Nonlinear Delay Integro-differential Equations Research Articles

The stability of nonlinear delay integro-differential equations in the sense of Hyers-Ulam

Abstract In this study, an initial-value problem for a nonlinear Volterra functional integro-differential equation on a finite interval was considered. The nonlinear term in the equation contains multiple time delays. In addition to giving some new theorems on the existence and uniqueness of solutions to the equation, the authors also prove the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of the equation. The proofs use several different tools including Banach’s fixed point theorem, the construction of a Picard operator, and an application of Pachpatte’s inequality. An example is provided to illustrate the existence, uniqueness, and stability properties of solutions.

Solution estimates and stability tests for nonlinear delay integro-differential equations

In this article, we examine various qualitative features of solutions of a nonlinear delay integro-differential equation. We prove three new theorems which include sufficient conditions on asymptotic stability (AS), integrability, and boundedness of solutions, using a suitable Lyapunov-Krasovskii functional. We present examples that show applications of our results.

Sumudu Lagrange-spectral methods for solving system of linear and nonlinear Volterra integro-differential equations

This paper presents new efficient numerical methods for solving Volterra integro-differential equations and a system of nonlinear delay integro-differential equations which arises in biology. The principal idea of these approaches is based on a careful blend of the Petrov-Galerkin technique and the Sumudu transform method. In the proposed methods, using Lagrange polynomials and zeros of Jacobi polynomials, the considered system of linear and nonlinear integro-differential equations, with their associated initial conditions are reduced to linear and nonlinear systems of algebraic equations in the unknown expansion coefficients. Solving the resulting algebraic systems by Gaussian elimination and Newton's methods respectively, approximate solutions of the integro-differential problems are constructed. Detailed error analysis of the proposed methods is carried out to establish and ascertain the reliability and effectiveness of the methods. The methods are then tested on several examples, and the results are compared with those obtained via existing methods in the literature. The numerical results showed that the proposed methods are accurate, efficient, and reliable for solving all kinds of integro-differential equations.

A Collocation Method for Numerical Solution of Nonlinear Delay Integro-Differential Equations for Wireless Sensor Network and Internet of Things.

Wireless sensor network and industrial internet of things have been a growing area of research which is exploited in various fields such as smart home, smart industries, smart transportation, and so on. There is a need of a mechanism which can easily tackle the problems of nonlinear delay integro-differential equations for large-scale applications of Internet of Things. In this paper, Haar wavelet collocation technique is developed for the solution of nonlinear delay integro-differential equations for wireless sensor network and industrial Internet of Things. The method is applied to nonlinear delay Volterra, delay Fredholm and delay Volterra–Fredholm integro-differential equations which are based on the use of Haar wavelets. Some examples are given to show the computational efficiency of the proposed technique. The approximate solutions are compared with the exact solution. The maximum absolute and mean square roots errors for distant number of collocation points are also calculated. The results show that Haar method is efficient for solving these equations for industrial Internet of Things. The results are compared with existing methods from the literature. The results exhibit that the method is simple, precise and efficient.

Population dynamics between a prey and a predator using spectral collocation method

The struggle for the existence of the biological species is a well-known Prey–Predator model study in the literature. In this study, we present an improved model of Jerri [J. Abdul, Introduction to Integral Equations with Applications, Vol. 10 (Wiley, New York, 1999)] by introducing the intra-species competition term between the same species in addition to the existing environmental changes and few other factors in the model. The demand from the existing (limited) resources and other requirements induces competition between the same species which may alter the survival tactics among themselves. This intra-species term provides strength to the model as it makes the model more realistic. The governing equations are a system of two nonlinear delay integro differential equations, which are solved using spectral collocation method. The role of intra-species coefficients denoting the logistic growth/decay of the two species and two other parameters affecting the population dynamics are analyzed with the three basis functions such as Chebyshev, Legendre and Jacobi polynomials. With the help of simple matrix analysis, the governing equations are converted into a system of nonlinear algebraic equations. Detailed error estimation is computed to compare our results with the existing methods. It is shown with the help of tables and figures that the present method is very efficient, has better accuracy and has least computational cost.

Continuability and Boundedness of Solutions for a Kind of Nonlinear Delay Integrodifferential Equations of the Third Order

In the paper, we consider a nonlinear integrodifferential equation of the third order with delay. We establish sufficient conditions guaranteeing the global existence and boundedness of the solutions of the analyzed equation. We use the Lyapunov second method to prove the main result. An example is also given to illustrate the applicability of our result. The result of this paper is new and improves previously known results.

Existence and Stability Results on Nonlinear Delay Integro-Differential Equations with Random Impulses

Existence and Stability Results on Nonlinear Delay Integro-Differential Equations with Random Impulses

Solving a Class of Nonlinear Delay Integro–differential Equations by Using Differential Transformation Method

In this paper, differential transformation method is used to find exact solutions of nonlinear delay integro– differential equations. Many theorems are presented that required for applying differential transformation method for nonlinear delay integro–differential equation. The validity and efficiency of the proposed method are demonstrated through several tests.

An exponential approach for the system of nonlinear delay integro-differential equations describing biological species living together

In this paper, we consider a system of nonlinear delay integro-differential equations with convolution kernels, which arises in biology. This problem characterizes the population dynamics for two separate species. We present an exponential approach based on exponential polynomials for solving this system. This technique reduces the model problem to a matrix equation, which corresponds to a system of nonlinear algebraic equations. Also, illustrative examples related to biological species living together are given to demonstrate the validity and applicability of technique. The comparisons are made with the existing results.

Strongly asymmetric square waves in a time-delayed system

Time-delayed systems are known to exhibit symmetric square waves oscillating with a period close to twice the delay. Here, we show that strongly asymmetric square waves of a period close to one delay are possible. The plateau lengths can be tuned by changing a control parameter. The problem is investigated experimentally and numerically using a simple bandpass optoelectronic delay oscillator modeled by nonlinear delay integrodifferential equations. An asymptotic approximation of the square-wave periodic solution valid in the large delay limit allows an analytical description of its main properties (extrema and square pulse durations). A detailed numerical study of the bifurcation diagram indicates that the asymmetric square waves emerge from a Hopf bifurcation.

Existence of solutions for some nonlinear delay integrodifferential equations with nonlocal initial conditions

Abstract The main purpose of this paper is the existence of mild solutions for a class of first-order nonlinear delay integrodifferential equations with nonlocal initial conditions in Banach spaces. We show that the solutions are given by the application of the theory of resolvent operators and the Sadovskii’s fixed point theorem. An example is presented in the end to show the applications of the obtained results.