Approximate Analytical Solution to Nonlinear Delay Differential Equations by Using Sumudu Iterative Method

In this study, an e ﬃ cient analytical method called the Sumudu Iterative Method (SIM) is introduced to obtain the solutions for the nonlinear delay di ﬀ erential equation (NDDE). This technique is a mixture of the Sumudu transform method and the new iterative method. The Sumudu transform method is used in this approach to solve the equation ’ s linear portion, and the new iterative method ’ s successive iterative producers are used to solve the equation ’ s nonlinear portion. Some basic properties and theorems which help us to solve the governing problem using the suggested approach are revised. The bene ﬁ t of this approach is that it solves the equations directly and reliably, without the prerequisite for perturbations or linearization or extensive computer labor. Five sample instances from the DDEs are given to con ﬁ rm the method ’ s reliability and e ﬀ ectiveness, and the outcomes are compared with the exact solution with the assistance of tables and graphs after taking the sum of the ﬁ rst eight iterations of the approximate solution. Furthermore, the ﬁ ndings indicate that the recommended strategy is encouraging for solving other types of nonlinear delay di ﬀ erential equations.


Introduction
The more general kind of differential equations (DEs) is called functional differential equations (FDEs), as well as the delay differential equation (DDE) is the simplest and may be the most natural class of FDEs [1]. DDEs constitute a large and significant class of dynamical systems. Time delays are natural components of the dynamic processes of biology, ecology, physiology, economics, epidemiology, and mechanics, and so a genuine model of these processes must comprise time delays. DDEs are a form of DEs in which the derivative of the unknown function at a specific period is provided in terms of values of the function at past periods. Announcing delays in models enriches the vitality of these models and allows a perfect interpretation of actual occurrences [2][3][4]. DDEs arise frequently in various physical occurrences. To be precise, they are vital once ordinary differential equation-(ODE-) based models are ineffective. In disparate ODEs, where preliminary conditions are stated at the initial point, DDEs need the history of the system over the delayed intervals which are then provided as preliminary conditions. DDEs are infinitedimensional and challenging to appraise since delay terms exist in the model [5]. Hence, solving DDEs is an important area of investigation for different scientists.
Recently, many researchers established and investigated various analytical and numerical methods to obtain approximate/exact solutions for nonlinear DEs as well as DDEs [6][7][8]. The variational iteration method (VIM) was employed by the authors of the publication [9][10][11][12] to discover a rough solution to nonlinear DDEs. To solve models of delayed vector-borne illnesses and delayed protein degradation, [13] incorporated the differential transform technique, Tarig transformation method, and Padè approximation in 2022. The Padè approximation is applied to this method to broaden the approximation solutions' convergence domain. Kumar and Methi [14] obtained the numerical result of some NDDEs with the support of Banach contraction method (BCM). To demonstrate that the suggested technique is fit for solving NDDEs, they also offered the numerical results, convergence theorem, and error analysis for various DDEs. In the publication [15], the authors debate an inverse problem for the NDDE that consists of approximating a solution by defining the beginning moment and delay parameter based on the experimental data. The inverse issue is seen as a nonlinear optimal control problem using the stated approach, for which the requisite criteria of optimality are established and shown. A method based on a better parallel evolutionary algorithm is used to solve the derived optimum control issue. In his publication [16], Familua proposes a straightforward method for solving specific second-order DDEs numerically without reducing them to systems of lower orders. Power series and an exponential function were combined to create the approaches utilizing a collocation approach. The first two grid points interpolate the approximation basis functions, and both grid and off-grid points collocate them. Additionally, the author combined the generated schemes and their derivatives to create block techniques, which enable the simultaneous direct solution of second-order DDEs without the laborious requirement of building separate predictors. To analyze the semianalytical solution of the DDEs, the writers of the effort [17] construct a novel technique called the Mohand homotopy perturbation transform method (MHPTM). This approach combines the Mohand transform with the homotopy perturbation method (HPTM). A novel class of neutral FDDEs using the generalized ψ-Caputo derivative on a partially ordered Banach space is investigated by the authors of the study [18]. The Dhage approach and the Banach contraction standard are used to demonstrate the presence and uniqueness of the solutions to the specified boundary value issue. The authors of the paper [19] considered a class of singularly perturbed advanced DDEs of convection-diffusion type. They use finite and hybrid difference schemes to solve the problem on piecewise Shishkin mesh. Readers can consult references [5,[20][21][22][23][24] for more work on DDEs that are linked to their interests.
Several intellectuals have created a variety of analytical integral transform techniques for precise and approximated answers during the past few years, including the Laplace transform method [25,26], Shehu transform method [27,28], Aboodh transform method [22], Sumudu transform method [29], Elzaki transform method [30], and Mohand transform method [17]. The Sumudu transform method is an integral like the Laplace transform method, introduced in the early 1990s by Watugala [31] to solve DEs and control engineering problems. This method converts a linear DE to an algebraic equation that can be resolved by consuming formal algebraic methods. The initial DE can then be resolved by employing the inverse Sumudu transformations. When matched to other methods, the Sumudu transform technique has the advantage of providing an accurate result quickly and without making any restricting assumptions about the result (see [29]). Unfortunately, this transform fails to crack nonlinear partial DEs, unlike the other integral transforms [32]. In these circumstances, the Sumudu transform method is frequently recycled with other well-used techniques such as the Adomian decomposition method (ADM), HPM, VIM, and the new iterative method [29,33]. Alternatively, we may frequently employ integral transform techniques such as Laplace, Elzaki, and Shehu in place of the Sumudu transform method. Likewise, techniques like HPM, VIM, and ADM may be employed in place of the new iterative procedure. The HPM, also known as the He-Laplace technique, was developed by Mishra and Tripathi [34] and combines the Laplace transform with He's polynomials to solve DDEs, which have a variety of applications in physics, digital image processing, signal processing, and applied disciplines. With the usage of the Laplace transform homotopy perturbation approach (He-Laplace method), the writers of the study [35] were able to acquire the approximate analytical solutions of four distinct kinds of conformable partial DEs. We encourage readers to look at the descriptions of a few integral transforms in Table 1 [36].
The major goal of this study is to suggest the Sumudu Iterative Method (SIM), a trustworthy analytical technique for obtaining precise answers to NDDEs. This strategy combines the Sumudu transform method with an iterative method, two potent approaches. An iterative method (IM) has been presented by Daftardar-Gejji and Jafari to solve linear and nonlinear functional equations [37][38][39][40]. The IM has been effectively applied in many kinds of investigation to solve some linear and nonlinear PDEs and ODEs, NDDEs, higher-order integro-DEs, two-dimensional nonlinear Sine Gordon equation (NLSGE), and Korteweg-de Vries equations [25,[41][42][43]. The author of the study [29] successfully integrated the double Sumudu transform with the iterative approach to produce an approximate analytical answer to the one-dimensional coupled NLSGE, which cannot be solved by utilizing the double Sumudu transform alone. In this approach, the linear part of the problem was handled Table 1: Some known integral transforms [17,22,36].

