On Some Modified Methods on Fractional Delay and Nonlinear Integro-Differential Equation

The fundamental objective of this work is to construct a comparative study of some modified methods with Sumudu transform on fractional delay integro-differential equation. The existed solution of the equation is very accurately computed. The aforesaid methods are presented with an illustrative example.

the proposed problem discussed. In Section four, special cases are stated. Finally, the comparative study is concluded in Section five.

Preliminaries
Here, we now briefly recall some necessary definitions, preliminaries and properties are used further in this work as follows: for a real number t t ∈ [0, ∞) the Sumudu transform of a function t u(t) over the set U(t) can be written as The relation between Laplace and Sumudu transforms are as (1) [2] ½2 S½t n ¼ u n Àðn þ 1Þ; (2) where ÀðnÞ ¼ R [5] The Sumudu transform of the Riemann-Liouville fractional integral of order α ∈ (0, ∞) is given as [6] The Caputo fractional derivative of order α > 0, defined for a continuous function by where n − 1 < α < n.

Modified Sumudu Homotopy Perturbation Method (MSHPM)
Taking the Sumudu operator S to both sides of Eq. (7), yields With the help of the linearity of Sumudu operator, Now, applying the property of the differentiation for Sumudu transform Lðt; sÞHðuðsÞÞds where c ¼ P kÀ1 i¼0 u ðiÞ ð0Þ v aÀi , then Lðt; sÞHðuðsÞÞds 2 4 3 5 : further, the solution u(t) and nonlinear functions can be described by infinite series as following: Eðuðt À sÞÞ ¼ GðuðtÞÞ ¼ X 1 n¼0 p n n n ðu À sÞ; HðuðtÞÞ ¼ X 1 n¼0 p n i n ðu À sÞ; substitute (13)- (16) in (12), we get On comparing of the two both sides of (20). Hence, we obtain p 0 : u 0 ¼ S À1 ½v a Sðf ðtÞÞ; Lðt; sÞi 0 ðuÞdsÞ þ v a Sð& 0 ðu À sÞÞ Kðt; sÞn 1 ðuÞdsÞ (21) in the same manner, we can calculate u n , n > 1. The approximate solution is given by the convergence of the series solutions are very easily with known methods.

Modified Sumudu Series Solution Method (MSSSM)
Evidently, on continuing the same fourth steps from Eqs. (9)- (12). As a result, by the aid of series solution method (12) uðtÞ ¼ GðuðtÞÞ ¼ Gð HðuðtÞÞ ¼ Hð in (12), taking (23)-(26) into consideration By the comparison of the coefficients on two both sides (27) and Taylor series. Consequently, the existed solution is in a closed form.

Modified Sumudu Homotopy Analysis Method (MSHAM)
In this section, Sumudu transform directly coupled with a homotopy analysis method. For an embedding parameter α ∈ [0, 1], we construct the nonlinear operator Lðt; sÞHðeðt; rÞÞdsÀv a S½Eðeðt À s; rÞ þ f ðtÞ; (28) By HAM, the deformation equation of order zero is constructed as follows: where r, c are the non zeros auxiliary parameter and functions respectively. ε(t;r) differ from u 0 (t) to u 1 (t). In particular, r = 0, by Taylor series, we can expand where u n ðtÞ ¼ ½ 1

Modified Sumudu Variational Iteration Method (MSVIM)
This section is based on the combination of variational iteration method with Sumudu transform.
Let us apply the inverse Sumudu transform of both sides (12).
Now, differential the preceding equations with respect to t, Due to the variational iteration method, the correct function can be rewritten as: Kðt; sÞGðu m ðsÞÞdsÞ þ S À1 ½v a ðc þ Sðf ðtÞÞ þ 2 S À1 ½v a Sð So, the limit of {u m (t)} m≥0 is equivalent to the exact solution.

Modified Sumudu Decomposition Method (MSDM)
Upon using the Sumudu decomposition method, definition of the solution u(t) and the nonlinear functions are given by the infinite series where, X j , Y j and Z j are the Adomian polynomials of u 0 , u 1 , … , u j . Now, the Adomian polynomials for the nonlinear functions are written as The new recursive relation becomes Finally, By Comparison of two both sides (45). The following iterative algorithm: Lðt; sÞZ 0 ds þ v a S½X 0 ðuðt À sÞÞ In general form, Suggest that R(t) is decomposed into two parts: RðtÞ ¼ R 1 ðtÞ þ R 2 ðtÞ; u 0 ðtÞ ¼ R 1 ðtÞ; n ! 1; Lðt; sÞZ 0 dsÞ: Lðt; sÞZ jÀ1 dsÞ: Example: with the initial condition: u(0) = 0, with the exact solution u(t) = t at τ = 1 and α = 0.75.
(4) By MSVIM: We construct the iteration formula as þ S À1 ½v a ðc þ Sð t 0:25 Àð1:25Þ Þ À 4S À1 ½v a Sð In particular, α = 0.75, upon using the iteration formula Hence, the general term um is obtained as u m = t agrees well with the exact solution as m tends to infinity.

Computational Scheme for the Proposed Problem
The estimate we will obtain in this section seems to be independent of interest. Without restriction of generality, we can assume α = 0.25, 0.75, 1.25 and α = 1.75 hold for t = 0.1, 0.2, …, 2. We compute the experimentally determined solution of Eq. (46) for the different values of α with many values of t. This yields information about the analogue between the exact and approximate value of u(t).

Special Cases
It is evident that this manuscript is more generalized from analogues papers. More remarkable special cases which are covered by a lot of last papers.
Remark 3 If the Eq. (7) has no delay item and it becomes Fractional integro-differential equation [11].

Conclusion
The benefit of modified several methods is that it successfully contributed to the rapidly progressing of the exact, approximate and numerical solutions. Also, modified methods are considered as a new powerful technique to solve a large range of equations such as differential, integral and integro-differential. At a certain value of α = 0.75 the approximate solution is equal to the exact solution. Also, the analogy between the exact and approximate solution differs from an example to others.