1 Introduction
In general aspect, functional differential equations (FDEs) describe a wide range of real life phenomena which are extremely valuable in scientific research fields. Specifically, delay differential equations (DDEs) and their applications perform widely in many different branches of medicine, engineering and life sciences. These fields have operations based on timedependent processes which can be easily described by DDEs [1], [2].
Economic models are easily described by using FDEs. Some arguments in the continuous systems are helpful to establish control problems and give us better understanding for the market models by using the timedelay contribution in equations. Multiple decision tasks, capable formulations are defined by FDEs with the idea of overlapping in time. Besides, competitive market scenarios are established with regard to the DDE models. In the theory of modern economic growth, continuoustime modeling by FDEs derives an outstanding task. For instance, the KaldorKalecki model, the Cournot oligopoly model and the Goodwin model are also wellknown models in economy which are described by the differential equations with timedelay [3][7].
In computer science, FDEs play an essential role. Recently, machine learning studies are very attractive in many applications. Timedelay differential equations establish a base for machine learning applications. Specifically, reservoir computing has been introduced recently as a machine learning pattern [8]. Its implementations with hardware accomplish processing of empirical data [9]. Besides, in machining processes chatter identification is described by FDEs [10]. In this area, the importance of FDEs has been described in current studies.
In electrodynamics, a device which is called as pantograph exists in an electric locomotive. This device collects the electric current and power is provided to the locomotive by a system from above cables. So, this system of electricity collection to trains or trams provides a contact with regard to the pantograph design [11][12]. This system is a physical model in electrodynamics and it is related to the time dependence. The modeling of this structure is formulated by DDEs. In Fig. 1, the pantograph above a tram is shown. This basic design of the pantograph is a physical motion and accomplishes DDEs with regard to the the vertical displacements of the pantograph at frame and head [13][15]
In the model, we can also consider some constants such as the damping coefficients for describing the motions from the head to the frame and other way around respectively with and . A simple representation of a pantograph and its trolley wire system is shown in Fig. 2 which is formulated as
(1)  
(2) 
Here, and values are the constants which are presenting bounce between frame and head of the pantograph, and symbolise the masses of head and frame, respectively, and are related to the displacement of head and frame, respectively, is the vertical component force, is the upward force and is the gravity of earth [17].
In biology, models which are described by the functional  differential equations have been established for determining a dynamics of some biological systems, and thermal science and mechanics systems. This wide usage of the FDEs supports mathematical biology studies and researchers in the field. For instance, recent studies on these type of models introduce a tool for explaining malaria, Dengue Fever epidemics and Covid19 virus spreading models which include very frequently time delays [18][21]. Furthermore, Mackey and Glass [22] proposed two possible models to describe the change of density of Hematopoietic cells in the blood that is circulating in the human body [23].
Particularly, in population dynamics DDEs are often used. Continuoustime models including delay term describe mainly structure of a population. In this research field, the rate of producing egg in a population, susceptibility to parasitism or a bacterial population and their relations with related to the time dependence and delay term have been used in many studies [24][25]. Also, population dynamics is described by DDEs with the examples of investigation on baleen whale populations, delay model for introducing the dynamics of human immunodeficiency virus (HIV) infection etc. [26][27].
Diversely, finding the solutions of these types of models analitically can be difficult. Due to this reason, numerical methods such as Adomian decompostion, finite element methods, homotopic perturbation techniques, RungeKutta type methods, direct twopoint block methods, collocation methods, variational iteration methods have been applied to find approximate solutions of these type of problems [28][35].
In this study, a class of FDEs is described as following
(3) 
with initial condition
(4) 
Herein are given as constants, is described for a delay term by using the name ”time delay”. Assuming that the approximate solutions of our problem, which has been given at (3)(4), defined in the form of Laguerre series as
(5) 
Here Laguerre polynomials are ;
(6) 
and , are expressed the unknown coefficients of Laguerre polynomials. Moreover, is an integer which is selected properly, i.e. [36].
Applications of such dynamics in numerical studies are of valuable contribution to the field which enable us to understand the dynamical structure comprehensively. Here, an algorithmic approximation in the interdisciplinary concept is introduced which leads to widen related studies. The aim of the study is to give an alternative solution scheme and different perspective than the previous works in the field. In this way, it is aimed by this study that addresses the development of many fields through an important dynamic.
The organisation of the manuscript is given as follows. In Section 2, a mathematical outlook and preliminaries are given. In Section 3, the numerical method is defined with its details. Accuracy of the numerical technique is described in Section 4. In Section 5, numerical simulations are demonstrated by figures and tables. A brief conclusion and future studies are presented in Section 6.
2 Preliminaries
Here some preliminaries about Laguerre polynomials and their properties on the delay differential equations has been introduced for the further applications. Besides, the recurrence relations of series and definitions with regard to the model are introduced to support the technical results.
