doi: 10.3934/jimo.2021069
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A computational approximation for the solution of retarded functional differential equations and their applications to science and engineering

Institute of Mathematics, Johannes Gutenberg-University Mainz, Germany

* Corresponding author: Burcu Gürbüz

Received  August 2020 Revised  December 2020 Early access April 2021

Delay differential equations are of great importance in science, engineering, medicine and biological models. These type of models include time delay phenomena which is helpful for characterising the real-world applications in machine learning, mechanics, economics, electrodynamics and so on. Besides, special classes of functional differential equations have been investigated in many researches. In this study, a numerical investigation of retarded type of these models together with initial conditions are introduced. The technique is based on a polynomial approach along with collocation points which maintains an approximated solutions to the problem. Besides, an error analysis of the approximate solutions is given. Accuracy of the method is shown by the results. Consequently, illustrative examples are considered and detailed analysis of the problem is acquired. Consequently, the future outlook is discussed in conclusion.

Citation: Burcu Gürbüz. A computational approximation for the solution of retarded functional differential equations and their applications to science and engineering. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021069
References:
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M. R. Abbott, Numerical method for calculating the dynamic behaviour of a trolley wire overhead contact system for electric railways, Comput. J., 13 (1970), 363-368.  doi: 10.1093/comjnl/13.4.363.

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A. N. Al-Mutib, Stability properties of numerical methods for solving delay differential equations, J. Comput. Appl. Math., 10 (1984), 71-79.  doi: 10.1016/0377-0427(84)90071-2.

[4]

H. Alıcı, The Laguerre pseudospectral method for the two-dimensional Schrödinger equation with symmetric nonseparable potentials, Hacet. J. Math. Stat., (2020), 1-14. doi: 10.15672/hujms.459593.

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H. I. Andrews, Third paper: Calculating the behaviour of an overhead catenary system for railway electrification, Proceedings of the Institution of Mechanical Engineers, 179 (1964), 809-846.  doi: 10.1243/PIME_PROC_1964_179_050_02.

[7] G. Arfken and H. J. Weber, Mathematical Methods for Physicists, Academic Press, San Diego, 1999. 
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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). doi: 10.1155/2013/413529.

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E. B. M. Bashier, Fitted numerical methods for delay differential equations arising in biology, Doctoral dissertation, University of the Western Cape, 2009.

[10] A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, New York, 2013. 
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[12]

G. Ben-Yu and Z. Xiao-Yong, A new generalized Laguerre spectral approximation and its applications, J. Comput. Appl. Math., 181 (2005), 342-363.  doi: 10.1016/j.cam.2004.12.008.

[13]

J. BenetN. CuarteroF. CuarteroT. RojoP. Tendero and E. Arias, An advanced 3D-model for the study and simulation of the pantograph catenary system, Transp. Res. Part C Emerg. Technol., 36 (2013), 138-156.  doi: 10.1016/j.trc.2013.08.004.

[14]

S. Bhalekar and J. Patade, Series solution of the pantograph equation and its properties, Fractal and Fractional, 1 (2017), 16. doi: 10.3390/fractalfract1010016.

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D. BorweinJ. M. Borwein and R. E. Crandall, Effective Laguerre asymptotics, SIAM Journal on Numerical Analysis, 46 (2008), 3285-3312.  doi: 10.1137/07068031X.

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M. ÇetinB. Gürbüz and M. Sezer, Lucas collocation method for system of high-order linear functional differential equations, J. Sci. Art., 4 (2018), 891-910. 

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X. Chen and L. Wang, The variational iteration method for solving a neutral functional-differential equation with proportional delays, Comput. Math. Appl., 59 (2010), 2696-2702.  doi: 10.1016/j.camwa.2010.01.037.

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C. W. Clark, A delayed-recruitment model of population dynamics with an application to baleen whale populations, J. Math. Biol., 3 (2000), 381-391.  doi: 10.1007/BF00275067.

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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 (2008), 065004. doi: 10.1088/0031-8949/78/06/065004.

