A New Technique for Solving Neutral Delay Differential Equations Based on Euler Wavelets

. An efective numerical scheme based on Euler wavelets is proposed for numerically solving a class of neutral delay diferential equations. Te technique explores the numerical solution via Euler wavelet truncated series generated by a set of functions and matrix inversion of some collocation points. Based on the operational matrix, the neutral delay diferential equations are reduced to a system of algebraic equations, which is solved through a numerical algorithm. Te efectiveness and efciency of the technique have been illustrated by several examples of neutral delay diferential equations. Te main advantages and key role of using the Euler wavelets in this work lie in the performance, accuracy, and computational cost of the proposed technique.


Introduction
We consider the following neutral delay diferential equation (NDDE) given by.
Te proposed problem 1 and (2) is a special class of DDE, which appears in a wide range of applications such as the mathematical modeling of ecology, electronics, and control of ships and aircraft. It is obvious that these kinds of NDDE cannot be solved exactly. Hence, it is crucial to develop efcient numerical techniques to simulate solutions of such equations. Many researchers reported several techniques in this development, for example, Runge-Kutta method has been used in [14], variational iteration method in [15], and other numerical schemes and properties can be found in [7,16,17].

The Numerical Approach Based on Euler Wavelet
Wavelet expansions and its generalization, such as framelets, have been successful in numerical simulation in many areas of applications in real-world phenomena [18]. Tis is largely due to the fact that wavelets have the right structure to capture the sparsity in linear systems. Note that the great hopes in developing wavelet-based representations and theory of turbulent fows have not been materialized. Euler wavelet approximation would have the right approximating structure to be efcient for a general NDDE due to the orthogonality of the envisaged wavelets in the proposed framework. To solve the proposed problem, we use Euler wavelets generated via Euler polynomials and presented in [19]. Te Euler polynomial U k (x) is defned based on the generating function that has the form.
Te frst few Euler polynomials are given below, and the graphical presentation of some of these functions is presented in Figure 1.
Let us defne the Euler polynomials U 1 (x), U 2 (x) and the needed functions in the following numerical scheme: Let Ξ be the set of all functions given in equations (3)- (8). For any function τ ∈ Ξ, we defne the function φ(x) as follows: where χ [0,1] is the indicator function on [0,1]. Assume that, We defne the following set of wavelets, where j, k ∈ Z as, Recall that, see for example., [20], a function τ ∈ L 2 (R) can be expanded using the following series, where, 3 U 6 (x) 2 Complexity In which 〈·, ·〉 denotes the usual inner product over the space L 2 (R) and w is a proper weight function.
One may truncate (16) by τ n,M where, Terefore, Notice that, Hence based on Bessel inequality, To solve the proposed problem, similar to the method in [19], we construct a vector Ξ τ of length M � 2 n+1 , n ∈ N, such that where, As an illustration, when n � 2, we have the following equations: (i) When φ τ � 1, κ � 1, we have the following equation: (ii) When φ τ � x, κ � j, we have the following equation: Complexity 3 (iii) When φ τ � x 2 /2, κ � j 2 , we have the following equation: Now, let us defne the solution of equations (1) and (2) via the form of matrices based on the above approach. Assume the Euler wavelet truncated expansion given by equation (10) to be defned as follows: Ten, Additionally, Note that from (28), we have the following equation: Assume that I ℓ j,k (t) � Te needed functions for the numerical scheme are ready to be used and so we defne, M � 2 1+n , n � 1, 2, . . . , As a collocation node where Hence, by substituting the proposed collocation points to the equation above, we get the following equation: 4 Complexity Note that implementing the collocation division generates a system of algebraic equations that can be solved easily, for example using Mathematica software 13.0 to produce the unknown coefcients d ℓ (j, k) needed to fnd an approximate solution given by (28).

Numerical Illustration
In this part, we illustrate some examples of the proposed NDDE problem based on the presented numerical scheme and obtain some maximum absolute errors. Te numerical evidences showed high accuracy compared with the exact ones. Six examples will be illustrated. For examples 1-3 from [21], we got an absolute error decreasing from 10 − 12 by 28 collocation points to 10 − 18 by 112 collocation points. In example 4, we got zero absolute error. Torough out the presented fgures and numerical results, the algorithm demonstrates the accuracy and excellent agreement between the numerical solutions and the exact ones. Example 1. Let's take the following NDDE du dt +(cos(t)) 1/2 u ′ t 1/2 − e t 1/2 Te initial condition for this formulation is given by, Te exact solution is defned as follows: Te exact and approximate solutions for diferent number of collocation points are presented in Figures 2 and  3, where the agreement between the approximate and exact functions and the error bounds are clearly improved gradually when the order of the partial sum of the truncated Euler series expansion enlarges. In Table 1, we show the error bound resulted for example 1.
where the initial condition is as follows: Te exact solution is as follows: Similarly, both the exact and approximate solutions for diferent number of collocation points are depicted in Figures 4 and 5 and we illustrate the error bound for example 1 in Table 2, where again the improvement of the numerical approximation is clearly shown in the graphical and numerical evidences.
where the initial condition for this formulation is given by, Te exact solution is given by, Another graphical and numerical evidences are presented in Figures 6, 7, and Table 3.

Example 4. Consider the following NDDE arising in electrodynamics
where the initial condition for this formulation is given by, Te exact solution is given by Te approximate solution u e for 7 collocation points is given by      (46) Lastly, we depict the graph of the approximate and exact solutions in Figure 8, where the accuracy is presented in Figure 9.

Conclusion
In this presented work, a new numerical scheme based on a certain wavelet settings generated by the Euler functions is proposed. Te collocation technique has been implemented on the neutral delay diferential equations. We demonstrated the Euler wavelet truncated expansions to convert the resulted equations to a system of algebraic equations. We illustrated the work by solving numerically a set of problems, which arise in many scientifc areas. Te numerical results achieved an exceptional absolute error among other methods known in the literature. Te error and approximate solutions of the given problems have been depicted by some fgures to show the accuracy of the algorithm.
Te main advantage of the presented algorithm is to shed some lights on the use of Euler wavelets and to develop a suitable computational technique for the numerical treatment of neural delay diferential equations on a side, and on the efectiveness, efciency of the technique in the performance, accuracy, and computational cost even with a small number of collocation points, on the other side.

Data Availability
No data were used to support this study.

Conflicts of Interest
Te authors declare that they have no conficts of interest.