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This paper deals the design and analysis of a parameter uniform finite difference algorithm for a class of parabolic singularly perturbed convection-diffusion problems. The discrete nearby problem is designed using the backward Euler scheme in time and upwind scheme on H (ℓ) mesh in space. We present the convergence analysis of the algorithm, which proves that it is parameter uniform in L ∞ norm. The numerical experiments support the theoretical estimates and demonstrates the efficiency of H (ℓ) mesh with the existing meshes, like Shishkin and Bakhvalov type meshes. [ABSTRACT FROM AUTHOR]

In this article, a numerical technique is developed and analyzed for singularly perturbed delay parabolic reaction–diffusion problems with Dirichlet boundary condition. An orthogonal spline collocation method with C 1 -cubic spline basis functions on a Shishkin mesh is used in the spatial direction. In the temporal direction Crank–Nicolson method on an equidistant mesh is used. An extensive analysis has been undertaken to establish the uniform convergence with respect to the perturbation parameter. To support the theoretical findings, numerical experiments have been presented. [ABSTRACT FROM AUTHOR]

This paper presents a non-symmetric interior penalty Galerkin (NIPG) finite element scheme for one-dimensional singularly perturbed reaction-diffusion equations with discontinuous diffusion. Such problems often arise in the modelling of phase transitions. The solution of the considered class of problem is known to exhibit boundary and interior layers. Priori bounds are given for the solution and its derivatives and NIPG finite element scheme is constructed for the numerical approximation of the given problem. Error estimates for the proposed scheme are derived in the energy norm as well as the balanced norm. Error bounds have been proved to be uniform with respect to the perturbation parameter ε . Numerical experiments are conducted in support of the theoretical estimates. [ABSTRACT FROM AUTHOR]

This article deals with the class of singularly perturbed convection-diffusion problem with time delay. A parameter uniform numerical method is developed, and its detailed analysis is done. To discretize the spatial domain, a harmonic mesh H(ℓ) is used, which gives more accurate results in comparison with Shishkin, S(ℓ) and Bakhvalov mesh. Numerical experiments are carried out to validate the proposed method. The computational results on H(ℓ) mesh have been compared with the other existing meshes like B-mesh, Shishkin mesh and S(ℓ) mesh. [ABSTRACT FROM AUTHOR]

The current work shows the efficacy of Dzyaloshinshkii–Moriya interaction to free the bound entanglement. Based on the work (Sharma et al. in Quant. Info. Proc. 15:1539 2016), we present further results in two qutrits bound entangled state proposed by Jurkowski et al. We consider a closed system of two qutrits and an auxiliary qutrit which interacts with either one of the two qutrits in a closed system. We erase the auxiliary qutrit from the system by doing partial trace operation and open system dynamics has been studied. We have found, the probability amplitude of auxiliary qutrit does not affect the system, while DM interaction plays a major role to govern the dynamics. The realignment and CCNR criteria have been used to detect the bound entanglement in the state, while for quantification of entanglement the negativity has been used. We explored the dynamics with all the possible cases of Jurkowski et al. bound entangled state. [ABSTRACT FROM AUTHOR]

In this article, we investigate the unitary dynamics of squashed entanglement and concurrence measures in Werner state and maximally entangled mixed states (MEMS) under two different Hamiltonians. The aim of the present study is twofold. The first part of the study deals with the dynamics under Heisenberg Hamiltonian and the second part deals under bi-linear bi-quadratic Hamiltonian which is the extension of the first Hamiltonian. In both parts, we investigate the dynamical trade-off and equilibrium points for squashed entanglement and concurrence. During the study, we also found the results of entanglement sudden death (ESD) with Heisenberg Hamiltonian in Werner state under concurrence measure. In the second part, we investigate the special result for the bi-linear bi-quadratic Hamiltonian which does not disturb squashed entanglement and concurrence in both the states and exhibits the robust character for both of the states. [ABSTRACT FROM AUTHOR]

