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doi: 10.3934/jimo.2021069

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 Published  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 & Management Optimization, doi: 10.3934/jimo.2021069
References:
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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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

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

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

[9]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[35]

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

[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. Google Scholar

[37]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[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.  Google Scholar

[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/. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[46]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[55]

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24]">Figure 1.  The pantograph of a tram (İstiklal Street, İstanbul) [24]
13]">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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