A numerical method with a control parameter for integro-differential delay equations with state-dependent bounds via generalized Mott polynomial

In this paper, we introduce a numerical method to obtain an accurate approximate solution of the integro-differential delay equations with state-dependent bounds. The method is based basically on the generalized Mott polynomial with the parameter-β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}, Chebyshev–Lobatto collocation points and matrix structures. These matrices are gathered under a unique matrix equation and then solved algebraically, which produce the desired solution. We discuss the behavior of the solutions, controlling their parameterized form via β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document} and so we monitor the effectiveness of the method. We improve the obtained solutions by employing the Mott-residual error estimation. In addition to comparing the results in tables, we also illustrate the solutions in figures, which are made up of the phase plane, logarithmic and standard scales. All results indicate that the present method is simple-structured, reliable and straightforward to write a computer program module on any mathematical software.


Introduction
Integro-differential equations (IDEs) and their delay types (IDDEs) govern many physical phenomena emerging in mathematics, mechanics, engineering, biology, economics, electrodynamics and oscillating magnetic field [1][2][3][4][5][6][7][8]. These varieties encourage many researchers in all over the world to give more attention than ever before. Because the sophisticated phenomena can be easily described with the aid of IDDEs. As these phenomena are evolved, finding the physical behavior of IDDEs becomes far more difficult. For example, state-dependent Riccati equation modeling vehicle state estimation [9], state-dependent delay Volterra equations considered in viscoelasticity theory [10] and the system of state-dependent delay differential equation describing forest growth [11] can be found in the literature. Thus, a difficult task appears while interpreting the physical responses of these complex structures, analytically. Therefore, several numerical methods have recently been focused and established more on IDEs and their various types. To this end, Kürkçü et al. [12,13] have solved IDEs and IDEs of difference type by means of Dickson matrix-collocation method. Reutskiy [14] has utilized the backward substitution method for solving the neutral Volterra-Fredholm IDEs. Chen ve Wang [15] have dealt with the neutral functional-differential equation with proportional delays using the variational iteration method. Bellen and Zennaro [4] have investigated the convergence and numerical solution of state-dependent delay differential equations. Gökmen et al. [16] have proposed Taylor polynomial method for solving the Volterra-type functional integral equations. Gülsu et al. [17] have used Chebyshev polynomial for delay differential equations. Savaşaneril and Sezer [18,19] have employed Taylor and Taylor-Lucas polynomial method for searching the solution of Fredholm IDEs and pantograph-type delay differential equations, respectively. Maleknejad and Mahmoidi [20] have obtained the Taylor and block-pulse numerical solutions of Fredholm integral equation. Rohaninasa et al. [21] have established Legendre collocation method to solve Volterra-Fredholm IDEs. Yüzbaşı [22] have approached the numerical solutions of pantograph-type Volterra IDES with the aid of Laguerre polynomial. Gümgüm at al. [23] have obtained the Lucas polynomial solutions of functional IDEs involving variable delays.
All above studies motivate us to develop a numerical method and deal with the highly stiff problems, such as the integro-differential delay equations with state-dependent bounds in this paper. In the literature, there is no study maintaining the numerical solution of such equations. By way of this study, we can investigate their physical responses numerically, evaluating the obtained values in tables and figures. This paper is organized as follows: "Fundamental properties of Mott polynomial" section mentions some properties of the Mott polynomials. "Constructing method of solution via matrix relations" section establishes new matrix relations and method solution. "Mott-residual error estimation" section gives Mott-residual error estimation as an algorithmic sense.
"Numerical examples" section includes stiff numerical examples solved with the aid of the present method. "Conclusions" section presents the discussions about the present method and its efficiency by taking into account the results in "Numerical examples" section. The functional integro-differential delay equations with state-dependent bounds are of the form subject to the initial conditions where y(t), P r (t) , g(t) and K q (t, s) are analytic functions on [a, b]; r and q are real constant delays r , q ≥ 0 ; q , c q , d q c q < d q and k are proper constants.
Our aim in this study is to efficiently obtain an accurate approximate solution of Eq. (1) by developing the Mott matrixcollocation method, which was previously introduced in [24]. Besides, the parameter-in the generalized Mott polynomial is used as a control parameter in the numerical approximations. Hence, we can control the obtained solutions in terms of their more consistent structures. The approximate solution comes out to be in the form (see [24]) where y n , n = 0, 1, … , N are unknown Mott coefficients to be calculated by the method and S n (t, ) is the generalized Mott polynomial [25]. Chebyshev-Lobatto collocation points used in the matrix systems are defined to be (see [23])

