Department of Mathematics

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In this paper, we consider the multi-Galerkin and multi-collocation methods for solving the Fredholm–Hammerstein integral equation with a smooth kernel, using Legendre polynomial bases. We show that Legendre multi-Galerkin and Legendre multi-collocation methods have order of convergence O(n−3r+34) and O(n−2r+12), respectively, in uniform norm, where n is the highest degree of Legendre polynomial employed in the approximation and r is the smoothness of the kernel. Also, one step of iteration method is used to improve the order of convergence and we prove that iterated Legendre multi-Galerkin and iterated Legendre multi-collocation methods have order of convergence O(n−4r) and O(n−2r), respectively, in uniform norm. Numerical examples are given to illustrate the theoretical results. © 2017 Elsevier B.V.

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Journal of Computational and Applied Mathematics

Superconvergence of Legendre spectral projection methods for Fredholm–Hammerstein integral equations

Stochastic orders are very useful tools to compare the lifetimes of two systems. Optimum lifetime of a series (resp. parallel) system with general standby component(s) depends on the allocation strategy of standby component(s) into the system. Here, we discuss three different models of one or more standby components. In each model, we compare different series (resp. parallel) systems (which are formed through different allocation strategies of standby component(s)) with respect to the usual stochastic order and the stochastic precedence order. The results related to the cold as well as the hot standby models are obtained as particular cases of the results discussed in this article because the model considered here is a general one. © 2017 Taylor & Francis Group, LLC.

Communications in Statistics - Theory and Methods

General standby allocation in series and parallel systems

In this paper, we consider series and parallel systems composed of n independent items drawn from a population consisting of m different substocks/subpopulations. We show that for a series system, the optimal (maximal) reliability is achieved by drawing all items from one substock, whereas, for a parallel system, the optimal solution results in an independent drawing of all items from the whole mixed population. We use the theory of stochastic orders and majorization orders to prove these and more general results. We also discuss possible applications and extensions. © 2017 Elsevier Inc.

Journal of Multivariate Analysis

On optimal grouping and stochastic comparisons for heterogeneous items

In this article, we study the Brezis–Nirenberg type problem of nonlinear Choquard equation involving the fractional Laplacian (Formula presented.), where Ω is a bounded domain in Rn with Lipschitz boundary, λ is a real parameter, s∈ (0 , 1) , n&gt; 2 s, 0 &lt; μ&lt; n and 2μ,s∗=(2n-μ)/(n-2s) is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. We obtain some existence, nonexistence and regularity results for weak solution of the above problem using variational methods. © 2017, Springer International Publishing AG.

Fulltext

Nonlinear Differential Equations and Applications

Fractional Choquard equation with critical nonlinearities

In this article, we study the following fractional elliptic equation with critical growth and singular nonlinearity: (Equation presented) where Ω is a bounded domain in ℝn with smooth boundary ∂Ω, n &gt; 2s, s ∈ (0, 1), λ &gt; 0, q &gt; 0 and 2∗s = 2n/n-2s. We use variational methods to show the existence and multiplicity of positive solutions with respect to the parameter λ. © 2017 Walter de Gruyter GmbH, Berlin/Boston.

Advances in Nonlinear Analysis

Positive solutions of fractional elliptic equation with critical and singular nonlinearity

We present a new fully spatially structured PDE metapopulation model for predator–prey dynamics in d≤3 space dimensions. A nonlinear reaction–diffusion system of Rosenzweig–MacArthur form models predator–prey dynamics in two ‘high’ quality patches embedded in a ‘low’ quality subdomain, where species can diffuse, convect and die. Our model substantially generalizes and improves earlier fully structured metapopulation models. After a nondimensionalization procedure, in order to approximate the metapopulation model we present a fully discrete Galerkin finite element method in two space dimensions, which is a generalization of the finite element method analyzed in a previous single patch predator–prey model. The numerical solutions are illustrated for some test cases using MATLAB. Numerical experiments demonstrate that the initial local extinction of predators in one patch leads to waves of recolonization from another patch. In an appendix we also give an outline for the proof of the well-posedness of the model. © 2016 Elsevier Ltd

Computers and Mathematics with Applications

Finite element approximation of a spatially structured metapopulation PDE model

The inviscid temporal stability analysis of two-fluid parallel shear flow with a free surface, down an incline, is studied. The velocity profiles are chosen as piecewise-linear with two limbs. The analysis reveals the existence of unstable inviscid modes, arising due to wave interaction between the free surface and the shear-jump interface. Surface tension decreases the maximum growth rate of the dominant disturbance. Interestingly, in some limits, surface tension destabilises extremely short waves in this flow. This can happen because of the interaction with the shear-jump interface. This flow may be compared with a corresponding viscous two-fluid flow. Though viscosity modifies the stability properties of the flow system both qualitatively and quantitatively, there is qualitative agreement between the viscous and inviscid stability analysis when the less viscous fluid is closer to the free surface. © 2016, Springer Science+Business Media Dordrecht.

Meccanica

Inviscid instability of two-fluid free surface flow down an incline

We study the allocation strategies for redundant components in the load-sharing series/parallel systems. We show that under the specified assumptions, the allocation of a redundant component to the stochastically weakest (strongest) component of a series (parallel) system is the best strategy to achieve its maximal reliability. The results have been studied under cumulative exposure model and for a general scenario as well. They have a clear intuitive meaning; however, the corresponding additional assumptions are not obvious, which can be seen from the proofs of our theorems. Copyright © Cambridge University Press 2016.

Probability in the Engineering and Informational Sciences

On stochastic comparisons for load-sharing series and parallel systems

Some partial orderings that compare probability distributions with the exponential distribution are found to be very useful to understand the phenomenon of ageing. Here, we introduce some new generalized partial orderings which describe the same kind of phenomenon of some generalized ageing classes. We give some equivalent conditions for each of the orderings. Inter-relations among the generalized orderings have also been discussed. © 2017 Taylor & Francis Group, LLC.

On some generalized ageing orderings

Generally, in watermarking techniques the size of the watermark is very small when compared to the host image. In other words, a little amount of watermark is embedded in the huge quantity of image pixels as the notice of legitimate ownership. Contrary to that idea, this is an attempt in which the capacity of watermarking is improved by embedding huge amount of watermark efficiently in the less quantity of image pixels. The core idea behind the proposed approach is to select watermarkable pixels from the host image based on the census transform and hamming distance followed by the embedding which is done by proposed spectral decompositions, i.e., Hankel, Circulant and Topelitz spectral decomposition. Finally, a reliable watermark extraction scheme is developed which is free from the false-positive detection problem of singular values. The experimental evaluation demonstrates that the proposed scheme is expeditiously able to withstand a variety of extreme attacks and highly suitable for covert communications. © 2016, Springer Science+Business Media New York.