Algebraic conditions of stability for Hopfield neural network

Using the relationship between the resistance, capacitance and current in Hopfield neural network, and the properties of sigmoid function, this paper gives the terse, explicit algebraical criteria of global exponential stability, global asymptotical stability and instability. Then this paper makes clear the essence of the stability that Hopfield defined, and provides a theoretical foundation for the design of a network.     

Intelligent computing for numerical treatment of nonlinear prey–predator models

In this study, a new computing paradigm is presented for evaluation of dynamics of nonlinear prey–predator mathematical model by exploiting the strengths of integrated intelligent mechanism through artificial neural networks, genetic algorithms and interior-point algorithm. In the scheme, artificial neural network based differential equation models of the system are constructed and optimization of the networks is performed with effective global search ability of genetic algorithm and its hybridization with interior-point algorithm for rapid local search. The proposed technique is applied to variants of nonlinear prey–predator models by taking different rating factors and comparison with Adams numerical solver certify the correctness for each scenario. The statistical studies have been conducted to authenticate the accuracy and convergence of the design methodology in terms of mean absolute error, root mean squared error and Nash-Sutcliffe efficiency performance indices.     

Pointwise uniformly convergent numerical treatment for the non-stationary Burger–Huxley equation using grid equidistribution

This article presents a numerical method for solving the singularly perturbed Burger–Huxley equation on a rectangular domain. That is, the highest-order derivative term in the equation is multiplied by a very small parameter. This small parameter is known as the perturbation parameter. When the perturbation parameter specifying the problem tends to zero, the solution of the perturbed problem exhibits layer behaviour in the outflow boundary region. Most conventional methods fail to capture this layer behaviour. For this reason, there is much current interest in the development of a robust numerical method that may handle the difficulties occurring due to the presence of the perturbation parameter and the nonlinearity of the problem. To solve both of these difficulties a numerical method is constructed. The first step in this direction is the discretization of the time variable using Euler's implicit method with a constant time step. This produces a nonlinear stationary singularly perturbed semidiscrete problem class. The problem class is then linearized using the quasilinearization process. This is followed by discretization in space, which uses the standard upwind finite difference operator. An extensive amount of analysis is carried out in order to establish the convergence and stability of the proposed method. Numerical experiments are carried out for model problems to illustrate graphically the theoretical results. The results indicate that the scheme faithfully mimics the dynamics of the differential equation.     

A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation

In this paper, an efficient numerical scheme based on uniform Haar wavelets and the quasilinearization process is proposed for the numerical simulation of time dependent nonlinear Burgers’ equation. The equation has great importance in many physical problems such as fluid dynamics, turbulence, sound waves in a viscous medium etc. The Haar wavelet basis permits to enlarge the class of functions used so far in the collocation framework. More accurate solutions are obtained by wavelet decomposition in the form of a multi-resolution analysis of the function which represents a solution of boundary value problems. The accuracy of the proposed method is demonstrated by three test problems. The numerical results are compared with existing numerical solutions found in the literature. The use of the uniform Haar wavelet is found to be accurate, simple, fast, flexible, convenient and has small computation costs.     

A nonstandard numerical scheme of predictor–corrector type for epidemic models

In this paper we construct and develop a competitive nonstandard finite difference numerical scheme of predictor–corrector type for the classical SIR epidemic model. This proposed scheme is designed with the aim of obtaining dynamical consistency between the discrete solution and the solution of the continuous model. The nonstandard finite difference scheme with Conservation Law (NSFDCL) developed here satisfies some important properties associated with the considered SIR epidemic model, such as positivity, boundedness, monotonicity, stability and conservation of frequency of the oscillations. Numerical comparisons between the NSFDCL numerical scheme developed here and Runge–Kutta type schemes show its effectiveness.     

