Variational Approximate Solutions of Fractional Delay Differential Equations with Integral Transform

The idea of the paper is to consolidate Mahgoub transform and variational iteration method (MTVIM) to solve fractional delay differential equations (FDDEs). The fractional derivative was in Caputo sense. The convergences of approximate solutions to exact solution were quick. The MTVIM is characterized by ease of application in various problems and is capable of simplifying the size of computational operations. Several non-linear (FDDEs) were analytically solved as illustrative examples and the results were compared numerically. The results for accentuating the efficiency, performance, and activity of suggested method were shown by comparisons with Adomian Decomposition Method (ADM), Laplace Adomian Decomposition Method (LADM), Modified Adomian Decomposition Method (MADM) and Homotopy Analysis Method (HAM).


Introduction
Fractional differential equations (FDEs) have a great effectiveness on modeling processes in fluid mechanics, mathematical biology, physics, population growth, and so on [1][2][3]. The best way to describe scientific problems has been the usage of differential equations with fractional derivatives. The main importance of this kind of equations is illustrated in pure mathematics and applications, such as water flow in pipe, blood flow, the analysis of ISSN: 0067-2904 Namah Iraqi Journal of Science, 2021, Vol. 62, No. 10, pp: 3679-3689 0863 pollution, and many other applications [4][5][6]. It is known that the description of standard models mathematically in integer order is not adequate in various cases. The theory of delay differential equations (DDEs) describes a kind of functional differential equations which takes into account the history of a phenomenon [7]. The DDEs can be used to describe dynamical models of numerous phenomena in life [8]. Therefore, the FDDEs have a large number of applications and present remarkably efficient models to formulate scientific problems [9,10]. It is important to note that finding exact solutions to non-linear FDDEs needs some new methods. The study of exact solution of FDDEs is a complex process. The approximate and numerical methods have been used to solve this class of equations, providing a good tool to solve some types of non-linear problems. In the last three decades, many researchers focused on approximate solution of the FDDEs. Some approximate methods have been modified, such as Mahgoub Adomian decomposition method [11], solving fractional delay differential equations by new approach [12], iterative decomposition method [13], DGJ method [14], Laplace Adomian decomposition method [15], and homotopy analysis method [16]. The proposed method in this paper consists of Mahgoub transform and the variational iteration method. Mahgoub Transform was previously used with ADM to solve FDDEs [11]. In fact, we formulate the Mahgoub Transform to be associated with the variation iteration method to solve a class of non-linear of FDDEs. The paper has the following structure ; Section 2 consists of main concepts of Mahgoub transform and fractional calculus. The MTVIM is presented in section 3. In section 4, numerical experiments of fractional order derivative are given to illustrate the efficiency of the considered method. Finally, in section 5, the conclusions are drawn.

Mahgoub Transform and Fractional Calculus
This section consists of essential concepts and definitions with features to Mahgoub transform and fractional calculus.
For . Definition 2.4 [17]: Let be the set of continuous functions with exponential order and defined by: Let ( ) then Mahgoub transform is defined as: To explain the main idea, we study the fractional delay differential equation: is derivative of order in Caputo sense, is a non-linear bounded operator, ( ) is a given continuous function, is the unknown function, and ( ) is the delay function. The main steps involved are given as follows: 1. Take the Mahgoub transform , -of eq. (6), , ( ) ( ( ) ( ( ))-By using Theorem (2.5), the initial conditions and linearity of the Mahgoub transform are then expressed as: By multiplying eq. (8) with Lagrange multipliers and using iteration formula, we have: Regarding the terms , ( ̃( ) ̃( ( ))as restricted varations, we make eq. (9) stationary with respect to :

Applications and Results
In this part, some applications to the usage of MVIM in solving FDDE are introduced.
The approximate solution in a series form of (14)-(15), when is given by ( ) ( ) which is the exact solution of (14) - (15). As shown in table (2), the proposed solutions using MVIM are better from the solutions acquire by ADM, LADM, and the exact solution.
The approximate solution in the series form of (33)-(33), when is given by

Conclusions
This paper comprises a new technique that involves the employment of Mahgoub transform with the variational iteration method to solve non-linear FDDEs. The aim of the algorithm is to acquire the not-required unreal suppositions. In example 1, table 2, we achieved the same results obtained in the exact solution. Also, the results were similar to those obtained by the ADM and, LADM. In Example 2, table 3, we reached the result that: the values of t from 0.1 to 0.5 have the same results obtained by the exact solution, while the values for 0.4 and 0.5 are better than those obtained by MADM. The values of t from 0.6 to 0.9 were closer to the values achieved by the exact solution than those by the MADM. Hence, our method showed better results than MADM and all other methods. In example 3, table 4, we found that MVIM gives the exact solution to the problem and the results are similar to those reached by the HAM and ADM, while it was better than MADM.
In conclusion, we demonstrated that the method has an excellent effectiveness in solving FDDEs.