Volume 23, Issue 3-4

The deterministic and stochastic synthetic drug transmission models with relapse are formulated. For the deterministic model, the basic reproduction number R 0 is derived. We show that if R 0 < 1, the drug-free equilibrium is globally asymptotically stable and if R 0 > 1, there exists a unique drug-addition equilibrium which is globally asymptotically stable. For the stochastic model, we show there exists a unique global positive solution of the stochastic model for any positive initial value. Then by constructing some stochastic Lyapunov functions, we show that the solution of the stochastic model is going around each of the steady states of the corresponding deterministic model under certain parametric conditions. The sensitive analysis of the basic reproduction number R 0 indicates that it is helpful to reduce the relapse rate of people who have a history of drug abuse in the control of synthetic drug spreading. Numerical simulations are carried out and approve our results.

We study the effects of the generic weak nonlinear loss on fast two-pulse interactions in linear waveguides. The colliding pulses are described by a system of coupled Schrödinger equations with a purely nonlinear coupling in the presence of the weak (2 m + 1)-order of nonlinear loss, for any m ≥ 1. We derive the analytic expression for the collision-induced amplitude shift in a fast two-pulse interaction. The analytic calculations are based on a generalization of the perturbation technique for calculating the effects of weak perturbations on fast collisions between solitons of the nonlinear Schrödinger equation. The theoretical predictions are confirmed by the numerical simulations with the full propagation model of coupled Schrödinger equations.

In this paper, we investigate the dissipativity of a class of BAM neural networks with both time-varying and distributed delays, as well as discontinuous activations. First, the concept of the Filippov solution is extended to functional differential equations with discontinuous right-hand sides via functional differential inclusions. Then, by constructing Lyapunov functional and employing a generalized Halanay inequality, several sufficient easy-to-test conditions are successfully obtained to guarantee the global dissipativity of the Filippov solution of the considered system. The derived results extend and improve some previous publications on conventional BAM neural networks. Meanwhile, the estimations of the positive invariant and globally attractive set are given. Finally, numerical simulations are provided to demonstrate the effectiveness of our proposed results.

In this paper, we consider the existence of a solution u ( x , t ) for the inverse backward problem for the nonlinear strongly damped wave equation with statistics discrete data. The problem is severely ill-posed in the sense of Hadamard, i.e., the solution does not depend continuously on the data. In order to regularize the unstable solution, we use the trigonometric method in non-parametric regression associated with the truncated expansion method. We investigate the convergence rate under some a priori assumptions on an exact solution in both L 2 and H q ( q > 0) norms. Moreover, a numerical example is given to illustrate our results.

In this paper, we propose a new nonlocal super-resolution (SR) model which is a combination of the nonlocal total variation (TV) regularization and the nonlocal p -Laplacian term (with p = 2). This choice is motivated by the success of the nonlocal TV term in preserving image edges and the efficiency of the nonlocal p -Laplacian term in preserving the image texture. To ensure the convergence of the proposed optimization SR problem, we prove the existence and uniqueness of a solution in a well-posed framework. In addition, to resolve the encountered minimization problem, we proposed a modified primal-dual algorithm and numerical results are also given to show the performance of the proposed approach.

By introducing two sets of Lenard recursion relations, we derive a hierarchy of modified shallow wave equations associated with a 3 × 3 matrix spectral problem with three potentials from the zero-curvature equation. The Baker–Akhiezer function and two meromorphic functions are defined on the trigonal curve which is introduced by utilizing the characteristic polynomial of the Lax matrix. Analyzing the asymptotic properties of the Baker–Akhiezer function and two meromorphic functions at two infinite points, we arrive at the explicit algebro-geometric solutions for the entire hierarchy in terms of the Riemann theta function by showing the explicit forms of the normalized Abelian differentials of the third kind.

This paper presents the singularity analysis of a 7-degrees of freedom (DOF) hybrid manipulator consisting of a closed-loop within it. From the past studies, it is well-known that the kinematic singularities play a significant role in the design and control of robotic manipulators. Kinematic singularities pose two-fold effects – first, they can induce the loss of one or more DOF of the manipulator and cause infinite joint rates at that particular joint, and second, they help to determine the trajectory or zone with high mechanical advantage. In current work, a 7-DOF hybrid manipulator is considered which is being developed at Council Of Scientific And Industrial Research–Central Scientific Instruments Organisation (CSIR–CSIO) Chandigarh to assist a surgeon during a medical-surgical task. To emulate the natural motion of a surgeon, the challenging configuration with redundant DOF is utilized. Jacobian has been computed analytically and analyzed at each instantaneous configuration with the evaluation of manipulability. Effect of a closed loop in the hybrid configurations is focused at, and utilizing the contour plots, good and worst working zones are identified in the workspace of the manipulator. The verification and validation of best and worst manipulability points (singularities) are done with the help of genetic algorithms, to determine locally and globally optimal configurations. Finally, on the basis of the singularity analysis, the present work concludes with few guidelines to the surgeon about the best and worst working zones for surgical tasks.