A conservative numerical method for the Cahn–Hilliard equation with Dirichlet boundary conditions in complex domains

In this paper we present a conservative numerical method for the Cahn–Hilliard equation with Dirichlet boundary conditions in complex domains. The method uses an unconditionally gradient stable nonlinear splitting numerical scheme to remove the high-order time-step stability constraints. The continuous problem has the conservation of mass and we prove the conservative property of the proposed discrete scheme in complex domains. We describe the implementation of the proposed numerical scheme in detail. The resulting system of discrete equations is solved by a nonlinear multigrid method. We demonstrate the accuracy and robustness of the proposed Dirichlet boundary formulation using various numerical experiments. We numerically show the total energy decrease and the unconditionally gradient stability. In particular, the numerical results indicate the potential usefulness of the proposed method for accurately calculating biological membrane dynamics in confined domains.     

Dissipative exponentially-fitted methods for the numerical solution of the Schrödinger equation

The first dissipative exponentially fitted method for the numerical integration of the Schr?dinger equation is developed in this paper. The technique presented is a nonsymmetric multistep (dissipative) method. An application to the bound-states problem and the resonance problem of the radial Schr?dinger equation indicates that the new method is more efficient than the classical dissipative method and other well-known methods. Based on the new method and the method of Raptis and Allison (Comput. Phys. Commun. 14 (1978) 1-5) a new variable-step method is obtained. The application of the new variable-step method to the coupled differential equations arising from the Schr?dinger equation indicates the power of the new approach.     

Three-dimensional DenseNet self-attention neural network for automatic detection of student’s engagement

Today, due to the widespread outbreak of the deadly coronavirus, popularly known as COVID-19, the traditional classroom education has been shifted to computer-based learning. Students of various cognitive and psychological abilities participate in the learning process. However, most students are hesitant to provide regular and honest feedback on the comprehensiveness of the course, making it difficult for the instructor to ensure that all students are grasping the information at the same rate. The students’ understanding of the course and their emotional engagement, as indicated via facial expressions, are intertwined. This paper attempts to present a three-dimensional DenseNet self-attention neural network (DenseAttNet) used to identify and evaluate student participation in modern and traditional educational programs. With the Dataset for Affective States in E-Environments (DAiSEE), the proposed DenseAttNet model outperformed all other existing methods, achieving baseline accuracy of 63.59% for engagement classification and 54.27% for boredom classification, respectively. Besides, DenseAttNet trained on all four multi-labels, namely boredom, engagement, confusion, and frustration has registered an accuracy of 81.17%, 94.85%, 90.96%, and 95.85%, respectively. In addition, we performed a regression experiment on DAiSEE and obtained the lowest Mean Square Error (MSE) value of 0.0347. Finally, the proposed approach achieves a competitive MSE of 0.0877 when validated on the Emotion Recognition in the Wild Engagement Prediction (EmotiW-EP) dataset.

     

New criteria for globally exponential stability of delayed Cohen–Grossberg neural network

This paper is concerned with analysis problem for the global exponential stability of the Cohen–Grossberg neural networks with discrete delays and with distributed delays. We first prove the existence and uniqueness of the equilibrium point under mild conditions, assuming neither differentiability nor strict monotonicity for the activation function. Then, we employ Lyapunov functions to establish some sufficient conditions ensuring global exponential stability of equilibria for the Cohen–Grossberg neural networks with discrete delays and with distributed delays. Our results are not only presented in terms of system parameters and can be easily verified and also less restrictive than previously known criteria. A comparison between our results and the previous results admits that our results establish a new set of stability criteria for delayed neural networks.     

Design of intelligent power controller for DC–DC converters using CMAC neural network

DC–DC converters are the devices which can convert a certain electrical voltage to another level of electrical voltage. They are very popularly used because of the high efficiency and small size. This paper proposes an intelligent power controller for the DC–DC converters via cerebella model articulation controller (CMAC) neural network approach. The proposed intelligent power controller is composed of a CMAC neural controller and a robust controller. The CMAC neural controller uses a CMAC neural network to online mimic an ideal controller, and the robust controller is designed to achieve L ₂ tracking performance with desired attenuation level. Finally, a comparison among a PI control, adaptive neural control and the proposed intelligent power control is made. The experimental results are provided to demonstrate the proposed intelligent power controller can cope with the input voltage and load resistance variations to ensure the stability while providing fast transient response and simple computation.     

An expert system for the numerical solution of the radial Schrödinger equation

An expert system for the numerical solution of the phase shift problem of the radial Schrödinger equation is developed in this paper.     

