Gut-Brain Axis as a Closed-Loop Molecular Communication Network

We study the numerical solutions of nonlinear fractional delay differential equations (DEs) and nonlinear fractional pantograph DEs. We introduce a new class of functions called fractional‐order generalized Taylor wavelets (FOGTW). We provide an exact formula for computing the Riemann‐Liouville fractional integral operator for FOGTW by using the regularized beta functions. By applying the formula and collocation method, we reduce the given nonlinear fractional delay DEs and nonlinear fractional pantograph DEs to a system of algebraic equations. The FOGTW method together with the exact formula is very efficient for solving the nonlinear fractional delay DEs and nonlinear fractional pantograph DEs and give very accurate results. Several examples are given to demonstrate the effectiveness of the present method.

The main result of this paper is that the oscillation and nonoscillation properties of a nonlinear impulsive delay differential equation are equivalent respectively to the oscillation and nonoscillation of a corresponding nonlinear delay differential equation without impulse effects. An explicit necessary and sufficient condition for the oscillation of a nonlinear impulsive delay differential equation is obtained.

A stability theory of nonlinear impulsive delay differential equations (IDDEs) is established. Existing algorithm may not converge when the impulses are variable. A convergent numerical scheme is established for nonlinear delay differential equations with variable impulses. Some stability conditions of analytical and numerical solutions to IDDEs are given by the properties of delay differential equations without impulsive perturbations.

Summary The nonlinear delay differential equation with exponential and quadratic nonlinearities is considered. It is assumed that the equation is exposed to stochastic perturbations of the white noise type, which are directly proportional to the deviation of the system state from the equilibrium point. Sufficient conditions for stability in probability of the zero and positive equilibriums of the considered system under stochastic perturbations are obtained. The research results are illustrated by numerical simulations. The proposed investigation procedure can be applied for arbitrary nonlinear stochastic delay differential equations with an order of nonlinearity higher than one. Copyright © 2016 John Wiley & Sons, Ltd.

Most of the physical phenomena located around us are nonlinear in nature and their solutions are of great significance for scientists and engineers. In order to have a better representation of these physical phenomena, fractional calculus is developed. Some of these nonlinear physical models can be represented in the form of delay differential equations of fractional order. In this article, a new method named Gegenbauer Wavelets Steps Method is proposed using Gegenbauer polynomials and method of steps for solving nonlinear fractional delay differential equations. Method of steps is used to convert the fractional nonlinear fractional delay differential equation into a fractional nonlinear differential equation and then Gegenbauer wavelet method is applied at each iteration of fractional differential equation to find the solution. To check the accuracy and efficiency of the proposed method, the proposed method is implemented on different nonlinear fractional delay differential equations including singular-type problems also.

Non-Linear Delay Differential Equations and Function Algebras

PurposeIn this article, the authors aims to introduce a novel Vieta–Lucas wavelets method by generalizing the Vieta–Lucas polynomials for the numerical solutions of fractional linear and non-linear delay differential equations on semi-infinite interval.Design/methodology/approachThe authors have worked on the development of the operational matrices for the Vieta–Lucas wavelets and their Riemann–Liouville fractional integral, and these matrices are successfully utilized for the solution of fractional linear and non-linear delay differential equations on semi-infinite interval. The method which authors have introduced in the current paper utilizes the operational matrices of Vieta–Lucas wavelets to converts the fractional delay differential equations (FDDEs) into a system of algebraic equations. For non-linear FDDE, the authors utilize the quasilinearization technique in conjunction with the Vieta–Lucas wavelets method.FindingsThe purpose of utilizing the new operational matrices is to make the method more efficient, because the operational matrices contains many zero entries. Authors have worked out on both error and convergence analysis of the present method. Procedure of implementation for FDDE is also provided. Furthermore, numerical simulations are provided to illustrate the reliability and accuracy of the method.Originality/valueMany engineers or scientist can utilize the present method for solving their ordinary or Caputo–fractional differential models. To the best of authors’ knowledge, the present work has not been used or introduced for the considered type of differential equations.