Interdisiplinary researches of the Laguerre polynomials are wellknown in many fields. The Laguerre polynomials are important in the applications of quantum theory in the hydrogen atom concept, chemical and mathematical physics [37]. They play an important role at the numerical methods in applied mathematics field such as Laguerre collocation method [38], [39], [40], [41], Laguerre spectral method [42][43], Laguerre pseudospectral method [44], and Laguerre wavelets collocation method [45]. Besides, they support the improvement of the numerical solutions by combining with other numerical techniques, i.e. LaguerreGalerkin methods [46], Laguerre Tau methods and so on [47].
Moreover, Laguerre polynomials are of a close relation with Hermite polynomials. Most of their properties and applications at the research area of orthogonal polynomials may help to find out characteristic of the Hermite polynomials. The Hermite polynomials are connected to Laguerre polynomials as following:
(7)  
(8) 
In Fig. 3, we can see the relations between the first few Laguerre and Hermite polynomials with and where Laguerre polynomials are presented by red lines and Hermite polynomials are shown by blue lines.
Definition 1
The first few Laguerre and Hermite polynomials are defined, respectively, as following:
(9)  
(10)  
(11)  
(12) 
Hermite polynomials have been applied on physical models, more specifically, at the solution of simple harmonic oscillator of quantum mechanics and Laguerre polynomials are seen in wave functions of hydrogen atom [48]. Usage of Laguerre polynomials are also well defined for the series solution. In this study, Laguerre polynomials give us an alternative series solution for the delay differential equations which is supported by following lemma.
Lemma 2.1
Suppose that and is defined on the interval which is a piecewise continuous function. Then for
(13) 
we have converges uniformly on to for any positive integer [47].
In this study, a special class of functional differential equations has been considered. Functional differential equations are of many different types including delay differential equations or more specifically retarded, neutral and advanced functional differential equations [49]. Now, the definitions with regard to these equations are introduced in order to complete the problem settlement.
Definition 2
Here the following equation is considered:
(14) 
which is given with st order in its derivative and difference. In this place, an equation in the form above is called as retarded type where .
3 Method
In this section, a general scheme with regard to Laguerre collocation technique is introduced. We provide a novel collocation structure by using the Laguerre polynomial approach directly with related to some fundamental relations between the matrices. Here, the idea of the implementation of the technique is presented together with collocation points. The truncation of the series is shown and algebraic equations are obtained. We have the series approach and show the approximate solutions with the help of these connections.
Our motivation here is to show that we expand the wellconditioned collocation method with regard to Laguerre matrix approach. The applications of the technique some realworld models give us a chance to investigate the models based on numerical concept.
3.1 Fundamental Relations
In this subsection, an approximate solution with related to Laguerre polynomials form (6) is presented. For our purpose, we consider a matrix form of Eq. (5) as
(15) 
in which
(16) 
Then, we use the matrix relation
(17) 
where
(18) 
Besides, the connections are defined between and . So, we write them as
(19) 
where
(20) 
Then from the relations in (15) and (19), we get
(21) 
So that, from (17), (19) and (21), we have
(22) 
(23) 
where
(24) 
Then we have
(25) 
Moreover, is replaced into (15) and get
(26) 
where
(27) 
Then
(28)  
(29)  
(30) 
are defined and in which
(31)  
(32)  
(33) 
By using the equations (28)(30), a matrix form of (3) is described as
(34) 
Here
(35) 
3.2 Method of Solution
In this subsection, collocation method is applied by using the collocation points which give us a pointwise approximation with regard to our truncation number together with step size . Then by replacing the collocation points
(36) 
into Eq. (34), so that the fundamental matrix equation is obtained and written as follows:
(37) 
Briefly,
(38) 
Correspondingly, the initial condition is written in the matrix form as
(39) 
or
(40) 
Therefore, the last rows of the augmented matrix (38) is reestablished by the row matrix (39) in order to get an approximate solution of Eq. (3) under the condition (4). Thus, the augmented matrix is obtained as
(41) 
Laguerre coefficients can be computed by solving the augmented matrix system [50][51]. Hence, the approximate solution is established as in the desired form of Laguerre series (6) as
(42) 
4 Accuracy
In this section, errors of approximation and solution strategy with regard to the its algorithm have been introduced. Error analysis has been proposed by using different norms in order to show the results more comprehensively. This lead us to see efficiency of our numerical technique. Besides, algorithm has been presented which is based on programming part of the construction of our method. Hereby the application of the numerical method by using a computer programming has been shared.
Accuracy is an important topic in approximation theory which has a reason to be introduced due to the errors occur in rounding and truncation of numerical solutions. When we use numerical algorithms with finite sensitivity, approximation errors, round or truncation error at the series solutions appears. Due to this reason, we should show the accuracy of the technique and how we deal with this approximation at the end of our application.
4.1 Error Estimation
In this subsection, a short presentation of the error estimation for the solutions with regard to Laguerre approach (
6) is given. Hereby accuracy of this technique is supported by this approximation. Let us first describe, , ”error function” for(43) 
where is appointed and we pay attention to the truncation limit which is an important argument for finding the approximate results. Accordingly, the difference becomes smaller and, in here, this is denoted by . Diversely, some particular norms are given in order to investigate the distinctive error functions for measuring errors. So, these are described as follows.