A variable step method for the numerical integration of the one-dimensional Schrödinger equation

Most numerical methods which have been proposed for the approximate integration of the one-dimensional Schrödinger equation use a fixed step length of integration. Such an approach can of course result in gross inefficiency since the small step length which must normally be used in the initial part of the range of integration to obtain the desired accuracy must then be used throughout the integration. In this paper we consider the method of embedding, which is widely used with explicit Runge-Kutta methods for the solution of first order initial value problems, for use with the special formulae used to integrate the Schrödinger equation. By adopting this technique we have available at each step an estimate of the local truncation error and this estimate can be used to automatically control the step length of integration. Also considered is the problem of estimating the global truncation error at the end of the range of integration. The power of the approaches considered is illustrated by means of some numerical examples.     

A solving method based on neural network for a class of multi-leader–follower games

In this paper, we present a new solving approach for a class of multi-leader–follower games. For the problem studied, we firstly propose a neural network model. Then, based on Lyapunov and LaSalle theories, we prove that the trajectory of the neural network model can converge to the equilibrium point, which corresponds to the Nash equilibrium of the problem studied. The numerical results show that the proposed neural network approach is feasible to the problem studied.

     

Artificial neural network–genetic algorithm for estimation of crop evapotranspiration in a semi-arid region of Iran

This study compares the daily potato crop evapotranspiration (ET_C) estimated by artificial neural network (ANN), neural network–genetic algorithm (NNGA) and multivariate nonlinear regression (MNLR) methods. Using a 6-year (2000–2005) daily meteorological data recorded at Tabriz synoptic station and the Penman–Monteith FAO 56 standard approach (PMF-56), the daily ET_C was determined during the growing season (April–September). Air temperature, wind speed at 2 m height, net solar radiation, air pressure, relative humidity and crop coefficient for every day of the growing season were selected as the input of ANN models. In this study, the genetic algorithm was applied for optimization of the parameters used in ANN approach. It was found that the optimization of the ANN parameters did not improve the performance of ANN method. The results indicated that MNLR, ANN and NNGA methods were able to predict potato ET_C at desirable level of accuracy. However, the MNLR method with highest coefficient of determination (R ² > 0.96, P value < 0.05) and minimum errors provided superior performance among the other methods.     

A family of Numerov-type exponentially fitted methods for the numerical integration of the Schrödinger equation

A family of predictor-corrector exponential Numerov-type methods is developed for the numerical integration of the one-dimensional Schrödinger equation. The Numerov-type methods considered contain free parameters which allow it to be fitted to exponential functions. The new fourth algebraic order methods are very simple and integrate more exponential functions than both the well-known fourth order Numerov-type exponentially fitted methods and the sixth algebraic order Runge-Kutta-type methods. Numerical results also indicate that the new methods are much more accurate than the other exponentially fitted methods mentioned above.     

Chebyshev polynomials for the numerical solution of fractal–fractional model of nonlinear Ginzburg–Landau equation

Engineering with Computers - This paper introduces a new version for the nonlinear Ginzburg–Landau equation derived from fractal–fractional derivatives and proposes a computational...     

An eighth order exponentially-fitted method for the numerical integration of the Schrödinger equation

An eighth order exponentially-fitted method is developed for the numerical solution of the Schrödinger equation. The formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions. An error analysis is also given. Numerical and theoretical results indicate that the new method is much more accurate than other classical and exponentially fitted methods.     

High order Bessel fitting methods for the numerical integration of the Schrödinger equation

In this paper we show how to construct explicit multistep algorithms for an accurate and efficient numerical integration of the radial Schr?dinger equation. The proposed methods are Bessel fitting, that is to say, they integrate exactly any linear combination of Bessel and Newman functions and ordinary polynomials. They are the first of the like methods that can achieve any order.     

On variable-step methods for the numerical solution of Schrödinger equation and related problems

In this paper we present a review for the construction of variable-step methods for the numerical integration of the Schr?dinger equation. Phase-lag and stability are investigated. The methods are variable-step because of a simple natural error control mechanism. Numerical results obtained for coupled differential equations arising from the Schr?dinger equation and for the wave equation show the validity of the approach presented.