Department of Mathematics

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This article deals with the study of the following Kirchhoff equation with exponential nonlinearity of Choquard type (see (KC) below). We use the variational method in the light of Moser–Trudinger inequality to show the existence of weak solutions to (KC). Moreover, analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to convex–concave problem (Pλ,M) below. © 2019 Elsevier Ltd

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Nonlinear Analysis, Theory, Methods and Applications

n-Kirchhoff–Choquard equations with exponential nonlinearity

In this article, we consider the multi-Galerkin method for solving the Fredholm–Hammerstein integral equations with weakly singular kernels, using piecewise polynomial bases. We show that multi-Galerkin method has order of convergence O(h r+α ) for the algebraic kernel, whereas for logarithmic kernel, it converges with the order O(log h h r+1 ) in uniform norm, where h is the norm of the partition and r is the smoothness of the solution. We also discuss the iterated multi-Galerkin method. We prove that iterated multi-Galerkin method has order of convergence O(h r+2α ) for the algebraic kernel and for logarithmic kernel, it has order of convergence of order O((log h) 2 h r+2 ) in uniform norm. Numerical examples are given to illustrate the theoretical results. © 2018, © 2019 Taylor & Francis Group, LLC.

Numerical Functional Analysis and Optimization

Superconvergence Results for Weakly Singular Fredholm–Hammerstein Integral Equations

In this paper, we consider Legendre multi-Galerkin methods to solve Fredholm integral equations of the second kind with weakly singular kernel and the corresponding eigenvalue problem. We obtain the convergence rates for the approximated solution and iterated solution in weakly singular Fredholm integral equations of the second kind in both L2 and infinity-norm. We also establish error bounds of approximated eigenelements with exact eigenelements in the eigenvalue problem of a compact integral operator with weakly singular kernel in both infinity and L2-norm. Numerical examples are presented to prove the theoretical estimates. © 2018 Elsevier B.V.

Journal of Computational and Applied Mathematics

Legendre multi-Galerkin methods for Fredholm integral equations with weakly singular kernel and the corresponding eigenvalue problem

Let X ⊂ AZd be a 2-dimensional subshift of finite type. We prove that any 2-dimensional subshift of finite type can be characterized by a square matrix of infinite dimension. We extend our result to a general d-dimensional case. We prove that the multidimensional shift space is non-empty if and only if the matrix obtained is of positive dimension. In the process, we give an alternative view of the necessary and sufficient conditions obtained for the non-emptiness of the multidimensional shift space. We also give sufficient conditions for the shift space X to exhibit periodic points. © AGT, UPV, 2019.

Applied General Topology

Matrix characterization of multidimensional subshifts of finite type

In this article a line search algorithm is proposed for solving constrained multi-objective optimization problems. At every iteration of the proposed method, a subproblem is formulated using quadratic approximation of all functions. A feasible descent direction is obtained as a solution of this subproblem. This scheme takes care some ideas of the sequential quadratically constrained quadratic programming technique for single objective optimization problems. A non-differentiable penalty function is used to restrict constraint violations at every iterating point. Convergence of the scheme is justified under the Slater constraint qualification along with some reasonable assumptions. The proposed algorithm is verified and compared with existing methods with a set of test problems. It is observed that this algorithm provides better results in most of the test problems. © 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group.

Engineering Optimization

A sequential quadratically constrained quadratic programming technique for a multi-objective optimization problem

The present study deals with the oblique wave trapping by a surface-piercing flexible porous barrier near a rigid wall in the presence of step-type bottoms under the assumptions of small amplitude water waves and the structural response theory in finite water depth. The modified mild-slope equation along with suitable jump conditions and the least squares approximation method are used to handle the mathematical boundary value problem. Four types of edge conditions, i.e., clamped-moored, clamped-free, moored-free, and moored-moored, are considered to keep the barrier at a desired position of interest. The role of the flexible porous barrier is studied by analyzing the reflection coefficient, surface elevation, and wave forces on the barrier and the rigid wall. The effects of step-type bottoms, incidence angle, barrier length, structural rigidity, porosity, and mooring angle are discussed. The study reveals that in the presence of a step bottom, full reflection can be found periodically with an increase in (i) wave number and (ii) distance between the barrier and the rigid wall. Moreover, nearly zero reflection can be found with a suitable combination of wave and structural parameters, which is desirable for creating a calm region near a rigid wall in the presence of a step bottom. © 2018, Harbin Engineering University and Springer-Verlag GmbH Germany, part of Springer Nature.

Journal of Marine Science and Application

Oblique Wave Trapping by a Surface-Piercing Flexible Porous Barrier in the Presence of Step-Type Bottoms

SYNOPTIC ABSTRACT: The Transformed-Transformer family of distributions are the resulting family of distributions as transformed from a random variable T through another transformer random variable X using a weight function ω of the cumulative distribution function of X. In this article, we study different stochastic ageing properties, as well as different stochastic orderings of this family of distributions. We discuss the results with several well known distributions. © 2018, © 2018 Taylor & Francis Group, LLC.

American Journal of Mathematical and Management Sciences

Some Reliability Properties of Transformed-Transformer Family of Distributions

Handbook of Multimedia Information Security: Techniques and Applications

SIE: An application to secure stereo images using encryption

In this paper, Legendre spectral projection methods are applied for the Volterra integral equations of second kind with a smooth kernel. We prove that the approximate solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the order O(n-r) in L2-norm and order O(n-r+12) in infinity norm, and the iterated Legendre Galerkin solution converges with the order O(n-2r) in both L2-norm and infinity norm, whereas the iterated Legendre collocation solution converges with the order O(n-r) in both L2-norm and infinity norm, n being the highest degree of Legendre polynomials employed in the approximation and r being the smoothness of the kernels. We have also considered multi-Galerkin method and its iterated version, and prove that the iterated multi-Galerkin solution converges with the order O(n-3r) in both infinity and L2 norm. Numerical examples are given to illustrate the theoretical results. © 2017, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.

Computational and Applied Mathematics

Superconvergence results of Legendre spectral projection methods for Volterra integral equations of second kind

CdTe quantum dots were synthesized using water as a solvent medium. Synthesized quantum dots were used to integrate into TiO2 nano-porous electrode using a combination of linker assisted direct adsorption and chemical bath deposition process. Sensitized electrodes were characterized to understand their physical and optical properties for photovoltaic application. © 2018 Elsevier Ltd.