corrugations on the scattered field in the shadow region of the structure. An optimally designed corrugated cylinder placed at the tip of a conducting wedge can effectively suppress electromagnetic scattering in the shadow region. Obtained numerical results using the proposed approach prove the above concept. These results were validated against numerical data obtained using a nodal finite element method. The aim of this research is to utilize these corrugated tips in horn antenna design for the reduction of side‐lobe level and the shaping of the respective E‐plane radiation pattern.
1404(2011); http://dx.doi.org/10.1063/1.3659910View Description Hide Description
During the last decade surface subdivision methods are leading in both industry and research interest due to their ability to generate surfaces defined by an arbitrary topology of vertices. A number of such methods have been developed and are used today in Computer Aided Geometric Design (CAGD). Most of them generate cubic or quadratic surfaces. Methods based on cubic polynomials are the most popular ones since they provide second derivative continuity on the surface, whereas quadratic methods have only continuity. There are many good reasons for using cubic polynomials: their theory is simple; they have a satisfactory continuity; and they are easier to be developed. This paper presents a curve and surface subdivision method with n‐degree polynomials. In the case of surfaces the method is generalized to handle surfaces defined by arbitrary topological meshes of vertices. It provides continuity and is fairly easy to be developed.
1404(2011); http://dx.doi.org/10.1063/1.3659911View Description Hide Description
Rapidly emerging micro‐electro‐mechanical devices create new potential microfluidic applications. A simulation of an internal and external gas flows with accurate boundary conditions for these devices is important for their design. In this paper we study influence of reservoirs used at the microchannel inlet and outlet on the characteristics of the gas flow in the microchannel. The problem is solved by using finite volume method SIMPLE‐TS (continuum approach), which is validated using Direct Simulation Monte Carlo (molecular approach). We investigate two cases: a microchannels with reservoirs and without reservoirs. We compare the microchannels with different aspect ratios and 50, where is the channel length, is the channel height. Comparisons of results obtained by using continuum approach for pressure driven flow in a microchannel with and without reservoirs at the channel ends are presented.
1404(2011); http://dx.doi.org/10.1063/1.3659912View Description Hide Description
A fractional oscillator with a power‐law memory kernel subjected to an external periodic force is considered. The influence of the fluctuating viscoelastic environment is modeled by a multiplicative dichotomous noise (fluctuating damping) and an additive internal noise. The main purpose of this work is to provide exact formulas for the analytic treatment of the dependence of spectral amplification on system parameters: viz. the noise correlation time, noise amplitude, memory exponent, and driving frequency. Based on those exact expressions we demonstrate that stochastic resonance is manifested in the dependence of the spectral amplification upon the noise parameters. Moreover, a critical memory exponent is found which marks the transitions between different dynamical regimes of the oscillator.
1404(2011); http://dx.doi.org/10.1063/1.3659913View Description Hide Description
The efficiency of energy transformation of overdamped Brownian particles in a tilted periodic sawtooth potential driven by a nonequilibrium three‐level noise and an additive thermal noise is considered analytically. All the physical results are computed by means of exact formulas. It is established that in certain parameter region the dependence of the efficiency of energy transformation on noise parameters exhibits a bell‐shaped form. Thus, in such parameters regions an increase of values of noise characteristics (temperature, noise flatness, correlation time, and noise amplitude) can facilitate conversion of noise energy into mechanical work. A connection such as resonance‐like behavior of efficiency at multiple current reversals is also discussed.
1404(2011); http://dx.doi.org/10.1063/1.3659914View Description Hide Description
The aim of this paper is to analyze the convection and diffusion of a solute in a porous medium, under the influence of non‐smooth chemical reactions taking place on the pore surfaces. Our model consists of a system of two coupled convection‐diffusion equations, one in the fluid part and another one on the boundaries of the grains of the porous medium. The coupling is made through a nonlinear reaction term, modeling the mass exchange between the bulk and, respectively, the surface concentration.
1404(2011); http://dx.doi.org/10.1063/1.3659915View Description Hide Description
Sterile Insect Technology (SIT) is a nonpolluting method of insect control that relies on the release of sterile males. We study the effectiveness of the application of SIT for control of Anopheles mosquito via mathematical modeling. The theoretical analysis of the mathematical model as a dynamical system leads to the formulation of possible strategies for control of the Anopheles mosquito, also illustrated by numerical simulations.
1404(2011); http://dx.doi.org/10.1063/1.3659916View Description Hide Description
We consider a quasilinear parabolic system to model mosquito displacement. In order to use efficiently vector control tools, like insecticides, and mechanical control, it is necessary to provide density estimates of mosquito populations, taking into account the environment and entomological knowledges. After a brief introduction to mosquito dispersal modeling, we present some theoretical results. Then, considering a compartmental approach, we get a quasilinear system of PDEs. Using the time splitting approach and appropriate numerical methods for each operator, we construct a reliable numerical scheme. Considering vector control scenarii, we show that the environment can have a strong influence on mosquito distribution and in the efficiency of vector control tools.
Analysis and Dynamically Consistent Numerical Schemes for the SIS Model and Related Reaction Diffusion Equation1404(2011); http://dx.doi.org/10.1063/1.3659917View Description Hide Description
The classical SIS epidemiological model is extended in two directions: (a) The number of adequate contacts per infective in unit time is assumed to be a function of the total population in such a way that this number grows less rapidly as the total population increases; (b) A diffusion term is added to the SIS model and this leads to a reaction diffusion equation, which governs the spatial spread of the disease. With the parameter representing the basic reproduction number, it is shown that is a forward bifurcation for the model (a), with the disease‐free equilibrium being globally asymptotic stable when is less than 1. In the case when is greater than 1, traveling wave solutions are found for the model (b). Nonstandard finite difference (NSFD) schemes that replicate the dynamics of the continuous models are presented. In particular, for the model (a), a nonstandard version of the Runge‐Kutta method having high order of convergence is investigated. Numerical experiments that support the theory are provided.