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D. J. Evans and K. R. Raslan, The Adomian decomposition method for solving delay differential equation, Int. J. Comput. Math., 82 (2005), 49-54.  doi: 10.1080/00207160412331286815.

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Istanbul Plans Third Heritage-Style Tramway, Report of Hong Kong SARS Expert Committee, 2019. Available from: https://www.railwaygazette.com/.

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G. Gilbert and H. E. H. Davies, Pantograph motion on a nearly uniform railway overhead line, Proceedings of the Institution of Electrical Engineers, 113 (1966). doi: 10.1049/piee.1966.0078.

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L. GrigoryevaJ. HenriquesL. Larger and J. P. Ortega, Optimal nonlinear information processing capacity in delay-based reservoir computer, Sci. Rep., 5 (2015), 1-11.  doi: 10.1038/srep12858.

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L. Grigoryeva, J. Henriques, L. Larger and J. P. Ortega, Time-delay reservoir computers and high-speed 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. doi: 10.1109/CSE-EUC-DCABES.2016.230.

[28]

M. GülsuB. GürbüzY. Öztürk and M. Sezer, Laguerre polynomial approach for solving linear delay difference equations, Appl. Math. Comput., 217 (2011), 6765-6776.  doi: 10.1016/j.amc.2011.01.112.

[29]

B. Y. Guo and J. Shen, Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval, Numer. Math., 86 (2000), 635-654.  doi: 10.1007/PL00005413.

[30]

B. Gürbüz and M. Sezer, Modified operational matrix method for second-order nonlinear ordinary differential equations with quadratic and cubic terms, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 10 (2020), 218-225.  doi: 10.11121/ijocta.01.2020.00827.

[31]

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.

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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.

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B. Gürbüz and M. Sezer, A numerical solution of parabolic-type Volterra partial integro-differential equations by Laguerre collocation method, International Journal of Applied Physics and Mathematics (IJAMP), 7 (2017), 49.

[34]

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.

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A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York, 6 (1966), 611-615.

[36]

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.

[37]

A. A. Keller, Generalized delay differential equations to economic dynamics and control, American-Math, 10 (2010), 278-286. 

[38]

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 (1927), 700-721. 

[39]

M. M. Khader, The use of generalized Laguerre polynomials in spectral methods for solving fractional delay differential equations, J. Comput. Nonlin. Dyn., 8 (2013). doi: 10.1115/1.4024852.

[40]

F. A. KhasawnehE. Munch and and J. A. Perea, Chatter classification in turning using machine learning and topological data analysis, IFAC-PapersOnLine, 51 (2018), 195-200.  doi: 10.1016/j.ifacol.2018.07.222.

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K. Kobayashi, An application of delay differential equations to market equilibrium, The Functional and Algebraic Method for Differential Equations (1996).

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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. doi: 10.1007/978-3-642-04023-8_11.

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J. D. Murray, Mathematical Biology 1: An Introduction, 3$^{rd}$ edition, Springer, Berlin, 2002.

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P. W. NelsonA. S. Perelson and J. D. Murray, Delay model for the dynamics if HIV infection, Math. Biosci., 163 (2000), 201-215.  doi: 10.1016/S0025-5564(99)00055-3.

[48]

R. M. Nisbet, Delay-differential equations for structured populations, in Structured-Population Models in Marine, Terrestrial, and Freshwater Systems, Springer, Boston, MA, 1997, 89-118. doi: 10.1007/978-1-4615-5973-3_4.

[49]

M. W. Sakdanupaph, A delay differential equation model for Dengue fever transmission in selected countries of South-East Asia, Doctoral dissertation, King Mongkut's University of Technology North Bangkok, 2007. doi: 10.1063/1.3225441.

[50]

E. Savku and G. W. Weber, A stochastic maximum principle for a markov regime-switching jump-diffusion model with delay and an application to finance, J. Optimiz. Theory App., 179 (2018), 696-721.  doi: 10.1007/s10957-017-1159-3.