For ; ,

For ; for ,

For ; .
Here also and are approximate and exact solutions of the problem, respectively.
4.2 Algorithm
In this section, Steps of algorithm for the present method:

Data: , and constants in Eq. (3).

Result: : approximate solutions.

Truncation is chosen as for ,

Construction of all the matrices,

Replacement of the fundamental matrix equation,

Apply the collocation points (colloc. pts.), and , to the fundamental matrix equation in S2.

Computation of the augmented matrix by Gauss elimination,

Construction of the initial conditions (ICs) in matrix forms ,

Replacement of the initial condition in matrix forms in S5. to the augmented matrix in S4. Then we get ,

Solution of the system in S6. and replacement in the truncated Laguerre series form in Eq. (5).

Stop.
We found an effective algorithmic approach for calculating an approximation procedure and to investigate the dynamics behind our model. This made a considerable impact on the dynamics and it has an excellent significance framework on approximation methodology.
Algorithms particularly connect the applicability with time standard. In order to increase the quality and to decrease the central processing unit (CPU) time, we may use an effective approach for coding. Due to this reason, straightforward techniques are of usable and effective to get the approximations. Then we can see in 4.2 that our numerical method may be applied with regard to this idea. Besides, the following flowchart of the algorithm shows us a standardized approach to find the approximate solutions in our problem (3)(4) [53].
5 Numerical Illustrations
In here, some illustrative numerical examples are presented to show applicability of this given present method. These examples have been chosen from science and engineering applications to be analysed. Besides, the algorithm, which has been explained 4.2, is applied on the problems. The problem has been introduced with the general formula at (3)(4). Maple and Matlab computer programmes have been used for the calculations for finding the results and plotting the figures [54][55].
Example 1.
Let us first consider a model from biology which is called as ”WazewskaCzyzewska and Lasota model”. This biological model describes a remainder red blood cells, specifically, in animals [31], [56]:
(44) 
and its initial condition
(45) 
Here, number of total red blood cells at time, denoted by , is expressed by . Moreover, which is presenting death rate of a unique red blood cell. Besides, and are denoted red blood cells generation per unit time and is time essential for generating a red blood cell. Specifically, , , are chosen. So that, we have (28) as
(46) 
Then we have the matrix form as
(47) 
By using the procedure, we obtain approximate solutions. The approximate solutions for different values can be seen in Figure 1. Different error norm results for is given by Table 1.
Error  Error  Error  

