Series solution for a delay differential equation arising in electrodynamics

2009 ◽  
Vol 25 (11) ◽  
pp. 1084-1096 ◽  
Author(s):  
Hüseyin Koçak ◽  
Ahmet Yıldırım
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Series Solution ◽  
Delay Differential

scholarly journals Analytical approximation solution for logistic delay differential equation

2020 ◽  
Vol 16 (3) ◽  
pp. 368-373
Author(s):  
Nurul Atiqah Talib ◽  
Normah Maan ◽  
Aminu Barde
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Delay Differential Equation ◽  
Series Solution ◽  
Homotopy Perturbation Method ◽  
Analytical Techniques ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Delay Differential ◽  
And Control

Although the non-linear analytical techniques are fast developing, they still do not entirely satisfy mathematicians and engineers. Many researchers have conducted the study to find the analytical solution for the logistic delay differential equation. However, for the time lags occasion, it is quite hard and tough to achieve analytical solution due to its limitation, and thus, we can only expect the approximate analytical solution. This paper describes the approximate analytical techniques, homotopy analysis method (HAM), and homotopy perturbation method (HPM) in order to indicate their ability in solving the logistic delay differential equation. HAM is one of the better approaches that can be used for solving this equation. The use of HAM will lead to obtaining the series solution that contains an auxiliary parameter  that can help to adjust and control the convergence and rate approximation for the series solution. Meanwhile, HPM is an analytical method with a combination of homotopy in topology and classical perturbation technique. Using the HPM technique, the logistic delay differential equation is reduced to a sufficiently simplified form, which usually becomes a linear equation that is easy to be solved. The comparison of numerical solution with -values of HAM has shown the influence of parameter  in the convergence of series solution. Using HAM and HPM, the relationship between the time-delay τ and the population size is obtained.  As a result, the higher the value of  the steeper the gradient of the population size . It is concluded that the parameter  helps to adjust and control the convergence and rate approximation for the series solution of HAM. Laterally, the comparison between HAM and HPM with numerical method is done to show that both methods are relatively approximate to the exact solution. Moreover, homotopy perturbation method (HPM) is a special case of homotopy analysis method (HAM) when and . Hence, using HAM and HPM techniques, two different kinds of series solutions of logistic delay differential equation are obtained.

scholarly journals Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 105
Author(s):  
Lokesh Singh ◽  
Dhirendra Bahuguna
Keyword(s):  
Differential Equation ◽  
Growth Rate ◽  
Invariant Manifold ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Stable Manifold ◽  
Exponential Dichotomy ◽  
Delay Differential ◽  
Stable Invariant ◽  
Nonuniform Exponential Dichotomy

In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt).

scholarly journals Oscillatory Behaviour of a First-Order Neutral Differential Equation in relation to an Old Open Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
K. C. Panda ◽  
R. N. Rath ◽  
S. K. Rath
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Oscillatory Behaviour ◽  
Neutral Delay ◽  
First Order ◽  
Neutral Delay Differential Equation ◽  
Delay Differential ◽  
Oscillation And Nonoscillation

In this paper, we obtain sufficient conditions for oscillation and nonoscillation of the solutions of the neutral delay differential equation yt−∑j=1kpjtyrjt′+qtGygt−utHyht=ft, where pj and rj for each j and q,u,G,H,g,h, and f are all continuous functions and q≥0,u≥0,ht<t,gt<t, and rjt<t for each j. Further, each rjt, gt, and ht⟶∞ as t⟶∞. This paper improves and generalizes some known results.

scholarly journals Maximum Likelihood Inference for Univariate Delay Differential Equation Models with Multiple Delays

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Ahmed A. Mahmoud ◽  
Sarat C. Dass ◽  
Mohana S. Muthuvalu ◽  
Vijanth S. Asirvadam
Keyword(s):  
Differential Equation ◽  
Maximum Likelihood ◽  
Delay Differential Equation ◽  
Information Matrix ◽  
Likelihood Inference ◽  
Multiple Delays ◽  
Unknown Parameters ◽  
Differential Equation Models ◽  
Delay Differential ◽  
Initial Starting

This article presents statistical inference methodology based on maximum likelihoods for delay differential equation models in the univariate setting. Maximum likelihood inference is obtained for single and multiple unknown delay parameters as well as other parameters of interest that govern the trajectories of the delay differential equation models. The maximum likelihood estimator is obtained based on adaptive grid and Newton-Raphson algorithms. Our methodology estimates correctly the delay parameters as well as other unknown parameters (such as the initial starting values) of the dynamical system based on simulation data. We also develop methodology to compute the information matrix and confidence intervals for all unknown parameters based on the likelihood inferential framework. We present three illustrative examples related to biological systems. The computations have been carried out with help of mathematical software: MATLAB® 8.0 R2014b.

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