A second-sound based, hyperbolic SIR model for high-diffusivity spread

We present an analytical study of one-dimensional (1D) kinematic wave phenomena under a hyperbolic SIR model based not on Fick's diffusion law, but rather on the inertial-type II flux law of second-sound theory. Unlike in the Ficken context, we are able to derive exact traveling wave solutions (TWS)s, as well as explicit asymptotic/approximate expressions, for both the susceptibles and infectives. We also determine, using singular surface theory, how shock-fronts resulting from initial jump discontinuities propagate and evolve under this model. In particular, critical values and special cases are examined and possible mitigation methods, which take the form of parameter-value manipulation(s), are noted.     

Existence and stability of traveling waves for an integro-differential equation for slow erosion

We study an integro-differential equation that describes the slow erosion of granular flow. The equation is a first order nonlinear conservation law where the flux function includes an integral term. We show that there exist unique traveling wave solutions that connect profiles with equilibrium slope at ±∞. Such traveling waves take very different forms from those in standard conservation laws. Furthermore, we prove that the traveling wave profiles are locally stable, i.e., solutions with monotone initial data approach the traveling waves asymptotically as t→+∞$t\to +\infty$.     

The traveling purchaser problem with stochastic prices: Exact and approximate algorithms

The paper formulates an extension of the traveling purchaser problem where multiple types of commodities are sold at spatially distributed locations with stochastic prices (each following a known probability distribution). A purchaser’s goal is to find the optimal routing and purchasing strategies that minimize the expected total travel and purchasing costs needed to purchase one unit of each commodity. The purchaser reveals the actual commodity price at a seller upon arrival, and then either purchases the commodity at the offered price, or rejects the price and visits a next seller. In this paper, we propose an exact solution algorithm based on dynamic programming, an iterative approximate algorithm that yields bounds for the minimum total expected cost, and a greedy heuristic for fast solutions to large-scale applications. We analyze the characteristics of the problem and test the computational performance of the proposed algorithms. The numerical results show that the approximate and heuristic algorithms yield near-optimum strategies and very good estimates of the minimum total cost.     

Scheduling periodic customer visits for a traveling salesperson

The problem considered in this paper deals with determining daily routes for a traveling salesperson who provides customers in Upper Austria with product range information of a large, global food wholesaler. Each customer has to be visited at least once a year, with some customers requiring up to one visit per month. Further, some customers may not be visited each day of the week. Our decision support system uses a commercial GIS software to extract customer data for input into the optimization procedure and to visualize the results obtained by the algorithm. The optimization approach is based on the variable neighborhood search algorithm which assigns customers to days and determines routes for the salesperson for each day with the primary objective to minimize the total travel time of the salesperson. Another objective studied is to minimize the number of days needed by the salesperson to visit all customers in a given month. Further we analyze the effects of changes in the business environment like increases in the amount or flexibility of the salesperson’s working time and variations in the possible days for customer visits. Finally, we enrich the objective function by considering periodicity requirements for customer visits. Specifically, we penalize irregular schedules, where the time between two successive customer visits varies.     

Effect of unidirectional solidification rate on Jc for current lead

The dependence of transport J_c value on the traveling rate and the nominal composition was investigated by taking different nominal compositions of YBa₂Cu₃O_6+x (Y123) + n mol% Y₂BaCuO₅ (Y211) (n = 10, 20, 30, 40 and 50) with addition of 0.5 wt% of Pt on samples processed by different unidirectional solidification rates, namely 1, 3 and 5 mm/h. The highest J_c was found in the sample with 30% Y211 addition by the higher traveling rate adjusted to prevent the formation of polycrystals. In this φ 1.56 × 60 mm sample the values of transport I_c and J_c were 1370 A and 71 700 A/cm², respectively, which were obtained by the conventional DC four-probe method with criterion of 1 μV/cm at 77 K and self-field.     

Solitary wave solution of the generalized Hirota–Satsuma coupled KdV system

In this research, we find the exact traveling wave solutions involving parameters of the generalized Hirota–Satsuma couple KdV system according to the modified simple equation method with the aid of Maple 16. When these parameters are taken special values, the solitary wave solutions are derived from the exact traveling wave solutions. It is shown that the modified simple equation method provides an effective and a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. Comparison between our results and the well-known results will be presented.     

Analytical examples of diffusive waves generated by a traveling wave

We construct analytical solutions for a system composed of a reaction–diffusion equation coupled with a purely diffusive equation. The question is to know if the traveling wave solutions of the reaction–diffusion equation can generate a traveling wave for the diffusion equation. Our motivation comes from the calcic wave, generated after fertilization within the egg cell endoplasmic reticulum, and propagating within the egg cell. We consider both the monostable (Fisher–KPP type) and bistable cases. We use a piecewise linear reaction term so as to build explicit solutions, which leads us to compute exponential tails whose exponents are roots of second-, third-, or fourth-order polynomials. These raise conditions on the coefficients for existence of a traveling wave of the diffusion equation. The question of positivity and monotonicity is only partially answered.