1  0.7560E05  0.5247E04  0.1000E06 
2  0.1164E05  0.3791E04  0.1502E05 
3  0.1550E04  0.5467E03  0.6855E04 
4  0.8259E03  0.7795E03  0.1752E03 
5  0.4643E04  0.5467E02  0.2916E05 
Example 2.
Here, the model is considered as in the form [57], [58]:
(48)  
(49) 
with initial conditions
(50) 
Herein, is chosen. For Eq. (39) , , and for Eq. (40) , are assigned. Moreover, for , and for , are defined in Eq. (40). Again, we follow the similar process by implementing Laguerre collocation method (LCM) for and truncation values on our problem. Besides, this results obtained by LCM, some other numerical methods such as RungeKutta method (RKM) with fourthorder and Hermite collocation method (HCM) with the same truncation values. The results can be seen by the figures and the tables.
LCM,  LCM,  HCM,  HCM,  

0.0  0.6250E02  0.1593E03  0.7081E02  0.2301E03 
1.2  0.4202E03  0.2490E04  0.5223E02  0.5100E03 
2.3  0.2664E03  0.2507E04  0.5708E02  0.6290E04 
4.5  0.8259E02  0.4510E03  0.4430E01  0.5291E03 
5.0  0.5531E02  0.7410E03  0.3548E01  0.8302E03 
LCM  