[51]

H. Y. Seong and Z. A. Majid, Solving second order delay differential equations using direct two-point block method, Ain. Shams. Eng. J., 8 (2017), 59-66.  doi: 10.1016/j.asej.2015.07.014.

[52]

F. Shakeri and M. Dehghan, Solution of delay differential equations via a homotopy perturbation method, Math. Comput. Model., 48 (2008), 486-498.  doi: 10.1016/j.mcm.2007.09.016.

[53]

L. F. Shampine and S. Thompson, Solving ddes in Matlab, App. Num. Math., 37 (2001), 441-458.  doi: 10.1016/S0168-9274(00)00055-6.

[54]

H. I. Siyyam, Laguerre Tau methods for solving higher-order ordinary differential equations, J. Comput. Anal. Appl., 3 (2001), 173-182.  doi: 10.1023/A:1010141309991.

[55]

H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.

[56]

H. L. SuW. 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.  doi: 10.1016/j.jmaa.2012.11.033.

[57]

B. Türkyılmaz, B. Gürbüz and M. Sezer, Morgan-Voyce polynomial approach for solution of high-order linear differential-difference equations with residual error estimation, Düzce Üniversitesi Bilim ve Teknoloji Dergisi, 4 (2016).

[58]

M. Wazewska-Czyzewska 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. 

[59]

Y. YangE. Ishiwata and H. Brunner, On the attainable order of collocation methods for pantograph integro-differential equations, J. Comput. Appl. Math., 152 (2003), 347-366.  doi: 10.1016/S0377-0427(02)00716-1.

[60]

Ş. YüzbaşıN. Şahin and M. Sezer, A Bessel collocation method for numerical solution of generalized pantograph equations, Numer. Meth. Part. D. E., 28 (2012), 1105-1123.  doi: 10.1002/num.20660.

[61]

F. Zhou and X. Xu, Numerical solutions for the linear and nonlinear singular boundary value problems using Laguerre wavelets, Adv. Differ. Equ-Ny, 1 (2016), 17. doi: 10.1186/s13662-016-0754-1.

show all references

References:
[1]

M. R. Abbott, Numerical method for calculating the dynamic behaviour of a trolley wire overhead contact system for electric railways, Comput. J., 13 (1970), 363-368.  doi: 10.1093/comjnl/13.4.363.

[2]

V. S. Aizenshtadt, I. K. Vladimir and A. S. Metel'skii, Tables of Laguerre Polynomials and Functions: Mathematical Tables Series, Elsevier, London, 2014.

[3]

A. N. Al-Mutib, Stability properties of numerical methods for solving delay differential equations, J. Comput. Appl. Math., 10 (1984), 71-79.  doi: 10.1016/0377-0427(84)90071-2.

[4]

H. Alıcı, The Laguerre pseudospectral method for the two-dimensional Schrödinger equation with symmetric nonseparable potentials, Hacet. J. Math. Stat., (2020), 1-14. doi: 10.15672/hujms.459593.

[5] R. M. Anderson, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, New York, 1992. 
[6]

H. I. Andrews, Third paper: Calculating the behaviour of an overhead catenary system for railway electrification, Proceedings of the Institution of Mechanical Engineers, 179 (1964), 809-846.  doi: 10.1243/PIME_PROC_1964_179_050_02.

[7] G. Arfken and H. J. Weber, Mathematical Methods for Physicists, Academic Press, San Diego, 1999. 
[8]

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). doi: 10.1155/2013/413529.

[9]

E. B. M. Bashier, Fitted numerical methods for delay differential equations arising in biology, Doctoral dissertation, University of the Western Cape, 2009.

[10] A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, New York, 2013. 
[11] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963. 
[12]

G. Ben-Yu and Z. Xiao-Yong, A new generalized Laguerre spectral approximation and its applications, J. Comput. Appl. Math., 181 (2005), 342-363.  doi: 10.1016/j.cam.2004.12.008.