Integral transform Expression
Elzaki transform Pourreza transform Ara transform Mohand transform Aboodh transform  [44]. The primary driving force behind the current study endeavor is the fact that the amalgamation of the Sumudu transform method and the new iterative SIM has not yet been investigated to solve NDDEs in the literature. The linear DDE was the subject of other research projects as well. This study work primarily focuses on NDDEs as a result, which is our second reason for introducing the novel approach SIM. The outcomes of the samples exhibit the accuracy and potency of this method, which does not require the imposition of any extra constraints to arrive at an analytical answer to the anticipated difficulties. It is a skilled scheme for reducing the number of calculations while keeping the answer more accurate and efficient.
In this work, we consider the following DDE in the form [22] subject to where d n y/dt n is the derivative of y of order n, P is the linear bounded operator, N is a nonlinear bounded operator, f ðtÞ is a given continuous function, and y = yðtÞ. The remaining parts of this paper are structured as follows: Section 2 presents the basic explanations and properties of the Sumudu transform method. In Section 3, we clarify the methodology of the new iterative method. Section 4 demonstrates how SIM is pragmatic to the proposed problem DDE. Section 5 is dedicated to illustrating the SIM to five problems. In Section 6, we talk over the numerical results and illustrate the accuracy and efficiency of the SIM. Lastly, concluding remarks are outlined in Section 7.

Sumudu Transform Method
The definitions, characteristics, and theorems of the Sumudu transform scheme that we employed in this work are offered in this section (refer to [29,[45][46][47][48][49][50][51][52]). Definition 1. For any real numbers t > 0, the function GðuÞ is defined as the Sumudu transform of a function f ðtÞ which is given by Definition 2. The inverse Sumudu transform of a function GðuÞ is denoted by the symbol S −1 ½GðuÞ = f ðtÞ and is defined by the Bromwich contour integral : : :

Some Properties of Sumudu Transform
Property 7. Over the set of function A: A = fðf ðtÞj∃M, τ 1 , Property 8 (linearity property). Let f ðtÞ and gðtÞ be any two functions whose Sumudu transforms exist. Then, for arbitrary constant a and b, we have Proof. Using the definition of Sumudu transform and property of integral, we have Hence, the proof.

Sumudu Iterative Method (SIM)
In this section, the general form of the n th -order DDE (1) with the initial value (2) is treated using the suggested method SIM.
Step 1. Applying the Sumudu transform on both sides of equation (1), we get Step 2. Theorem4is used to obtain where C = ∑ n−1 k=0 f k ð0Þ:(24) Replacing (24) into (21) and shortening, we obtain Step 3. Applying the inverse Sumudu transform on both sides of equation (22), we get Step 4. Now, we surprise the iterative method. We assume that yðtÞ = ∑ ∞ n=0 y n ðtÞ is the solution of equation (1).