3  1.140 
4  1.258 
6 Conclusion
In this study, numerical investigation of a class of FDEs has been considered. Besides, a technique based on matrices together with collocation points has been introduced. With the help of computer programmes, Maple and Matlab, numerical approximation of Laguerre collocation method has been compared with the techniques such as RungeKutta and Hermite collocation methods. Moreover, the technique together with an error analysis have been implemented on some illustrations to see the applicability as well as efficiency of the method. Then the results have been seen by figures and tables. We have the advantages of the method such as straightforward applicability and coding come along with the proven convergency by some authors [60], [61], [62]. As a future plan, this technique can be improved and applied on some other mathematical models after completing some modifications.
References
 1. A. Bellen and M. Zennaro, Numerical methods for delay differential equations, Oxford University Press, New York, 2013.
 2. A. Halanay, Differential equations: Stability, oscillations, time lags, Academic Press, New York, 6 (5) (1966), 611–615.
 3. A.A. Keller, Generalized delay differential equations to economic dynamics and control, AmericanMath, 10 (2010), 278–286.
 4. K. Kobayashi, An application of delay differential equations to market equilibrium, The Functional and Algebraic Method for Differential Equations (1996).
 5. R. Boucekkine, O. Licandro and C. Paul, Differentialdifference equations in economics: on the numerical solution of vintage capital growth models, J. Econ. Dyn. Control, 21 (23) (1997), 347–362.
 6. A. Matsumoto and F. Szidarovszky, Delay differential nonlinear economic models, in Nonlinear dynamics in economics, finance and social sciences, Springer, Berlin, Heidelberg, (2010), 195–214.
 7. E. Savku and G.W. Weber, A stochastic maximum principle for a markov regimeswitching jumpdiffusion model with delay and an application to finance, J. Optimiz. Theory App., 179 (2) (2018), 696–721.
 8. L. Grigoryeva, J. Henriques, L. Larger and J.P. Ortega, Optimal nonlinear information processing capacity in delaybased reservoir computer, Sci. Rep., 5 (1) (2015), 1–11.
 9. L. Grigoryeva, J. Henriques, L. Larger and J.P. Ortega, Timedelay reservoir computers and highspeed information processing capacity, in 2016 IEEE Intl Conference on Computational Science and Engineering (CSE) and IEEE Intl Conference on Embedded and Ubiquitous Computing (EUC) and 15th Intl Symposium on Distributed Computing and Applications for Business Engineering (DCABES), (2016), 492–495.
 10. F.A. Khasawneh, E. Munch, and J.A. Perea, Chatter classification in turning using machine learning and topological data analysis, IFACPapersOnLine, 51 (14) (2018), 195–200.
 11. S. Bhalekar and J. Patade, Series solution of the pantograph equation and its properties, Fractal and Fractional, 1 (1) (2017), 16.
 12. M. Dehghan and F. Shakeri, The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics, Phys. Scr., 78 (6) (2008), 065004.
 13. M.R. Abbott, Numerical method for calculating the dynamic behaviour of a trolley wire overhead contact system for electric railways, Comput. J., 13 (4) (1970), 363–368.
 14. H.I. Andrews, Third paper: Calculating the behaviour of an overhead catenary system for railway electrification, Proceedings of the institution of mechanical engineers, 179 (1) (1964), 809–846.
 15. G. Gilbert and H.E.H. Davies, Pantograph motion on a nearly uniform railway overhead line, Proceedings of the Institution of Electrical Engineers, 113 (3) (1966).
 16. Istanbul plans third heritagestyle tramway, Report of Hong Kong SARS Expert Committee, 2019. Available from: https://www.railwaygazette.com/.
 17. J. Benet, N. Cuartero, F. Cuartero, T. Rojo, P. Tendero and E. Arias, An advanced 3Dmodel for the study and simulation of the pantograph catenary system, Transp. Res. Part C Emerg. Technol., 36 (2013), 138–156.
 18. R.M. Anderson, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, New York, 1992.
 19. J.D. Murray, Mathematical Biology 1: An Introduction, 3 edition, Springer, Berlin, 2002.
 20. M.W. Sakdanupaph, A delay differential equation model for Dengue Fever Transmission in selected countries of SouthEast Asia, Doctoral dissertation, King Mongkut’s University of Technology North Bangkok, 2007.
 21. L. Dell’Anna, Solvable delay model for epidemic spreading: the case of Covid19 in Italy, preprint, arXivmath/2003.13571.
 22. M.C. Mackey and L. Glass, Oscillation chaos in physiological control systems, Science, New Series, 197 (4300) (1977), 287–289.
 23. E.B.M. Bashier, Fitted numerical methods for delay differential equations arising in biology, Doctoral dissertation, University of the Western Cape, 2009.
 24. R.M. Nisbet, Delaydifferential equations for structured populations, in Structuredpopulation models in marine, terrestrial, and freshwater systems, Springer, Boston, MA, (1997), 89–118.
 25. O. Diekmann, S.A. Van Gils, S.M.V. Lunel and H.O. Walther, Delay equations: functional, complex, and nonlinear analysis, Springer Science & Business Media, 110 2012.
 26. P.W. Nelson, A.S. Perelson and J.D. Murray, Delay model for the dynamics if HIV infection, Math. Biosci., 163 (2000), 201–215.
 27. C.W. Clark, A delayedrecruitment model of population dynamics with an application to baleen whale populations, J. Math. Biol., 3 (2000), 381–391.
 28. H.L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, 2011.
 29. D.J. Evans and K.R. Raslan, The Adomian decomposition method for solving delay differential equation, Int. J. Comput. Math., 82 (1) (2005), 49–54.
 30. A.N. AlMutib, Stability properties of numerical methods for solving delay differential equations, J. Comput. Appl. Math., 10 (1984), 71–79.
 31. H.L. Su, W. Li and X. Ding, Numerical dynamics of a nonstandard finite difference method for a class of delay differential equations, J. Math. Anal. Appl., 400 (2013), 25–34.
 32. F. Shakeri and M. Dehghan, Solution of delay differential equations via a homotopy perturbation method, Math. Comput. Model., 48 (2008), 486–498.
 33. X. Chen and L. Wang, The variational iteration method for solving a neutral functionaldifferential equation with proportional delays, Comput. Math. Appl., 59 (2010), 2696–2702.
 34. H.Y. Seong and Z.A. Majid, Solving second order delay differential equations using direct twopoint block method, Ain. Shams. Eng. J., 8 (2017), 59–66.
 35. Ş. Yüzbaşı, N. Şahin and M. Sezer, A Bessel collocation method for numerical solution of generalized pantograph equations, Numer. Meth. Part. D. E., 28 (4) (2012), 1105–1123.
 36. V.S. Aizenshtadt, I.K. Vladimir and A.S. Metel’skii, Tables of Laguerre Polynomials and Functions: Mathematical Tables Series, Elsevier, London, 39 (2014).
 37. D. Borwein, J.M. Borwein and R.E. Crandall, Effective Laguerre asymptotics, SIAM Journal on Numerical Analysis, 46 (6) (2008), 3285–3312.

