Solving neutral delay differential equations using least square method based on successive integration technique

The main objective of this work is to propose the Least square method (LSM) using successive integration technique for solving Neutral delay differential equations (NDDEs). Continuous LSM and Discrete LSM have been presented by adopting different orthogonal polynomials as weighted basis functions. In this study, the most widely used classical orthogonal polynomials, namely, the Bernoulli polynomial, the Chebyshev polynomial, the Hermite polynomial, and the Fibonacci polynomial are considered. Numerical examples of linear and nonlinear NDDEs have been provided to demonstrate the efficiency and accuracy of the method. Approximate solutions obtained by the proposed method are well comparable with exact solutions. From the results it is observed that the accuracy of the numerical solutions by the proposed method increases as N (order of the polynomial) increases. The proposed method is very effective, simple, and suitable for solving the linear and nonlinear NDDEs in real-world problems.


Introduction
Neutral delay differential equations are a type of delay differential equations (DDEs) in which the highest-order derivative of the unknown function occurs with delay.DDEs and NDDEs arise in the fields of signal processing, digital images, control systems, epidemiology, chemical kinetics, etc.Some notable applications of DDEs and NDDEs are in electrochemical biosensor [1], cancer cells growth [2] and population model [3], human balancing models [4], quasistatic piezoelectric beams [5].
The Least square method is a kind of weighted residual method to solve ordinary differential equations (ODEs).Daniele [15] has applied least square method to initial and boundary value problems of ODEs.Siti Farhana et al. [16] have solved ODEs by using LSM with an implementation of gradient method.Salisu [17] has investigated LSM for finding approximate solutions to ODEs.Parth et al. [18] have examined the performance of LSM on solving first order ODEs.Salisu and Abdulnasir [19] have used continuous LSM to solve second order ODEs.
In this study, two kinds of LSM, namely, Continuous LSM (CLSM) and Discrete LSM (DLSM) based on successive integration technique have been presented for solving NDDEs.We adopted four different orthogonal polynomials for weighted basis functions.Numerical examples are considered for testing the efficiency of the proposed method.In section 2, basic definition of polynomials is given.The description of discrete and continuous LSM for solving NDDEs are provided in section 3.In section 4, illustrative examples are provided.

Basic definition of polynomials
In this study, we consider the most widely used classical orthogonal polynomials, namely, the Hermite polynomial, the Bernoulli polynomial, the Chebyshev polynomial and the Fibonacci polynomial.

Hermite Polynomial
The Hermite polynomial   () of order n is defined on the interval (−∞, ∞).There are different ways to define for Hermite polynomial, one of them is the so-called Rodrigues' formula From Eqn. (1), the recurrence relation for the polynomials can be derived as 0 () can be obtained from Eqn. (1) and the remaining terms are determined by using the recursion relation Eqn.(2).Thus, we have the following sequence of polynomials: and so on.The  ℎ order Hermite polynomial   () has a leading coefficient 2  .

Bernoulli Polynomial
The Bernoulli polynomial is named after Jacob Bernoulli which combines the Bernoulli numbers and binomial coefficients.The generating function of  ℎ order Bernoulli polynomial is defined by The Bernoulli polynomial is explicitly written as: for n ≥ 0.
0 () can be obtained from Eqn. (3) and the remaining terms are determined by using the recursion relation.Thus, we have few terms of the Bernoulli polynomials as:

Chebyshev Polynomial
The Chebyshev polynomial related to cosine functions on the interval [−1, 1] of order n is defined as The recursion relation of Chebyshev polynomial is: 0 () and  1 () can be obtained from Eqn. (5).Then the remaining terms are determined by from Eqn. (6).Thus, we have the following sequence of polynomials:

Fibonacci Polynomial
The Fibonacci polynomials are generated by Fibonacci numbers.The recurrence relation of Fibonacci polynomial is: Using this relation, we have the following sequence of polynomials:

Continuous Least Square Method
In CLSM, we make the residue function R tend to zero by minimizing the error function for   Ω.
To obtain an optimum solution with minimal error E, we differentiate the Eqn.( 12) with respect to   and then equate to zero.Thus, we have = 0, for  = 1, 2, … , …………………… (13) This yields an algebraic system of linear and nonlinear equations subject to the linear and nonlinear terms involving in the Eqn.(7).By solving this system of equations, we get the respective polynomial co-efficient   's from which the solution of the NDDE ( 7) can be obtained.

Discrete Least Square Method
In DLSM, we consider the residuals at the points   , 1 ≤  ≤ .Let ………………… ( 14) To obtain an optimum solution with minimal error E, we differentiate the Eqn.( 14) respect to   and then equate to zero.Thus, we have This yields an algebraic system of linear and nonlinear equations.By solving this system of equations, we get the respective polynomial coefficients   's from which the solution of the NDDE (7) can be obtained.

Numerical Examples
Three examples of NDDEs have been solved by using CLSM and DLSM based on successive integration technique with four orthogonal polynomials, namely Hermite, Bernoulli, Chebyshev, and Fibonacci.Here, for convenience, in the case of CLSM, we denote them as H-CLSM, B-CLSM, C-CLSM and F-CLSM respectively.Similarly, in the case of DLSM, we denote them as H-DLSM, B-DLSM, C-DLSM and F-DLSM respectively.
Exact solution is () = sin() − cos ().The numerical solutions obtained by using the proposed methods H-CLSM and H-DLSM with N = 7 are compared with the exact solution.The results are given in Table 1.The solution graphs obtained by using the proposed methods with N = 7 are presented in Figure 1. with initial condition (0) = 0.
For this example, the error results of the proposed methods CLSM and DLSM using different polynomials with different values for N are presented in Tables 2 and 3.The given initial conditions are The solution graphs obtained by using the proposed CLSM and DLSM with N = 7 are compared with Analytical algorithm presented in [20].They are given in Figure 2.

Conclusion
In this study, a new approach of continuous and discrete Least square methods based on successive integration technique is proposed for solving Neutral delay differential equations.Numerical examples of linear and nonlinear NDDEs with constant, state-dependent and pantograph delays have been considered to demonstrate the efficiency of the proposed method.
The numerical results demonstrates that the proposed least square method gives results with good precision.Also, the accuracy of the results improves with increasing N (order of polynomial).Hence it is evident that the proposed method is very effective, simple, and suitable for solving linear and nonlinear NDDEs in real world problems.

Figure 1
Figure 1 Solution Graphs for Example 1

Figure 2
Figure 2 Comparison of Solutions for Example 3

Table 1
Solutions and Absolute Error results for Example 1

Table 2
Error Results in CLSM for Example 2

Table 3
Error Results in DLSM for Example 2