Bifurcation analysis of a cellular nonlinear network model via neural network approach

In this paper, receptor-based cellular nonlinear network model is studied. By applying neural network method, the ordinary differential equations being equivalent to the partial differential equations of the model are resulted. Also, the bifurcation analysis of the transformed system is presented. To support our theoretical results, some numerical examples are given.     

Neural network method: delay and system of delay differential equations

In the present article, delay and system of delay differential equations are treated using feed-forward artificial neural networks. We have solved multiple problems using neural network architectures with different depths. The neural networks are trained using the extreme learning machine algorithm for the satisfaction of delay differential equations and associated initial/boundary conditions. Further, numerical rates of convergence of the proposed algorithm are reported based on variation of error in the obtained solution for different number of training points. Emphasis is on analysing whether deeper network architectures trained with extreme learning machine algorithm can perform better than shallow network architectures for approximating the solutions of delay differential equations.

     

A Hopfield neural network with multi-compartmental activation

The Hopfield network is a form of recurrent artificial neural network. To satisfy demands of artificial neural networks and brain activity, the networks are needed to be modified in different ways. Accordingly, it is the first time that, in our paper, a Hopfield neural network with piecewise constant argument of generalized type and constant delay is considered. To insert both types of the arguments, a multi-compartmental activation function is utilized. For the analysis of the problem, we have applied the results for newly developed differential equations with piecewise constant argument of generalized type beside methods for differential equations and functional differential equations. In the paper, we obtained sufficient conditions for the existence of an equilibrium as well as its global exponential stability. The main instruments of investigation are Lyapunov functionals and linear matrix inequality method. Two examples with simulations are given to illustrate our solutions as well as global exponential stability.

     

A novel meshless collocation solver for solving multi-term variable-order time fractional PDEs

This paper presents a novel meshless collocation method to solve multi-term variable-order time fractional partial differential equations (VOTFPDEs). In the proposed method, it employs the Fourier series expansion for spatial discretization, which transforms the original multi-term VOTFPDEs into a sequence of multi-term variable-order time fractional ordinary differential equations (VOTFODEs). Then, these VOTFODEs can be solved using the recent-developed backward substitution method. Several numerical examples verify the accuracy and efficiency of the proposed numerical approach in the solution of multi-term VOTFPDEs.

     

Artificial neural networks for solving ordinary and partialdifferential equations

We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial/boundary conditions and contains no adjustable parameters. The second part is constructed so as not to affect the initial/boundary conditions. This part involves a feedforward neural network containing adjustable parameters (the weights). Hence by construction the initial/boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ordinary differential equations (ODE), to systems of coupled ODE and also to partial differential equations (PDE). In this article, we illustrate the method by solving a variety of model problems and present comparisons with solutions obtained using the Galerkin finite element method for several cases of partial differential equations. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.     

Adaptive neural network control for a nonlinear Euler‐Bernoulli beam in three‐dimensional space with unknown control direction

In this paper, an adaptive neural network control system is developed for a nonlinear three‐dimensional Euler‐Bernoulli beam with unknown control direction. The Euler‐Bernoulli beam is modeled as a combination of partial differential equations (PDEs) and ordinary differential equations (ODEs). Adaptive radial basis function–based neural network control laws are designed to determine approximation of disturbances. A projection mapping operator is adopted to realize bounded approximation of disturbances. A Nussbaum function is introduced to compensate for the unknown control direction. The goal of this study is to suppress the vibrations of the Euler‐Bernoulli beam in three‐dimensional space. In addition, unknown control direction problem and bounded disturbances are considered to guarantee that the signals of the system are uniformly bounded. Numerical simulations demonstrate the effectiveness of the proposed method.     

On parameter identification for distributed systems using Galerkin's criterion

A method is presented for estimating parameters in distributed parameter systems. The system is assumed to be modeled by a set of partial differential equations whose form is known to within a set of unknown constant parameters. Galerkin's Method is used to transform the partial differential equations into a set of ordinary differential equations. The approach to the identification problem is given in a step by step procedure. Three optimization schemes for estimating the unknown parameters are discussed. They are a steepest descent method, a search technique, and nonlinear filtering.     