38.
B. Gürbüz and M. Sezer,
Modified operational matrix method for secondorder nonlinear ordinary differential equations with quadratic and cubic terms,
An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 10 (2) (2020), 218–225.  39. B. Gürbüz and M. Sezer, A Modified Laguerre Matrix Approach for Burgers  Fisher Type Nonlinear Equations, Numerical Solutions of Realistic Nonlinear Phenomena, Springer, Cham., (2020), 107–123.
 40. B. Gürbüz and M. Sezer, Laguerre Matrix  Collocation Method to Solve Systems of Pantograph Type Delay Differential Equations, International Conference on Computational Mathematics and Engineering Sciences, Springer, Cham., (2019), 218–225.
 41. B. Gürbüz and M. Sezer, A numerical solution of parabolictype Volterra partial integrodifferential equations by Laguerre collocation method, International Journal of Applied Physics and Mathematics (IJAMP), 7 (1) (2017), 49.
 42. G. BenYu and Z. XiaoYong, A new generalized Laguerre spectral approximation and its applications, J. Comput. Appl. Math., 181 (2) (2005), 342–363.
 43. D. Baleanu, A.H. Bhrawy and T.M. Taha, A modified generalized Laguerre spectral method for fractional differential equations on the half line, Abst. Appl. Anal., 2013 (2013).
 44. H. Alıcı, The Laguerre pseudospectral method for the twodimensional Schrödinger equation with symmetric nonseparable potentials, Hacet. J. Math. Stat., (2020), 1–14.
 45. F. Zhou and X. Xu, Numerical solutions for the linear and nonlinear singular boundary value problems using Laguerre wavelets, Adv. Differ. EquNy, 1 (2016), 17.

46.
B.Y. Guo and J. Shen,
LaguerreGalerkin method for nonlinear partial differential equations on a semiinfinite interval,
Numer. Math., 86 (4) (2000), 635–654.  47. H.I. Siyyam, Laguerre Tau methods for solving higherorder ordinary differential equations, J. Comput. Anal. Appl., 3 (2) (2001), 173–182.
 48. G. Arfken and H.J. Weber, Mathematical methods for physicists, Academic Press, San Diego, 1999.
 49. R. Bellman and K.L. Cooke, Differentialdifference equations, Academic Press, New York, 1963.
 50. M. Çetin, B. Gürbüz and M. Sezer, Lucas collocation method for system of highorder linear functional differential equations, J. Sci. Art., 4 (2018), 891–910.
 51. M. Gülsu, B. Gürbüz, Y. Öztürk and M. Sezer, Laguerre polynomial approach for solving linear delay difference equations, Appl. Math. Comput., 217 (15) (2011), 6765–6776.
 52. B. Türkyılmaz, B. Gürbüz and M. Sezer, MorganVoyce polynomial approach for solution of highorder linear differentialdifference equations with residual error estimation, Düzce Üniversitesi Bilim ve Teknoloji Dergisi, 4 (1) (2016).
 53. B. Gürbüz, H. Mawengkang, I. Husein, G. W. Weber and M. Sezer, Rumour propagation: an operational research approach by computational and information theory, Central European Journal of Operations Research, 1–21.
 54. Maple 18 Release 1, Waterloo Maple Inc., 450 Phillip St., Waterloo, ON N2L 5J2, Canada, 2014. Available from: https://www.maplesoft.com/products/maple/history/.
 55. MATLAB 8.4, The MathWorks Inc., 3 Apple Hill Dr., Natick, MA 01760, 2014. Available from: https://de.mathworks.com/products/compiler/matlabruntime.html.
 56. M. WazewskaCzyzewska and A. Lasota, Mathematical problems of the dynamics of the red blood cells system, Ann. Polish Math. Soc. Ser. III, Appl. Math., 17 (1976), 23–40.
 57. S.I. Jumaa, Solving Linear First Order Delay Differential Equations by MOC and Steps Method Comparing with Matlab Solver, Ph.D thesis, Near East University in Nicosia, 2017.
 58. L.F. Shampine and S. Thompson, Solving ddes in Matlab, App. Num. Math., 37 (4) (2001), 441–458.
 59. W.O. Kermack and A.G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the royal society of London. Series A, 115 (772) (1927), 700–721.
 60. M.M. Khader, The use of generalized Laguerre polynomials in spectral methods for solving fractional delay differential equations, J. Comput. Nonlin. Dyn., 8 (4) (2013).
 61. Y. Muroya and S. Thompson, Solving ddes in Matlab, App. Num. Math., 37 (4) (2001), 441–458.
 62. Y. Yang, E. Ishiwata and H. Brunner, On the attainable order of collocation methods for pantograph integrodifferential equations, J. Comput. Appl. Math., 152 (12) (2003), 347–366.