A reaction-diffusion equation and its traveling wave solutions

In the present paper, we study a non-linear reaction-diffusion equation, which can be considered as a generalized Fisher equation. An exact solution and traveling wave solutions to the generalized Fisher equation are obtained by means of the Cole-Hopf transformation and the Lie symmetry method.     

Reoptimization of minimum and maximum traveling salesman's tours

In this paper, reoptimization versions of the traveling salesman problem (TSP) are addressed. Assume that an optimum solution of an instance is given and the goal is to determine if one can maintain a good solution when the instance is subject to minor modifications. We study the case where nodes are inserted in, or deleted from, the graph. When inserting a node, we show that the reoptimization problem for MinTSP is approximable within ratio 4/3 if the distance matrix is metric. We show that, dealing with metric MaxTSP, a simple heuristic is asymptotically optimum when a constant number of nodes are inserted. In the general case, we propose a 4/5-approximation algorithm for the reoptimization version of MaxTSP.     

Simpler analysis of LP extreme points for traveling salesman and survivable network design problems

We consider the Survivable Network Design Problem (SNDP) and the Symmetric Traveling Salesman Problem (STSP). We give simpler proofs of the existence of a -edge and 1-edge in any extreme point of the natural LP relaxations for the SNDP and STSP, respectively. We formulate a common generalization of both problems and show our results by a new counting argument. We also obtain a simpler proof of the existence of a -edge in any extreme point of the set-pair LP relaxation for the element connectivitySurvivable Network Design Problem ().     

Exponential decay rate of solutions toward traveling waves for the Cauchy problem of generalized Benjamin-Bona-Mahony-Burgers equations

In this paper, we investigate the exponential time decay rate of solutions toward traveling waves for the Cauchy problem of generalized Benjamin-Bona-Mahony-Burgers equations

(E)     

A numerical investigation of blow-up in reaction-diffusion problems with traveling heat sources

This paper studies the numerical solution of a reaction-diffusion differential equation with traveling heat sources. According to the fact that the locations of heat sources are known, we add auxiliary mesh points exactly at heat sources and present a novel moving mesh algorithm for solving the problem. Several examples are provided to demonstrate the efficiency of the new moving mesh method, especially in the case of two or three traveling heat sources. Moreover, numerical results illustrate that the speed of the movement of the heat source is critical for blow-up when there is only one traveling heat source. For the case of two traveling heat sources, blow-up depends not only on the speed but also on the distance between the two traveling heat sources.     

Embedding learning capability in Lagrangean relaxation: An application to the travelling salesman problem

This paper presents an effective procedure that finds lower bounds for the travelling salesman problem based on the 1-tree using a learning-based Lagrangian relaxation technique. The procedure can dynamically alter its step-size depending upon its previous iterations. Along with having the capability of expansion–contraction, the procedure performs a learning process in which Lagrange multipliers are influenced by a weighted cost function of their neighbouring nodes. In analogy with simulated annealing paradigm, here a learning process is equivalent to escaping local optimality via exploiting the structure of the problem. Computational results conducted on Euclidean benchmarks from the TSPLIB library show that the procedure is very effective.     

Spreading speeds and traveling waves for abstract monostable evolution systems

This paper is devoted to the development of the theory of spreading speeds and traveling waves for abstract monostable evolution systems with spatial structure. Under appropriate assumptions, we show that the spreading speeds coincide with the minimal wave speeds for monotone traveling waves in the positive and negative directions. Then we use this theory to study the spatial dynamics of a parabolic equation in a periodic cylinder with the Dirichlet boundary condition, a reaction-diffusion model with a quiescent stage, a porous medium equation in a tube, and a lattice system in a periodic habitat.     

Quasi-Stationary Regime of a Branching Random Walk in Presence of an Absorbing Wall

A branching random walk in presence of an absorbing wall moving at a constant velocity v undergoes a phase transition as the velocity v of the wall varies. Below the critical velocity v _c , the population has a non-zero survival probability and when the population survives its size grows exponentially. We investigate the histories of the population conditioned on having a single survivor at some final time T. We study the quasi-stationary regime for v<v _c when T is large. To do so, one can construct a modified stochastic process which is equivalent to the original process conditioned on having a single survivor at final time T. We then use this construction to show that the properties of the quasi-stationary regime are universal when v→v _c . We also solve exactly a simple version of the problem, the exponential model, for which the study of the quasi-stationary regime can be reduced to the analysis of a single one-dimensional map.     

The Millimeter Wave Radiation of a Traveling Wave Sinusoidal Wire Antenna

In this paper, investigation of radiation properties of the traveling-wave sinusoidal wire antennas is extended to the millimeter-wave frequencies (Ka-band) for the antennas whose geometrical dimensions vary in a wide range. Far-field patterns and S-parameters of composed three antenna sets were measured. A mathematical model was constructed for the structure and a MATLAB code based on this theoretical approach was written to calculate patterns, phase and attenuation constants of all investigated antennas. Frequency characteristics and the relation of antenna dimensions with wave parameters were investigated. Measured and calculated patterns were also compared with the constructed far-field patterns obtained by MoM (method of moments) and the MoM current distributions were used to explain the loss mechanisms of antennas. A directive, undistorted and smooth radiation can be achieved only choosing small undulated antennas whose peak-to-peak amplitude to period ratio κ is smaller than 0.4 (κ < 0.4). It is shown that wavelength of broadside radiation is not equal to antenna period for all antennas, except for very small undulated antennas (κ < 0.2). This antenna type can be used as a frequency-scan antenna for millimeter wave radars.

Elimination of Anti-spiral Waves by Local Inhomogeneity in Oscillatory Systems

采用系统中的局部不均匀性消除振荡系统中的反螺旋波. 该不均匀性在系统中成为一个波源,不断产生稳定的相波. 研究结果发现,不均匀性的尺寸大小存在一个临界值,如果低于此临界值,则系统将无法激发任何相波. 根据不均匀性形状的不同,系统分别产生靶波和行波. 此外,实验还发现靶波与反螺旋波之间以及行波与反螺旋波之间存在着不同的动态竞争. 数值计算表明,对于行波,无论是低频行波还是高频行波,都可以成功地消除系统中的反螺旋波;而对于靶波,只有低频靶波才可以消除反螺旋波. 此控制方法简单易行,且同样适用于消除向外传播的螺旋波