Delay differential equations (DDEs), as well as neutral delay differential equations (NDDEs), are often used as a fundamental tool to model problems arising from various areas of sciences and engineering. However, NDDEs particularly the systems of these equations are special transcendental in nature; it has therefore, become a challenging task or times almost impossible to obtain a convergent approximate analytical solution of such equation. Therefore, this study introduced an analytical method to obtain solution of linear and nonlinear systems of NDDEs. The proposed technique is a combination of Homotopy analysis method (HAM) and natural transform method, and the He's polynomial is modified to compute the series of nonlinear terms. The presented technique gives solution in a series form which converges to the exact solution or approximate solution. The convergence analysis and the maximum estimated error of the approach are also given. Some illustrative examples are given, and comparison for the accuracy of the results obtained is made with the existing ones as well as the exact solutions. The results reveal the reliability and efficiency of the method in solving systems of NDDEs and can also be used in various types of linear and nonlinear problems.

Neutral delay differential equations (NDDEs) are a type of delay differential equations (DDEs) that arise in numerous areas of applied sciences and play a vital role in mathematical modelling of real-life phenomena. Some techniques have experienced difficulties in finding the approximate analytical solution which converges rapidly to the exact solution of these equations. In this paper, an analytical approach is proposed for solving linear and nonlinear NDDEs with proportional and constant delays based on the homotopy analysis method (HAM) and natural transform method where the nonlinear terms are simply calculated as a series of, He's polynomial. The proposed method produces solutions in the form of a rapidly convergent series which leads to the exact solution from only a few numbers of iterations. Some illustrative examples are solved, and the convergence analysis of the proposed techniques was also provided. The obtained results reveal that the approach is very effective and efficient in handling both linear and nonlinear NDDEs with proportional and constant delays and can also be used in various types of linear and nonlinear problems.

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.

This work proposed a new analytical approach for solving a famous model from mathematical physics, namely, advanced Lorenz system. The method combines the Natural transform and Homotopy analysis method, and it’s have been suggested for the solution of different types of nonlinear systems of delay differential equations. This technique gives solution in a series form where the He’s polynomial is adjusted for the series calculation of nonlinear terms of Lorenz system. By choosing an optimal value of auxiliary parameters the more precise approximate Solution of this model is obtained from only three iterations number of terms. Some figures are used to demonstrate the accuracy of the result based on the residual error function. Therefore, the approach gives rise to an easy and straightforward means of solving these models analytically. Hence, it can be used in finding solutions to other forms of nonlinear problems.

Delay differential equation (DDEs) is a type of functional differential equation arising in numerous applications from different areas of studies, for example biology, engineering population dynamics, medicine, physics, control theory, and many others. However, determining the solution of delay differential equations has become a difficult task more especially the nonlinear type. Therefore, this work proposes a new analytical method for solving non-linear delay differential equations. The new method is combination of Natural transform and Homotopy analysis method. The approach gives solutions inform of rapid convergence series where the nonlinear terms are simply computed using He's polynomial. Some examples are given, and the results obtained indicate that the approach is efficient in solving different form of nonlinear DDEs which reduces the computational sizes and avoid round-off of errors.