[13]

J. BenetN. CuarteroF. CuarteroT. RojoP. Tendero and E. Arias, An advanced 3D-model for the study and simulation of the pantograph catenary system, Transp. Res. Part C Emerg. Technol., 36 (2013), 138-156.  doi: 10.1016/j.trc.2013.08.004.

[14]

S. Bhalekar and J. Patade, Series solution of the pantograph equation and its properties, Fractal and Fractional, 1 (2017), 16. doi: 10.3390/fractalfract1010016.

[15]

D. BorweinJ. M. Borwein and R. E. Crandall, Effective Laguerre asymptotics, SIAM Journal on Numerical Analysis, 46 (2008), 3285-3312.  doi: 10.1137/07068031X.

[16]

R. BoucekkineO. Licandro and C. Paul, Differential-difference equations in economics: On the numerical solution of vintage capital growth models, J. Econ. Dyn. Control, 21 (1997), 347-362.  doi: 10.1016/S0165-1889(96)00935-9.

[17]

M. ÇetinB. Gürbüz and M. Sezer, Lucas collocation method for system of high-order linear functional differential equations, J. Sci. Art., 4 (2018), 891-910. 

[18]

X. Chen and L. Wang, The variational iteration method for solving a neutral functional-differential equation with proportional delays, Comput. Math. Appl., 59 (2010), 2696-2702.  doi: 10.1016/j.camwa.2010.01.037.

[19]

C. W. Clark, A delayed-recruitment model of population dynamics with an application to baleen whale populations, J. Math. Biol., 3 (2000), 381-391.  doi: 10.1007/BF00275067.

[20]

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 (2008), 065004. doi: 10.1088/0031-8949/78/06/065004.

[21]

L. Dell'Anna, Solvable delay model for epidemic spreading: The case of Covid-19 in Italy, preprint, arXiv: 2003.13571. doi: 10.1038/s41598-020-72529-y.

[22]

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, 2012. doi: 10.1007/978-1-4612-4206-2.

[23]

D. J. Evans and K. R. Raslan, The Adomian decomposition method for solving delay differential equation, Int. J. Comput. Math., 82 (2005), 49-54.  doi: 10.1080/00207160412331286815.

[24]

Istanbul Plans Third Heritage-Style Tramway, Report of Hong Kong SARS Expert Committee, 2019. Available from: https://www.railwaygazette.com/.

[25]

G. Gilbert and H. E. H. Davies, Pantograph motion on a nearly uniform railway overhead line, Proceedings of the Institution of Electrical Engineers, 113 (1966). doi: 10.1049/piee.1966.0078.

[26]

L. GrigoryevaJ. HenriquesL. Larger and J. P. Ortega, Optimal nonlinear information processing capacity in delay-based reservoir computer, Sci. Rep., 5 (2015), 1-11.  doi: 10.1038/srep12858.

[27]

L. Grigoryeva, J. Henriques, L. Larger and J. P. Ortega, Time-delay reservoir computers and high-speed 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. doi: 10.1109/CSE-EUC-DCABES.2016.230.

[28]

M. GülsuB. GürbüzY. Öztürk and M. Sezer, Laguerre polynomial approach for solving linear delay difference equations, Appl. Math. Comput., 217 (2011), 6765-6776.  doi: 10.1016/j.amc.2011.01.112.

[29]

B. Y. Guo and J. Shen, Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval, Numer. Math., 86 (2000), 635-654.  doi: 10.1007/PL00005413.

[30]

B. Gürbüz and M. Sezer, Modified operational matrix method for second-order nonlinear ordinary differential equations with quadratic and cubic terms, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 10 (2020), 218-225.  doi: 10.11121/ijocta.01.2020.00827.

[31]

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.

[32]

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.

[33]

B. Gürbüz and M. Sezer, A numerical solution of parabolic-type Volterra partial integro-differential equations by Laguerre collocation method, International Journal of Applied Physics and Mathematics (IJAMP), 7 (2017), 49.

[34]

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.

[35]

A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York, 6 (1966), 611-615.

[36]

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.