This article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs)by a set of ordinary differential equations (ODEs).We work in Hilbert spaces endowed with a natural inner product including a point mass,and introduce polynomials orthogonal with respect to such an inner product that live in the domain ofthe linear operator associated with the underlying DDE. Thesepolynomials are then used to design a general Galerkin scheme for whichwe derive rigorous convergence results and show that it can benumerically implemented via simple analytic formulas.The scheme so obtained is applied to three nonlinear DDEs, two autonomous and one forced:(i) a simple DDE with distributed delays whose solutions recall Brownian motion;(ii) a DDE with a discrete delay that exhibits bimodal and chaotic dynamics;and (iii) a periodically forced DDE with two discrete delays arising in climate dynamics.In all three cases, the Galerkin scheme introduced in this articleprovides a good approximation by low-dimensional ODE systemsof the DDE's strange attractor, as well as of the statistical features that characterize its nonlinear dynamics.

Global solutions of a singular initial value problem to second order nonlinear delay differential equations

The paper proposes a time-delayed hyperchaotic system composed of multiscroll attractors with multiple positive Lyapunov exponents (LEs), which are described by a three-order nonlinear retarded type delay differential equation (DDE). The dynamical characteristics of the time-delayed system are far more complicated than those of the original system without time delay. The three-order time-delayed system not only generates hyperchaotic attractors with multiscroll but also has multiple positive LEs. We observe that the number of positive LEs increases with increasing time delay. Through numerical simulations, the time-delayed system exhibits a larger number of scrolls than the original system without time delay. Moreover, different numbers of scrolls with variable delay and coexistence of multiple attractors with a variable number of scrolls are also observed in the time-delayed system. Finally, we setup electronic circuit of the proposed system, and make Pspice simulations to it. The Pspice simulation results agree well with the numerical results.

This paper is concerned with obtaining the exact solutions of linear and nonlinear delay differential equations (DDEs) via a combination of the Laplace transform and variational iteration method. In this approach, a correction functional is constructed by a general Lagrange multiplier, which is determined by using the Laplace transform with the variational theory. Numerical examples are given to elucidate the solution process, the simplicity, efficiency and reliability of the new approach.

This paper aims to study two approximate analytical methods for solving linear and nonlinear delay differential equations. Approximate approaches are shown in the Variational iteration method and the Adomian decomposition method. Through the conversion of some instances, including linear and nonlinear delay differential equations with initial values, by comparing different approximate methods. The results show that this procedure is accurate and adequate for DDE. Comparisons between the Adomian Decomposition Method and Variational Iteration Method results demonstrate the accuracy of the results obtained from both mentioned methods. The variational iteration method is efficient and convenient based on comparing the results and exact solutions.

Time series analysis with nonlinear delay differential equations (DDEs) is a powerful tool since it reveals spectral as well as nonlinear properties of the underlying dynamical system. Here global DDE models are used to analyze electrocardiography recordings (ECGs) in order to capture distinguishing features for different heart conditions such as normal heart beat, congestive heart failure, and atrial fibrillation. To capture distinguishing features of the different data types the number of terms and delays in the model as well as the order of nonlinearity of the DDE model have to be selected. The DDE structure selection is done in a supervised way by selecting the DDE that best separates different data types. We analyzed 24 h of data from 15 young healthy subjects in normal sinus rhythm (NSR) of 15 congestive heart failure (CHF) patients as well as of 15 subjects suffering from atrial fibrillation (AF) selected from the Physionet database. For the analysis presented here we used 5 min non-overlapping data windows on the raw data without any artifact removal. For classification performance we used the Cohen Kappa coefficient computed directly from the confusion matrix. The overall classification performance of the three groups was around 72–99 % on the 5 min windows for the different approaches. For 2 h data windows the classification for all three groups was above \(95\,\%\).

Numerical and theoretical treatment for solving linear and nonlinear delay differential equations using variational iteration method

We consider a 'data assimilation problem' for nonlinear delay differential equations. Our problem is to find an initial function that gives rise to a solution of a given nonlinear delay differential equation, which is a close fit to observed data. A role for adjoint equations and fundamental solutions in the nonlinear case is established. A 'pseudo-Newton' method is presented. Our results extend those given by the authors in [(C. T. H. Baker and E. I. Parmuzin, Identification of the initial function for delay differential equation: Part I: The continuous problem & an integral equation analysis. NA Report No. 431, MCCM, Manchester, England, 2004.), (C. T. H. Baker and E. I. Parmuzin, Analysis via integral equations of an identification problem for delay differential equations. J. Int. Equations Appl. (2004) 16, 111–135.)] for the case of linear delay differential equations.