Analytical approximation for logistic delay differential equation

Abstract

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 ℎ 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 ℎ-values of HAM has shown the influence of parameter ℎ 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 ττ, the steeper the gradient of the population size xx. It is concluded that the parameter ℎ 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 HH(tt) = 1 and ℎ = −1. Hence, using HAM and HPM techniques, two different kinds of series solutions of logistic delay differential equation are obtained.