[37]

A. A. Keller, Generalized delay differential equations to economic dynamics and control, American-Math, 10 (2010), 278-286. 

[38]

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 (1927), 700-721. 

[39]

M. M. Khader, The use of generalized Laguerre polynomials in spectral methods for solving fractional delay differential equations, J. Comput. Nonlin. Dyn., 8 (2013). doi: 10.1115/1.4024852.

[40]

F. A. KhasawnehE. Munch and and J. A. Perea, Chatter classification in turning using machine learning and topological data analysis, IFAC-PapersOnLine, 51 (2018), 195-200.  doi: 10.1016/j.ifacol.2018.07.222.

[41]

K. Kobayashi, An application of delay differential equations to market equilibrium, The Functional and Algebraic Method for Differential Equations (1996).

[42]

M. C. Mackey and L. Glass, Oscillation chaos in physiological control systems, Science, New Series, 197 (1977), 287-289.  doi: 10.1126/science.267326.

[43]

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/.

[44]

MATLAB 8.4, The MathWorks Inc., 3 Apple Hill Dr., Natick, MA 01760, 2014. Available from: https://de.mathworks.com/products/compiler/matlab-runtime.html.

[45]

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. doi: 10.1007/978-3-642-04023-8_11.

[46]

J. D. Murray, Mathematical Biology 1: An Introduction, 3$^{rd}$ edition, Springer, Berlin, 2002.

[47]

P. W. NelsonA. S. Perelson and J. D. Murray, Delay model for the dynamics if HIV infection, Math. Biosci., 163 (2000), 201-215.  doi: 10.1016/S0025-5564(99)00055-3.

[48]

R. M. Nisbet, Delay-differential equations for structured populations, in Structured-Population Models in Marine, Terrestrial, and Freshwater Systems, Springer, Boston, MA, 1997, 89-118. doi: 10.1007/978-1-4615-5973-3_4.

[49]

M. W. Sakdanupaph, A delay differential equation model for Dengue fever transmission in selected countries of South-East Asia, Doctoral dissertation, King Mongkut's University of Technology North Bangkok, 2007. doi: 10.1063/1.3225441.

[50]

E. Savku and G. W. Weber, A stochastic maximum principle for a markov regime-switching jump-diffusion model with delay and an application to finance, J. Optimiz. Theory App., 179 (2018), 696-721.  doi: 10.1007/s10957-017-1159-3.

[51]

H. Y. Seong and Z. A. Majid, Solving second order delay differential equations using direct two-point block method, Ain. Shams. Eng. J., 8 (2017), 59-66.  doi: 10.1016/j.asej.2015.07.014.

[52]

F. Shakeri and M. Dehghan, Solution of delay differential equations via a homotopy perturbation method, Math. Comput. Model., 48 (2008), 486-498.  doi: 10.1016/j.mcm.2007.09.016.

[53]

L. F. Shampine and S. Thompson, Solving ddes in Matlab, App. Num. Math., 37 (2001), 441-458.  doi: 10.1016/S0168-9274(00)00055-6.

[54]

H. I. Siyyam, Laguerre Tau methods for solving higher-order ordinary differential equations, J. Comput. Anal. Appl., 3 (2001), 173-182.  doi: 10.1023/A:1010141309991.

[55]

H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.

[56]

H. L. SuW. 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.  doi: 10.1016/j.jmaa.2012.11.033.

[57]

B. Türkyılmaz, B. Gürbüz and M. Sezer, Morgan-Voyce polynomial approach for solution of high-order linear differential-difference equations with residual error estimation, Düzce Üniversitesi Bilim ve Teknoloji Dergisi, 4 (2016).

[58]

M. Wazewska-Czyzewska 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. 

[59]

Y. YangE. Ishiwata and H. Brunner, On the attainable order of collocation methods for pantograph integro-differential equations, J. Comput. Appl. Math., 152 (2003), 347-366.  doi: 10.1016/S0377-0427(02)00716-1.