An LPV approach to the robust control of a class of quasi-linear propagation processes

In this paper the trajectory tracking control problem for a certain class of propagation processes modeled as quasi-linear parameter varying systems is considered. The propagation physical models are generally described by means of partial differential equations (PDEs). However in real world control problems the PDE models are usually converted into ordinary differential equations (ODEs) models adopting numerical and/or physical approximations. In many practical problems it happens that the propagation dynamics are linear, while the boundary conditions are described by nonlinear algebraic equations. A trajectory following control scheme is proposed for this class of systems together with a robust performance analysis based on the concept of quadratic stability with an H_∞ norm bound. An LMI based observer synthesis procedure is also proposed to increase the closed loop system performance.     

Darcy–Forchheimer flow and heat transfer augmentation of a viscoelastic fluid over an incessant moving needle in the presence of viscous dissipation

The main focus of the present study is to analyse the effect of viscous dissipation Darcy–Forchheimer flow and heat transfer augmentation of a viscoelastic fluid over an incessant moving needle. The governing partial differential equations of the defined problem are reduced into a set of nonlinear ordinary differential equations using adequate similarity transformations. Obtained set of similarity equations are then solved with the help of efficient numerical method fourth fifth order RKF-45 method. The effects of different flow pertinent parameters on the flow fields like velocity and temperature are shown in the form of graphs and tables. The detailed analysis of the problem is carried out based on the plotted graphs and tables.

     

Theory and application of stability for stochastic reaction diffusion systems

So far, the Lyapunov direct method is still the most effective technique in the study of stability for ordinary differential equations and stochastic differential equations. Due to the shortage of the corresponding It6 formula, this useful method has not been popularized in stochastic partial differential equations. The aim of this work is to try to extend the Lyapunov direct method to the It6 stochastic reaction diffusion systems and to establish the corresponding Lyapunov stability theory, including stability in probablity, asymptotic stability in probablity, and exponential stability in mean square. As the application of the obtained theorems, this paper addresses the stability of the Hopfield neural network and points out that the main results ob- tained by Holden Helge and Liao Xiaoxin et al. can be all regarded as the corollaries of the theorems presented in this paper.     

Single Layer Chebyshev Neural Network Model for Solving Elliptic Partial Differential Equations

The purpose of the present study is to solve partial differential equations (PDEs) using single layer functional link artificial neural network method. Numerical solution of elliptic PDEs have been obtained here by applying Chebyshev neural network (ChNN) model for the first time. Computations become efficient because the hidden layer is eliminated by expanding the input pattern by Chebyshev polynomials. Feed forward neural network model with unsupervised error back propagation principle is used for modifying the network parameters and to minimize the computed error function. Numerical efficiency and accuracy of the ChNN model are investigated by three test problems of elliptic partial differential equations. The results obtained by this method are compared with the existing methods and are found to be in good agreement.     

A comparison of numerical methods for the solution of quasilinear partial differential equations

Various methods of approximating a parabolic partial differential equation by a system of ordinary differential equations are discussed. The methods are compared using the results of numerical experiments with a highly efficient integration procedure designed for this type of problem.     

Multilayer perceptron neural networks with novel unsupervised training method for numerical solution of the partial differential equations

In this paper by using MultiLayer Perceptron and Radial Basis Function (RBF) neural networks, a novel method for solving both kinds of differential equation, ordinary and partial differential equation, is presented. From the differential equation and its boundary conditions, the energy function of the network is prepared which is used in the unsupervised training method to update the network parameters. This method was implemented to solve the nonlinear Schrodinger equation in hydrogen atom and triangle-shaped quantum well. Comparison of this method results with analytical solution and two well-known numerical methods, Runge–kutta and finite element, shows the efficiency of Neural Networks with high accuracy, fast convergence and low use of memory for solving the differential equations.     

Robust adaptive boundary control of a perturbed hybrid Euler-Bernoulli beam with coupled rigid and flexible motion

In this paper, a robust adaptive boundary controller is proposed to stabilize the coupled rigid-flexible motion of an Euler-Bernoulli beam in presence of boundary and distributed perturbations. Applying Hamilton’s principle, the dynamics of the hybrid beam model, including the actuators hub and the payload at its ends, is represented through four nonhomogeneous nonlinear partial differential equations (PDEs) subject to ordinary differential equations (ODEs) of boundary conditions. Using a Lyapunov-based control synthesis procedure, a robust nonlinear boundary controller is established that asymptotically stabilizes the perturbed beam vibration while regulating the rigid motion coordinates. A redesign of the proposed control laws produces a robust adaptive boundary controller that achieves control objectives in the presence of both parametric and modelling uncertainties. Control design is directly based on system PDEs without truncating the model so that instabilities from spillover effects are mitigated. The control inputs to the beam consist of three forces/torque applied to the actuators hub and a transverse force applied to the tip payload. Simulation results are used to investigate the efficiency of the proposed approach.