Illustrative Examples
In this part, four nonlinear DDEs are used to demonstrate the effectiveness and validity of the SIM.
Example 1 (see [44]). Consider a first-order nonlinear DDE: subject to the initial condition The analytical solution is given by yðtÞ = sin t:.
Solution: taking the Sumudu transform on both sides of equation (29), we get Taking the place of equation (30) into equation (29), we achieve

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Subsequently, taking the inverse Sumudu transform of equation (32) implies that By using the properties of Sumudu transforms listed in Table 2, equation (33) becomes Now, applying the new iterative method to equation (34) in a vision of equations (27)-(31), we attain the components of the solution as follows: Hence, y 1 ðtÞ = −ðt 3 /3!Þ and y 1 ðt/2Þ = −ðt 3 /48Þ: Therefore, , Likewise, we can obtain the leftover values by means of the recurrence relation y n+1 ðtÞ = −2S −1 ½uSð∑ ∞ n=1 fNð∑ k n=0 y n 2 ðt/2ÞÞ − Nð∑ k−1 n=0 y n 2 ðt/2ÞÞgÞ, n ≥ 1:. Now, in vision of (28), the solution of Example 1 is Example 2. Consider the following nonlinear second-order DDE [23]: with initial condition The analytical solution is given by yðtÞ = e −2t :. Solution: taking the Sumudu transform on both sides of equation (39), we get Replacing equation (40) into equation (39), we get Advances in Mathematical Physics Subsequently, taking the inverse Sumudu transform of equation (42) suggests that By using the properties of Sumudu transforms listed in Table 2, equation (43) becomes Now, applying the new iterative method to equation (44) in the opinion of equations (27)-(31), we achieve the components of the answer as follows: As a result,

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Similarly, we can obtain the continuing values by using the recurrence relation.
Now, in sight of (28), the solution of Example 2 is with initial condition The analytical solution is given by yðtÞ = e t . Solution: taking the Sumudu transform on both sides of equation (49), we acquire Replacing equation (50) into equation (49), we get Consequently, taking the inverse Sumudu transform of equation (52) implies that By using the properties of Sumudu transforms listed in Table 2, equation (53) becomes Now, applying the new iterative method to equation (54) in the understanding of equations (27)-(31), we attain the components of the answer as follows: Similarly, we can get the continuing values by consuming the recurrence relation with initial condition The analytical solution is given by yðtÞ = e t 2 . Solution: taking the Sumudu transform on both sides of equation (58), we get Substituting equation (59) into equation (58), we get Subsequently, taking the inverse Sumudu transform of equation (61) implies that By using the properties of Sumudu transforms listed in Table 2, equation (62) becomes Now, applying the new iterative method to equation (63) in the opinion of equations (27)-(31), we acquire the components of the solution as follows: Hence, y 1 ðtÞ = t 2 and y 1 ðt/2Þ = t 2 /4. Therefore, , Similarly, we can gain the remaining values by using the recurrence relation Now, in the understanding of (28), the solution of Example 4 is 9 Advances in Mathematical Physics with initial condition The exact solution is given by Solution: taking the Sumudu transform on both sides of equation (69), we get Replacing equation (70) into equation (72), we get Consequently, taking the inverse Sumudu transform of equation (73) implies that By using the properties of Sumudu transforms listed in Table 2, equation (74) becomes Similarly, we can obtain the continuing values by consuming the recurrence relation Now, in view of (28), the solution of Example 5 is 6. Discussion      Moreover, the proposed approach gives a small error neighboring t = 0, but the error increases as jtj grows. This is to means that a greater approximation can be achieved for small values of time t. Additionally, the recurrence and relative errors of the SIM approximation solution for instances 1 − 5 are generated to verify the correctness of our technique, as shown in Tables 8-12. It is clear from the findings that the current approach is a useful and efficient solution for solving specific classes of nonlinear DDEs with a minimum of computations and iterations.

Conclusion
This study introduced SIM, a hybridization of the Sumudu transform method with the new iterative method. First, the linear component of DDEs is solved using the Sumudu transform method. To simplify the complexity of the novel term from the nonlinear term, a posttreatment new iterative method is employed as illustrated in Section 3. We provide the fundamental definitions and terminology for the DDEs, the new iterative approach, and the Sumudu transform method. The validity and consistency of SIM have been verified with the help of five significant problems. The absolute, recurrence, and relative errors of all considered examples are interpreted graphically and numerically, for different values of the time variable t. From the demonstrative examples, the results reveal that the current technique SIM generates a decent approximation that is extremely close to the precise answers with a low amount of error. Therefore, SIM is quite valuable, as it allows us to improve accuracy and efficiency and provides a mathematical tool for nonlinear DDEs. Finally, we hope that this work is a step toward examining this approach to tackling some exciting problems in various areas of science and engineering, given the ongoing use of nonlinear DDEs as models in several different disciplines.

Data Availability
No data were used to support the study.

Conflicts of Interest
There are no conflicts of interest declared by the authors.