[60]

Ş. YüzbaşıN. Şahin and M. Sezer, A Bessel collocation method for numerical solution of generalized pantograph equations, Numer. Meth. Part. D. E., 28 (2012), 1105-1123.  doi: 10.1002/num.20660.

[61]

F. Zhou and X. Xu, Numerical solutions for the linear and nonlinear singular boundary value problems using Laguerre wavelets, Adv. Differ. Equ-Ny, 1 (2016), 17. doi: 10.1186/s13662-016-0754-1.

Figure 1.  The pantograph of a tram (İstiklal Street, İstanbul) [24]
Figure 2.  A representation of a pantograph and its trolley wire system [13]
Figure 3.  $ L_n(t) $ and $ H_n(t) $ values for $ n = 0, 1, 2, 3 $ and $ t\in[0, 2] $
Figure 4.  Exact solution comparison with some approximate solutions for $ N = 6 $ and $ 10 $
Figure 5.  Runge-Kutta method (RKM) solution for $ u_{1}(t) $ of Example 2
Figure 6.  Laguerre collocation method (LCM) solution for $ u_{1}(t) $, $ N = 3 $ of Example 2
Figure 7.  Runge-Kutta method (RKM) solution for $ u_{2}(t) $ of Example 2
Figure 8.  Laguerre collocation method (LCM) solution for $ u_{2}(t) $, $ N = 3 $ of Example 2
Figure 9.  Comparison between RKM and LCM solutions for $ N = 3, 4 $ and $ u_{1}(t) $ of Example 2
Figure 10.  Comparison between RKM and LCM solutions for $ N = 3, 4 $ and $ u_{2}(t) $ of Example 2
Table 1.  $ L_\infty $, $ L_2 $ and $ RMS $ errors for $ N = 3 $
$ t $ $ L_2 $-Error $ L_\infty $-Error $ RMS $-Error
1 0.7560E-05 0.5247E-04 0.1000E-06
2 0.1164E-05 0.3791E-04 0.1502E-05
3 0.1550E-04 0.5467E-03 0.6855E-04
4 0.8259E-03 0.7795E-03 0.1752E-03
5 0.4643E-04 0.5467E-02 0.2916E-05
$ t $ $ L_2 $-Error $ L_\infty $-Error $ RMS $-Error
1 0.7560E-05 0.5247E-04 0.1000E-06
2 0.1164E-05 0.3791E-04 0.1502E-05
3 0.1550E-04 0.5467E-03 0.6855E-04
4 0.8259E-03 0.7795E-03 0.1752E-03
5 0.4643E-04 0.5467E-02 0.2916E-05
Table 2.  $ L_2 $ errors of LCM and HCM for $ N = 3 $ and $ N = 4 $ of Example 2
$ t $ LCM, $ N=3 $ LCM, $ N=4 $ HCM, $ N=3 $ HCM, $ N=4 $
0.0 0.6250E-02 0.1593E-03 0.7081E-02 0.2301E-03
1.2 0.4202E-03 0.2490E-04 0.5223E-02 0.5100E-03
2.3 0.2664E-03 0.2507E-04 0.5708E-02 0.6290E-04
4.5 0.8259E-02 0.4510E-03 0.4430E-01 0.5291E-03
5.0 0.5531E-02 0.7410E-03 0.3548E-01 0.8302E-03
$ t $ LCM, $ N=3 $ LCM, $ N=4 $ HCM, $ N=3 $ HCM, $ N=4 $
0.0 0.6250E-02 0.1593E-03 0.7081E-02 0.2301E-03
1.2 0.4202E-03 0.2490E-04 0.5223E-02 0.5100E-03
2.3 0.2664E-03 0.2507E-04 0.5708E-02 0.6290E-04
4.5 0.8259E-02 0.4510E-03 0.4430E-01 0.5291E-03
5.0 0.5531E-02 0.7410E-03 0.3548E-01 0.8302E-03
Table 3.  CPU comparisons of Example 2
$ N $ LCM
3 1.140
4 1.258
$ N $ LCM
3 1.140
4 1.258
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