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 is devoted to the study of singularly perturbed delay differential equations with or without a turning point. The solution of the considered class of problems may exhibit boundary or interior layer(s) due to the presence of the perturbation parameter, the turning point, and the delay term. Some a priori estimates are derived on the solution and its derivatives. To solve the problem numerically, a finite difference scheme on piecewise uniform Shishkin mesh along with interpolation to tackle the delay term is proposed. The solution is decomposed into regular and singular components to establish parameter uniform error estimate. It is shown that the proposed scheme converges to the solution of the continuous problem uniformly with respect to the singular perturbation parameter. The numerical experiments corroborate the theoretical findings. [ABSTRACT FROM AUTHOR]

In the present article, the robustness of entanglement in two qubits maximally entangled mixed states (MEMS) have been studied under quantum decoherence channels. Here we consider bit flip, phase flip, bit-phase-flip, amplitude damping, phase damping and depolarization channels. To quantify the entanglement, the concurrence has been used as an entanglement measure. During this study interesting results have been found for sudden death and birth of entanglement under bit flip and bit-phase-flip channels. While amplitude damping channel produces entanglement sudden death and does not allow re-birth of entanglement. On the other hand, two qubits MEMS exhibit the robust character against the phase flip, phase damping and depolarization channels. The elegant behavior of all the quantum channels have been investigated with varying parameter of quantum state MEMS in different cases. [ABSTRACT FROM AUTHOR]

In this paper, an attempt is made to present a method of quantum cost minimization or optimization technique for quantum reversible circuits using proposed merger rules in Exclusive Sum of Product (ESOP) method. These modified ESOP methods are used to minimize the quantum circuits. We found that the quantum cost is drastically decreased than the previous ESOP method. It will be easy to find the quantum cost and quantum gate optimized quantum circuits implementation. It will also reduce quantum error while the quantum circuit is executed in real quantum processor. [ABSTRACT FROM AUTHOR]

The motive of the present work is to develop a parameter robust numerical scheme for the class of problems involving singularly perturbed parabolic differential-difference equations with delay, which often arise in computational neuroscience. The numerical schemes developed prior to this work are restricted either to the case of small values of delay argument or linear convergence with restriction on the mesh generation. In practice, the delay argument can be of arbitrary size. Parameter ε may take small enough values e.g., viscosity coefficient in Navier-Stokes equation for fluids with high Reynolds number. It is required to construct a higher order parameter robust numerical scheme without any restriction on the mesh generation for singularly perturbed parabolic differential-difference equations with state dependent delay of arbitrary size. A new class of non-standard finite difference method based on interpolation, θ-method and Micken's techniques is constructed to approximate the solution of singularly perturbed parabolic differential-difference equations with arbitrary values of delay. It is shown that proposed numerical scheme is parameter uniform convergent. It is proved that this method is unconditionally stable and is convergent for 12≤θ≤1, without having any restriction on the mesh. Some numerical experiments are provided to illustrate the performance of the method. [ABSTRACT FROM AUTHOR]

In this paper, an expanded mixed finite element method with lowest order Raviart Thomas elements is developed and analyzed for a class of nonlinear and nonlocal parabolic problems. After obtaining some regularity results for the exact solution, a priori error estimates for the semidiscrete problem are established. Based on a linearized backward Euler method, a complete discrete scheme is proposed and a variant of Brouwer’s fixed point theorem is used to derive an existence of a fully discrete solution. Further, a priori error estimates for the fully discrete scheme are established. Finally, numerical experiments are conducted to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we study thermal quantum correlations as quantum discord and entanglement in bipartite system imposed by external magnetic field with Herring-Flicker coupling, i.e., J(R)=1.642e-2RR5/2+O(R2e-2R). The Herring-Flicker coupling strength is the function of R, which is the distance between spins and systems carry XXX Heisenberg interaction. By tuning the coupling distance R, temperature and magnetic field quantum correlations can be scaled in the bipartite system. We find the long sustainable behavior of quantum discord in comparison with entanglement over the coupling distance R. We also investigate the situations, where entanglement totally dies but quantum discord exists in the system. [ABSTRACT FROM AUTHOR]