In this article, the transport of a passive pollutant by a flow modeled by shallow water equations is numerically investigated. The kinetic flux-vector splitting (KFVS) scheme is extended to solve the one and two-dimensional equations. The first two equations of the considered model are mass and momentum equations and the third equation is the transport equation. The suggested scheme focuses on the direct splitting of the macroscopic flux functions at the cell interfaces. It achieves second-order accuracy by using MUSCL-type initial reconstruction and the Runge–Kutta time stepping technique. Several numerical test problems from literature are considered to check the efficiency and performance of the scheme. The results of the proposed scheme are compared to the central scheme for validation. It is found that the results of both the schemes are in close agreement with each other. However, our suggested KFVS scheme resolves the sharp discontinuous profiles precisely.

The conditions to have a stable limit cycle by an Andronov–Hopf bifurcation in a tritrophic model are given. A generalized logistic growth function for the prey is considered, and a general family of functional responses, including the Holling type, are taken for the predators. Some results obtained in previous works for tritrophic models, which consider logistic growth in the prey and Holling functional responses, are generalized.

Based upon full domain decomposition, local and parallel stabilized finite element methods for the stationary Stokes equations are proposed and analysed, where the quadratic equal-order finite elements are employed for the velocity and pressure approximations, and a stabilized term based on two local Gauss integrations is used to offset the discrete pressure space to circumvent the discrete inf-sup condition. In the proposed parallel method, all of the computations are performed on the locally refined global grids that are fine around the interested subdomain and coarse elsewhere, making the method easy to implement based on a sequential solver with low communication cost. Stability and optimal error estimates of the present methods are deduced. Numerical results on examples including a problem with known analytic solution, lid-driven cavity flow, backward-facing step flow and flow around a cylinder are given to verify the theoretical predictions and demonstrate the high efficiency of the method. Results show that our parallel method can provide an approximate solution with the convergence rate of the same order as the solution computed by the standard stabilized finite element method, with a substantial reduction in computational time.

Theoretical and experimental large deflection and stress analysis of a master leaf spring considering stress concentration effect of clamping is reported. The non-uniformly curved master leaf spring under three point bending subjected to moving boundaries is modeled. Geometrically nonlinear strain-displacement relations, as necessary for the theoretical analysis, are derived through visualization of physics behind the large deformation problem. An embedded curvilinear coordinate system is considered, to study the combined effects of non-uniform curvature, bending, stretching and shear deformation including cross-sectional warping. Governing equation of the non-uniformly curved beam system is derived in variational form using energy method, based on linear material constitutive relations and the derived nonlinear kinematic relations. An iterative solution scheme through successive geometry updation is developed and executed in MATLAB ® software to solve the governing equation involving strong geometric nonlinearity together with complicating moving boundary effect. Experimental deflection profiles under static loading are obtained through manual image processing technique using AutoCAD ® software. Whereas, strain measurements are performed using strain gauges with data acquisition system (HBM-MX840B). Comparison between the theoretical and experimental results lead towards observation on stress concentration effect due to presence of geometric discontinuity in form of a small hole in the physical system. A modified formulation is proposed using domain decomposition method incorporating effect of geometric discontinuity through an equivalent curved beam geometry of variable cross-section. The modified theoretical model is validated successfully with the experimental results, and observations on stress characteristics and effect of hole diameter to beam width ratio are made.

In this paper, nonlinear vibration and frequency response analysis of double walled piezoelectric nanoshell (DWPENS) is investigated using nonclassical approach of the Gurtin–Murdoch surface/interface (GMSIT) theory. The piezoelectric nanoshell is simultaneously subjected to visco-Pasternak medium, the nonlinear van der Waals and electrostatic forces. Hamilton’s principles, the assumed mode method combined with Lagrange–Euler’s are used for the governing equations and boundary conditions. Complex averaging method combined with Arc-length continuation is used to achieve the nonlinear frequency response and stability analysis of the DWPENS. It is found that the electrostatic and piezoelectric voltages, the length to radius ratio, the nanoshell gap width, van der Waals (vdW) coefficients and other parameters can effectively change the flexural rigidity of the system which in turn affects the nonlinear frequency response. And also, increasing or decreasing of some parameters lead to increasing or decreasing the resonance amplitude, resonant frequency, the system’s instability, nonlinear behavior, and bandwidth.

