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In this work, a compact finite difference approach is constructed for singularly perturbed parabolic reaction diffusion problems with a retarded term. The time and space derivatives have been discretized using the θ-method and a compact fourth-order finite difference method on a Shishkin mesh, respectively. Parameter uniform error estimates have been calculated in the $ L_\infty $ L ∞ norm. Some numerical examples have been considered to corroborate the theoretical results and compare the numerical results with existing methods in the literature. It is shown that the present approach provides improved results till date for the problem considered in this paper. [ABSTRACT FROM AUTHOR]

The authors present numerical analysis of singularly perturbed parabolic problems. The previous papers in this direction focussed on the convection-diffusion problems with regular type of boundary layers. It is very difficult and almost impossible as stated in the literature, e.g, in Chapter 14 of 'Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions' (1996) by Miller, John JH and O'Riordan, Eugene and Shishkin, Grigorii I, to capture singularities due to parabolic layers and develop a robust scheme for reaction-diffusion problems with general values of space shifts. The work is in progress now and this is the first paper in this direction. In this work, we are investigating such problems and develop a robust numerical scheme using Mickens's non-standard finite difference scheme and special mesh. Furthermore, numerical examples have been presented to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

Modern industrialisation and the need for low-density material initiated the up-gradation of material for the necessity of sustainability, and the same goes for engineering materials as well. Right from ancient times, composites have an important place in the field of engineering materials, but now demand for such materials, especially fibre-reinforced polymer (FRP) composites, proliferates just to improve the quality of materials as well as the quality of life. The present work provides a comprehensive literature review on hybrid FRP composites, techniques involved in its fabrication, machining and variation in mechanical properties due to hybridisation. The main focused fibre-reinforced composite are glass fibre, carbon fibre, basalt fibre and Kevlar fibre-reinforced polymeric composites. The paper includes recent work that showcases the hybridisation behaviour and their field of application. [ABSTRACT FROM AUTHOR]

This article focuses on the investigation of two-dimensional elliptic singularly perturbed problems that incorporate positive and negative shifts, the solution of this class of problems may demonstrate regular/parabolic/degenerate or interior boundary layers. The goal of this article is to establish the development of numerical techniques for two-dimensional elliptic singularly perturbed problems with positive and negative shifts having regular boundary layers. The three numerical schemes are proposed to estimate the solution of this class of problems based on the fitted operator and fitted mesh finite-difference methods. The fitted operator finite difference method is analysed for convergence. The effect of shift terms on the solution behaviour is demonstrated through numerical experiments. The paper concludes by providing several numerical results that demonstrate the performance of proposed numerical schemes. [ABSTRACT FROM AUTHOR]

In this work, we embarked on the research of linear two-dimensional elliptic singularly perturbed problems having positive and negative shifts in both directions of the reaction terms, whose solution exhibits characteristic boundary layers. The study of this class of problems was started in the mid-eighties, but all the studies were restricted to 1D. The Taylor series expansion is used to estimate decelerated terms of the problem and the emerging problem is discretized using the fitted mesh finite difference method to establish parameter uniform error estimates. The effect of positive and negative shifts on the behaviour of the solution is explained by executing numerical experiments on two test examples. [ABSTRACT FROM AUTHOR]

This paper presents the nonsymmetric interior penalty Galerkin (NIPG) finite element method for a class of one-dimensional convection dominated diffusion problems with discontinuous coefficients. The solution of the considered class of problem exhibits boundary and interior layers. Piecewise uniform Shishkin-type meshes are used for the spatial discretization. The error estimates in the energy norm have been derived for the proposed schemes. Theoretical results are supported by conducting numerical experiments. It is established that the errors are uniform with respect to the perturbation parameter . The uniformness of the error estimates with the perturbation parameter has also been established numerically for L ∞ - norm. [ABSTRACT FROM AUTHOR]

In the present paper, we study spin squeezing under decoherence in the superposition of tripartite maximally entangled GHZ and W states. Here we use amplitude damping, phase damping and depolarisation channel. We have investigated the dynamics of spin squeezing with the interplay of super-position and decoherence parameters with different directions of the mean spin vector. We have found the mixture of GHZ and W states is robust against spin squeezing generation for amplitude damping and phase damping channels for certain directions of the mean spin vector. However, the depolarisation channel performs well for spin squeezing generation and generates permanent spin squeezing in the superposition of GHZ and W states.

In this article, we derive one-parameter family of Schroder's method based on Gupta et al.'s (K.C. Gupta, V. Kanwar, and S. Kumar, "A family of ellipse methods for solving non-linear equations", Int. J. Math. Educ. Sci. Technol. 40 (2009), pp. 571-575) family of ellipse methods for the solution of nonlinear equations. Further, we introduce new families of Schroder-type methods for multiple roots with cubic convergence. Proposed families are derived from modified Newton's method for multiple roots and one-parameter family of Schroder's method. Numerical examples are also provided to show that these new methods are competitive to other known methods for multiple roots. (Contains 2 tables.)

Investigating the influence of quantum information (QI) scrambling on quantum correlations in a physical system is an interesting problem. In this article, we establish the mathematical connections among the quantifiers known as quantum information scrambling, Uhlmann fidelity, Bures metric and bipartite concurrence. We study these connections via four-point out-of-time-order correlation function used for quantum information scrambling. Further, we study the dynamics of all the quantifiers and investigate the influence of QI scrambling on entanglement in two qubits prepared in Bell states. We also investigate the QI scrambling and entanglement balancing points in Bell states under Ising Hamiltonian. [ABSTRACT FROM AUTHOR]

This paper continues the authors's study of robust numerical scheme for generalized Stein's model of neuronal variability, which is a singularly perturbed parabolic partial differential–difference equation with general values of shift arguments. The work on this class of problems is initiated in the papers [K. Bansal and K.K. Sharma, Parameter-robust numerical scheme for time dependent singularly perturbed reaction- diffusion problem with large delay, Numer. Funct. Anal. Optim. 39(2) (2018), pp. 127–154] for unit shift argument and in [K. Bansal and K.K. Sharma, Parameter uniform numerical scheme for time dependent singularly perturbed convection- diffusion-reaction problems with general shift arguments, Numer. Algorithms 75(1) (2017), pp. 113–145] for general shift arguments. In the present paper, this work is further extended to the development of numerical scheme based on Mickens techniques, interpolation and θ scheme. The advantage of this work over the previous research is that it deals with the problem having general values of the shift arguments with higher order of convergence and without any constraints on the number of intervals. [ABSTRACT FROM AUTHOR]

In modern industries, basalt and glass are the most commonly used fibers for the fabrication of various engineering components. Present work is focused on the fabrication of hybrid fiber reinforced plastic (FRP) composites, comprises of basalt and glass fibers. The aim of the work is to fabricate a composite justifying the current requirement of the era followed by the identification of the capabilities of fabricated composite by investigating its mechanical properties. Further, the machining of the fabricated composite has also been explored in order to limit the common problems in machining like fiber pull-out and delamination. From the results, it has been perceived that the fabricated composite can be machined flawlessly using laser beam machine subjected to the selection of input parameters. The proposed methodology seems helpful for researchers in fabricating the FRP composite and in identifying the range of input parameters suitable for machining. [ABSTRACT FROM AUTHOR]