An Approximation Technique for Fractional Order Delay Differential Equations

In this research article, an Iterative Decomposition Method is applied to approximate linear and non-linear fractional delay differential equation. The method was used to express the solution of a Fractional delay differential equation in the form of a convergent series of infinite terms which can be effortlessly computable. The method requires neither discretization nor linearization. Solutions obtained for some test problems using the proposed method were compared with those obtained from some methods and the exact solutions. The outcomes showed the proposed approach is more efficient and correct.


Introduction
Fractional Differential Equations (FDE), that are a generalization of the differential equation were applied to model many physical phenomena very appropriately [1]. They have been used drastically in areas like thermal engineering, acoustics, electromagnetism, robotics, viscoelasticity, sign processing and populace dynamics [2]. Delay Differential equations are differential equations involving memory and heritage properties. In real life system, delays can be recognized and thus interest in the study of delay differential has been growing over the years, these include modeling of processes of the system in engineering, sciences and biological systems that involved gestation and maturation as well as population models and mining problems [3].
Although Fractional Delay Differential equations (FDDE) are current, there nevertheless exist a variety of materials in the literature, as an example, [4] described the situations for the existence of the solution of Fractional Delay Differential Equations with Riemann-Liouinlle derivatives. In [5,6], the existence of positive solution for a class of singleterm Fractional Delay Differential Equations are taken into consideration, whilst [6 -9]  It is famous that fractional differential equations do no longer have an analytical solution. Several authors have taken into consideration the approximation of Delay Differential Equations of integer order and these consist of the researches of [10 -14] even as [15 -16] taken into consideration the method of splines. [3,17] implemented the Adomian Decomposition Method (ADM), [18] ISSN: 0067-2904 implemented an Iterative Decomposition Method and [19] used the Differential Transform Method. Fractional Delay Differential equations have been approximated using very many strategies. [20] applied a combination of the trapezoidal and Simpson rule to find a series solution of FDDEs, while [21] implemented the Chebyshev Wavelet approach. Another Wavelet approach, the Hermite Wavelet approach was used by [22]. [23] applied a finite differencebased approach and [24] carried out the Adams -Bashforth -Moulton algorithm. [25] had in advance applied the Homotopy Analysis Method. Then, this research article will be used to implement the application of the Iterative Decomposition Method which has been carried out effectively for integer order differential equations consisting of Delay Differential Equations [18]. The method is devoid of any form of linearization or discretization [1].

Materials and Methods
There are several definitions of a fractional derivate of order . Examples include Riemann -Liouville, Grunwald -Letnikov, Jumarie, Caputo, Generalised approach derivatives. The most commonly implemented definitions are those of Riemann-Liouville and Caputo. However, the Caputo derivative draws more attention for its ease of adaptability, especially for physical problems. Some relevant basic definitions and relevant properties as they relate to fractional calculus will be mentioned as utilized in [1].
is termed the Riemann -Liouville fractional derivative of order . Definition (2.3): The Riemann -Liouville fractional integral operator defined on L 1 [a,b] of order of a function is described as Properties of the operator are located in [2] and encompassed the following For ( (  14) where N is a non-linear operator, which in this situation may be replaced by the Riemann -Liouville integral operator in equation (4).

Results and Discussion Numerical Examples
Some numerical examples will be illustrated to decide the adaptability of the postulated approach. Example 1: Consider the Fractional Delay Differential equation For the particular case ( (33) For the particular case ( For the particular case ( + … (41)

Conclusion
A numerical algorithm termed the Iterative Decomposition Method has been carried out to resolve linear and nonlinear Fractional Delay Differential. The result compared favorably with other previous outcomes from acknowledged approaches. The results obtained suggest that the IDM is a method that can be used to obtain solutions of fractional DDE's even when the exact solutions are unknown.