Fundamental properties of Mott polynomial
In this section, we briefly describe some fundamental properties of Mott polynomial, which is used as a basis of the matrixcollocation method. In 1932, Mott [26] originally introduced the polynomial while monitoring the roaming behaviors of electrons for a problem in the theory of electrons. After this exploration, Erdèlyi et al. [27] established the explicit formula of the polynomial S n (t) as follows: In 1984, Roman [28] presented both an associated Sheffer sequence and a generating function for the polynomial as follows: 16 . On the other hand, a triangle coefficient matrix of the polynomial can be found in A137378 of OEIS [29]. In 2014, Kruchinin [25] converted the polynomial to a generalized form with a parameter-: where the Mott polynomial is obtained for = 0.5 . For further properties of the polynomial, the reader can refer to [25][26][27][28].

Constructing method of solution via matrix relations
In this section, the fundamental matrix relations are presented to construct method of solution. Let us first state the solution form (3) in the matrix relation [24] (4) where Now, inserting t → t − r into the matrix relation (5) (7) and (9) that where (5) y(t) = S(t, )Y and y (r) (t) = S (r) (t, )Y, Recalling the matrix relations (8) and (10) and collocation points (4), we thus write the combined matrix relation as More briefly, we can construct the matrix relation (11) as the fundamental matrix equation where Using the matrix relation (5), we similarly state the matrix relation of the initial conditions (2) as the following: By the matrix equation (12), we are now ready to constitute the method of solution

Mott-residual error estimation
The residual error analysis has successfully been employed in [7,8,12,13,16,22,30]. For this motivation, we introduce the Mott-residual error estimation technique including the Mott polynomial and a residual function to improve the Mott polynomial solution (3) of Eq. (1). Algorithmic procedure of this technique can be described for the present method as Step 1:

Numerical examples
In this section, we apply the present method to solve some stiff problems concerned with Eq.  (14), we construct the fundamental matrix equation as After applying the described procedure to the equation above, we easily get the augmented matrix , we solve the problem by using both the present method and Taylor collocation method [16,18] to compare the obtained results. We later employ the Mott-residual error estimation to improve the solution. It is important to state that we investigate the effects of and the delays on the Mott polynomial solutions. Therefore, the following discussion is made: • Table 1 shows the absolute errors for fixed 2 = 0 = 0.01 and = 1.5 . Also in there, the better numerical results are obtained in comparison with Taylor collocation method [16,18].  respect to in the logarithmic scaled plot shown in Fig. 3.

Example 3
Consider the second-order external forced oscillatory differential equation exposing to single time-delayed effect subject to the initial conditions y(0) = 0 and y � (0) = 1 . A under-damped parameter | | < 1 , an external force F, a nonresonance excitation ≠ 1 [5]. Here, L = {1, 15} and also the exact solution of the problem is unknown, but it can be approached numerically with the aid of Mathematica as Previously, Kalmar-Nagy and Balachandran [5] have studied the linear oscillator differential equation with external forcing, under-damped system and non-resonance excitation. They have determined the steady-state response and magnification factor. In this example, by exposing the linear oscillator equation [5] to the delayed effect 1 , let us seek the numerical solutions for different N, M, 1 , , , and the fixed F = = 2 . Thus, • By increasing N and M, we demonstrate the absolute errors for 1 = 0 and = 1 in Table 3. This indicates that N and M enable us to enhance the accuracy of the method. is demonstrated with respect to the control parameter in Fig. 8.
In addition, we draw the attention to the fact that both and 1 have a different effect on the Mott polynomial solution.

Conclusions
An efficient numerical method based on the generalized Mott polynomial, Chebyshev-Lobatto collocation points and the matrix structures has been proposed to solve stiff IDDEs with state-dependent bounds, which are introduced for the first time with this paper. Thanks to the simplicity of the present method, the obtained solutions have been accurately approximated to the exact and Mathematica solutions.
Controlling the optimum value of the parameterin the    and reliable for solving other well-known phenomena, such as partial differential and fractional differential equations after making some required modifications on the proposed method.
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