In this paper, we have approached a complicated nonlinear wave equation which links the Rosenau–Kawahara equation to the regularized long wave (RLW) equation. Taking advantages from the sine–cosine method as well as from the generalized auxiliary equation method, we have successfully reached to three types of traveling wave solutions: periodic, hyperbolic and exponential ones. Our results do constitute themselves as a challenge to apply the mentioned techniques in order to solve other generalized dynamical models, for example, the ones which involve phenomena such as a fully nonlinear dispersion and a fully nonlinear convection.

A predictor–corrector compact finite difference scheme is proposed for a nonlinear partial integro-differential equation. In our method, the time direction is approximated by backward Euler scheme and the Riemann–Liouville (R–L) fractional integral term is treated by means of first order convolution quadrature suggested by Lubich. Meanwhile, a two-step predictor–corrector (P–C) algorithm called MacCormack method is used. A fully discrete scheme is constructed with space discretization by compact finite difference method. Numerical experiment presents the scheme is in good agreement with the theoretical analysis.

Stochastic gene expression in regulatory networks is conventionally modelled via the chemical master equation (CME). As explicit solutions to the CME, in the form of so-called propagators, are oftentimes not readily available, various approximations have been proposed. A recently developed analytical method is based on a separation of time scales that assumes significant differences in the lifetimes of mRNA and protein in the network, allowing for the efficient approximation of propagators from asymptotic expansions for the corresponding generating functions. Here, we showcase the applicability of that method to simulated data from a ‘telegraph’ model for gene expression that is extended with an autoregulatory mechanism. We demonstrate that the resulting approximate propagators can be applied successfully for parameter inference in the non-regulated model; moreover, we show that, in the extended autoregulated model, autoactivation or autorepression may be refuted under certain assumptions on the model parameters. These results indicate that our approach may allow for successful parameter inference and model identification from longitudinal single cell data.

In this paper, a new chaotic system has been introduced and the fundamental properties of the system were investigated and presented using a bifurcation diagram, max Lyapunov exponent (LE) and phase portraits. The synchronization of the drive and response system has been done using the threshold control parameter. Later the differential chaos shift keying (DCSK) modulation scheme has been carried out for the system as it is the most efficient modulation scheme. The demodulator detects the data without the use of chaotic signal phase recovery, as it uses the non-coherent detection technique. The results were compared with other modulation schemes using the bit error rate (BER) graph. It reveals that the proposed chaos-based system could be used for secure communication. The system has been implemented using the MATLAB Simulink technique.

This paper presents a new technique for solving linear and nonlinear boundary value problems subject to linear or nonlinear conditions. The technique is based on the blending of the Chebyshev pseudospectral method. The rapid convergence and effectiveness are verified by several linear and nonlinear examples, and results are compared with the exact solutions. Our results show a remarkable improvement in the convergence of the results when compared with exact solutions.

This article aims to solve the nonlinear Allen–Cahn equation numerically. The diagonally implicit fractional-step θ -(DIFST) scheme is used for the discretization of the time derivative term while the space derivative is discretized by the conforming finite element method. The computational efficiency of the DIFST scheme in terms of CPU time and temporal error estimation is computed and compared with other time discretization schemes. Several test problems are presented to show the effectiveness of the DIFST scheme.

According to the homoclinic breather limit method, we obtain the homoclinic breather wave and rational wave of a nonlinear evolution differential equation. The n -soliton wave solutions are derived by utilizing the Hirota method. In addition, the graphs of these solutions are shown by selecting the appropriate parameters.

The (3 + 1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like equation has certain advantages in solving engineering problems. In this paper, based on the generalized bilinear form, we successfully derived the diversity of exact solutions under certain constraints by using the symbolic computation Maple. These solutions have interaction wave solitons, cross-kink wave solitons, and bright-dark solitons. To ensure the accuracy of these solutions, we made a special selection of the parameters involved and made a three-dimensional graph, density graph, and contour graph to illustrate the dynamics of the solutions. The resulting solutions can be used for the study of certain phenomena in physics.