     

Bifurcation Analysis of Delayed Complex-Valued Neural Network with Diffusions

In this paper, a class of delayed complex-valued neural network with diffusion under Dirichlet boundary conditions is considered. By using the properties of the Laplacian operator and separating the neural network into real and imaginary parts, the corresponding characteristic equation of neural network is obtained. Then, the dynamical behaviors including the local stability, the existence of Hopf bifurcation of zero equilibrium are investigated. Furthermore, by using the normal form theory and center manifold theorem of the partial differential equation, the explicit formulae which determine the direction of bifurcations and stability of bifurcating periodic solutions are obtained. Finally, a numerical simulation is carried out to illustrate the results.

     

Artificial neural network approach for solving fuzzy differential equations

The current research attempts to offer a novel method for solving fuzzy differential equations with initial conditions based on the use of feed-forward neural networks. First, the fuzzy differential equation is replaced by a system of ordinary differential equations. A trial solution of this system is written as a sum of two parts. The first part satisfies the initial condition and contains no adjustable parameters. The second part involves a feed-forward neural network containing adjustable parameters (the weights). Hence by construction, the initial condition is satisfied and the network is trained to satisfy the differential equations. This method, in comparison with existing numerical methods, shows that the use of neural networks provides solutions with good generalization and high accuracy. The proposed method is illustrated by several examples.     

Analytical solution for suction and injection flow of a viscoplastic Casson fluid past a stretching surface in the presence of viscous dissipation

In this study, steady two-dimensional flow of a viscoplastic Casson fluid past a stretching surface is considered under the effects of thermal radiation and viscous dissipation. Both suction and injection flows situations are considered. The partial differential governing equations are transformed into ordinary differential equations and solved analytical. Analytical solutions for velocity and temperature are obtained in terms of hypergeometric function and discussed graphically. Moreover, numerical results are also obtained by Runge–Kutta–Fehlberg fourth–fifth-order (RKF45) method and compared with the analytical results. The results showed that the injection and suction parameter can be used to control the direction and strength of flow. The effects of Casson parameter on the temperature and velocity are quite opposite. The effects of thermal radiation on the temperature are much more stronger in case of injection. The heat transfer coefficient shows higher value for Casson fluid while for Newtonian fluid is the lowest.

     

Dynamic analysis of thick laminated shell panel with piezoelectric layer based on three dimensional elasticity solution

Elasticity solution is presented for infinitely long, simply-supported, orthotropic, piezoelectric shell panel under dynamic pressure excitation. The direct and inverse piezoelectric effects are considered. The highly coupled partial differential equations (p.d.e.) are reduced to ordinary differential equations (o.d.e.) with variable coefficients by means of trigonometric function expansion in circumferential direction. The resulting ordinary differential equations are solved by the finite element method. Numerical examples are presented for [0/90/P] lamination, where P indicates the piezoelectric layer. Finally the results are compared with the published results.     

Transient analysis of structural members by the CSDT riccati transfer matrix method

A method for the direct integration of the dynamic governing partial differential equations of motion for structural members is developed. This technique is called the continuous-space discrete-time (CSDT) Riccati transfer matrix method. This formulation transforms a boundary value problem of governing partial differential equations of motion into a boundary value problem of ordinary differential equations. First, a standard procedure such as finite differences is employed to discretize the time derivatives. Then, a line solution technique such as the Riccati transfer matrix method is utilized to integrate the spatial derivatives. The stability and accuracy of the CSDT Riccati transfer matrix method using the Newmark generalized acceleration formulation for time discretization is studied. For a particular class of governing equations, it is shown that the method is unconditionally stable without amplitude decay error for particular parameter values in the Newmark formulation. The method, however, exhibits period elongation error as a function of the time step. Numerical results for bar and beam example problems indicate that this may well be a viable method for calculating the dynamic response of linear structural members.     

(2+1)维KD方程的解及分岔行为

通过行波变换,将非线性偏微分方程化为常微分方程,利用辅助常微分方程的解来构造偏微分方程的精确解,获得了(2+1)维Konopelchenko-Dubrovsky方程的孤波解和周期解.然后直接研究变换以后的常微分方程,揭示该方程控制的动力系统的鞍结分岔行为,画出了系统的分岔图.