Online Facility Location in Evolving Metrics

The Dynamic Facility Location problem is a generalization of the classic Facility Location problem, in which the distance metric between clients and facilities changes over time. Such metrics that develop as a function of time are usually called “evolving metrics”, thus Dynamic Facility Location can be alternatively interpreted as a Facility Location problem in evolving metrics. The objective in this time-dependent variant is to balance the trade-off between optimizing the classic objective function and the stability of the solution, which is modeled by charging a switching cost when a client’s assignment changes from one facility to another. In this paper, we study the online variant of Dynamic Facility Location. We present a randomized O(logm+logn)-competitive algorithm, where m is the number of facilities and n is the number of clients. In the first step, our algorithm produces a fractional solution, in each timestep, to the objective of Dynamic Facility Location involving a regularization function. This step is an adaptation of the generic algorithm proposed by Buchbinder et al. in their work “Competitive Analysis via Regularization.” Then, our algorithm rounds the fractional solution of this timestep to an integral one with the use of exponential clocks. We complement our result by proving a lower bound of Ω(m) for deterministic algorithms and lower bound of Ω(logm) for randomized algorithms. To the best of our knowledge, these are the first results for the online variant of the Dynamic Facility Location problem.

PIECEWISE OPTIMAL FRACTIONAL REPRODUCING KERNEL SOLUTION AND CONVERGENCE ANALYSIS FOR THE ATANGANA–BALEANU–CAPUTO MODEL OF THE LIENARD’S EQUATION

In this paper, an attractive reliable analytical technique is implemented for constructing numerical solutions for the fractional Lienard’s model enclosed with suitable nonhomogeneous initial conditions, which are often designed to demonstrate the behavior of weakly nonlinear waves arising in the oscillating circuits. The fractional derivative is considered in the Atangana–Baleanu–Caputo sense. The proposed technique, namely, reproducing kernel Hilbert space method, optimizes numerical solutions bending on the Fourier approximation theorem to generate a required fractional solution with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some applications. The acquired results are numerically compared with the exact solutions in the case of nonfractional derivative, which show the superiority, compatibility, and applicability of the presented method to solve a wide range of nonlinear fractional models.

Calorimetric Investigation for Thermal Plate of Casson Fluid via Fractional Derivative

The manuscript reveals the collective effects of the moving plate of Casson fluid in which magnetic, porous outcomes are under consideration. Thermal stratification is investigated to disclose the hidden phenomenon of mass concentration and temperature distribution. Fractional operator has been applied on the fundamental equations of Casson fluid namely Caputo-Fabrizio fractional operator based on sufficient memory operator. For the exact analysis of basic fractional governing equations of velocity profile, temperature distribution and mass concentration the integral transforms have been employed. The solutions of the dimensionless equations have been described in terms special functions in convoluted form. For the sake of non-fractional solution of the basic equations of the Casson fluid X = 1 in the obtained solutions has been implemented for the four type's fluid models. Finally, thermal conductivity of the fluid has been analyzed by estimating various different parametric values which result the increment in velocity profile along with porous permeability but reverse in transvers magnetic field on the flow.

Discrete fractional solution of a nonhomogeneous non-Fuchsian differential equations

In this article, we also present new fractional solutions of the non-homogeneous and homogeneous non-Fuchsian differential equation by using nabla-discrete fractional calculus operator ??(0 < ? < 1). So, we acquire new solution of these equation in the discrete fractional form via a newly developed method.

Adaptation of ant supercolony behavior to solve route assignment problem in integers

Purpose Until now, the algorithms used to compute an equilibrate route assignment do not return an integer solution. This disagreement constitutes a non-negligible drawback. In fact, it is shown in the literature that a fractional solution is not a good approximation of the integer one. The purpose of this paper is to find an integer route assignment. Design/methodology/approach The static route assignment problem is modeled as an asymmetric network congestion game. Then, an algorithm inspired from ant supercolony behavior is constructed, in order to compute an approximation of the Pure Nash Equilibrium (PNE) of the considered game. Several variants of the algorithm, which differ by their initializing steps and/or the kind of the provided algorithm information, are proposed. Findings An evaluation of these variants over different networks is conduced and the obtained results are encouraging. Indeed, the adaptation of ant supercolony behavior to solve the problem under consideration shows interesting results, since most of the algorithm’s variants returned high-quality approximation of PNE in more than 91 percent of the treated networks. Originality/value The asymmetric network congestion game is used to model route assignment problem. An algorithm with several variants inspired from ant supercolony behavior is developed. Unlike the classical ant colony algorithms where there is one nest, herein, several nests are considered. The deposit pheromone of an ant from a given nest is useful for the ants of the other nests.

Identifying Codes in Vertex-Transitive Graphs and Strongly Regular Graphs

We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and $2\ln(|V|)+1$ where $V$ is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order $|V|^{\alpha}$ with $\alpha \in \{\frac{1}{4},\frac{1}{3},\frac{2}{5}\}$. These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs.

Space-Time Fractional Diffusion-Advection Equation with Caputo Derivative

An alternative construction for the space-time fractional diffusion-advection equation for the sedimentation phenomena is presented. The order of the derivative is considered as0<β,γ≤1for the space and time domain, respectively. The fractional derivative of Caputo type is considered. In the spatial case we obtain the fractional solution for the underdamped, undamped, and overdamped case. In the temporal case we show that the concentration has amplitude which exhibits an algebraic decay at asymptotically large times and also shows numerical simulations where both derivatives are taken in simultaneous form. In order that the equation preserves the physical units of the system two auxiliary parametersσxandσtare introduced characterizing the existence of fractional space and time components, respectively. A physical relation between these parameters is reported and the solutions in space-time are given in terms of the Mittag-Leffler function depending on the parametersβandγ. The generalization of the fractional diffusion-advection equation in space-time exhibits anomalous behavior.

Frequency-Dependent Transmission Line Fractional Model and its Solution Based on Skin Effect

Due to the continuous increasing of operating frequency in the power system and the transmission speed, under the high frequencies of the transmission line calculation and simulation process, it is necessary to consider the frequency-dependent properties. At present, the frequency-dependent transmission line modeling has a variety of methods, but in the modeling and calculation of frequency variable term, processing is relatively complicated. This article will introduce transmission line equation of fractional calculus, intuitive representation of frequency varying parameters, and by a time-domain fractional solution, simplify the operation, improve the computational efficiency. Application of this algorithm for fractional differential equations can be obtained the voltage and current responses at any point in the transmission line. Thesis also by comparison with actual example, confirmed the validity and feasibility of the algorithm. At the same time, proposed algorithm can be extended to the multiple conductor transmission lines of fractional order model, also has certain applicability.