On the Solution of the Delay Differential Equation via Laplace Transform

Erkan Cimen, Sevket Uncu

Abstract


In this paper, we consider the initial-value problem for a linear second order delay differential equation. We use Laplace transform method for solving this problem. Furthermore, we present examples provided support the theoretical results.


Keywords


Delay differential equation; Initial-value problem; Laplace transform method

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References


C. T. H. Baker, G. A. Bocharov, C. A. H. Paul and F. A. Rihan, Modelling and analysis of time-lags in some basic patterns of cell proliferation, Journal of Mathematical Biology 37(4) (1998), 341 – 371, DOI: 10.1007/s002850050133.

B. Balachandran, T. K. Nagy and D. E. Gilsinn, Delay Differential Equations, New York, Springer (2009), URL: https://www.springer.com/gp/book/9780387855943.

A. Bellen, S. Maset, M. Zennaro and N. Guglielmi, Recent trends in the numerical solution of retarded functional differential equations, Acta Numerica 18 (2009), 1 – 110, DOI: 10.1017/S0962492906390010.

A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford, Oxford University Press (2003), URL: https://global.oup.com/academic/product/numerical-methods-for-delay-differential-equations-9780198506546.

M. Benchohra and S. Abbas, Advanced Functional Evolution Equations and Inclusions, Switzerland, Springer (2015), URL: https://www.springer.com/gp/book/9783319177670.

L. Berezansky, A. Domoshnitsky, M. Gitman and V. Stolbov, Exponential stability of a second order delay differential equation without damping term, Applied Mathematics and Computation 258 (2015), 483 – 488, DOI: 10.1016/j.amc.2015.01.114.

A. Beuter, J. Belair and C. Labrie, Feedback and delays in neurological diseases: a modeling study using dynamical systems, Bulletin of Mathematical Biology 55(3) (1993), 525 – 541, DOI: 10.1007/BF02460649.

E. Cimen, A first order convergent numerical method for solving the delay differential problem, International Journal of Mathematics and Computer Science 14(2) (2019), 387 – 402, URL: http://ijmcs.future-in-tech.net/14.2/R-ErkanCimen.pdf.

I. V. Curato, M. E. Mancino and M. C. Recchioni, Spot volatility estimation using the Laplace transform, Econometrics and Statistics 6 (2018), 22 – 43, DOI: 10.1016/j.ecosta.2016.07.002.

R. D. Driver, Ordinary and Delay Differential Equations, New York, Springer-Verlag (1977), URL: https://www.springer.com/gp/book/9780387902319.

E. Eljaoui, S. Melliani and L. S. Chadli, Aumann fuzzy improper integral and its application to solve fuzzy integro-differential equations by Laplace transform method, Advances in Fuzzy Systems 2018 (2018), Article ID 9730502, 10 pages, DOI: 10.1155/2018/9730502.

L. E. El’sgolts and S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments, New York, Academic Press (1973), URL: https://www.sciencedirect.com/bookseries/mathematics-in-science-and-engineering/vol/105.

H. Fatoorehchi and M. Alidadi, The extended Laplace transform method for mathematical analysis of the Thomas-Fermi equation, Chinese Journal of Physics 55(6) (2017), 2548 – 2558, DOI: 10.1016/j.cjph.2017.10.001.

C. Foley and M. C. Mackey, Dynamic hematological disease: a review, Journal of Mathematical Biology 58(1-2) (2009), 285 – 322, DOI: 10.1007/s00285-008-0165-3.

P. Getto and M. Waurick, A differential equation with state-dependent delay from cell population biology, Journal of Differential Equations 260(7) (2016), 6176 – 6200, DOI: 10.1016/j.jde.2015.12.038.

D. E. Gilsinn, Estimating critical Hopf bifurcation parameters for a second-order delay differential equation with application to machine tool chatter, Nonlinear Dynamics 30(2) (2002), 103 – 154, DOI: 10.1023/A:1020455821894.

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Netherlands, Springer (1992), URL: https://www.springer.com/gp/book/9780792315940.

S. Guo and W. Ma, Global behavior of delay differential equations model of HIV infection with apoptosis, Discrete & Continuous Dynamical Systems B 21(1) (2016), 103 – 119, DOI: 10.3934/dcdsb.2016.21.103.

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, New York, Springer-Verlag (1993), URL: https://www.springer.com/gp/book/9780387940762.

J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, Proceedings of the National Academy of Sciences of the United States of America 81(10) (1984), 3088 – 3092, DOI: 10.1073/pnas.81.10.3088.

V. A. Ilea and D. Otrocol, Some properties of solutions of a functional-differential equation of second order with delay, The Scientific World Journal 2014 (2014), Article ID 878395, 8 pages, DOI: 10.1155/2014/878395.

V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Netherlands, Kluwer Academic Publishers (1999), URL: https://www.springer.com/gp/book/9780792355045.

W. Li, C. Huang and S. Gan, Delay-dependent stability analysis of trapezium rule for second order delay differential equations with three parameters, Journal of the Franklin Institute 347(8) (2010), 1437 – 1451, DOI: 10.1016/j.jfranklin.2010.06.013.

E. Liz and G. Röst, Global dynamics in a commodity market model, Journal of Mathematical Analysis and Applications 398(2) (2013), 707 – 714, DOI: 10.1016/j.jmaa.2012.09.024.

D. S. Mitrinovic, J. E. Pecaric and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Dordrecht, Springer (1991), URL: https://www.springer.com/gp/book/9780792313304.

H. Rezaei, S. M. Jung and T. M. Rassias, Laplace transform and Hyers-Ulam stability of linear differential equations, Journal of Mathematical Analysis and Applications 403(1) (2013), 244 – 251, DOI: 10.1016/j.jmaa.2013.02.034.

F. A. Rihan, S. Lakshmanan and H. Maurer, Optimal control of tumour-immune model with timedelay and immuno-chemotherapy, Applied Mathematics and Computation 353 (2019), 147 – 165, DOI: 10.1016/j.amc.2019.02.002.

J. L. Schiff, The Laplace Transform Theory and Applications, New York, Springer-Verlag (1999), URL: https://www.springer.com/gp/book/9780387986982.

H. Y. Seong and Z. A. Majid, Solving second order delay differential equations using direct two-point block method, Ain Shams Engineering Journal 8(1) (2017), 59 – 66, DOI: 10.1016/j.asej.2015.07.014.

R. Vallee, M. Dubois, M. Cote and C. Delisle, Second-order differential-delay equation to describe a hybrid bistable device, Physical Review A 36(3) (1987), 1327 – 1332, DOI: 10.1103/PhysRevA.36.1327.

M. Villasana and A. Radunskaya, A delay differential equation model for tumor growth, Journal of Mathematical Biology 47(3) (2003), 270 – 294, DOI: 10.1007/s00285-003-0211-0.

Y. Wang, H. Lian and W. Ge, Periodic solutions for a second order nonlinear functional differential equation, Applied Mathematics Letters 20(1) (2007), 110 – 115, DOI: 10.1016/j.aml.2006.02.028.

S. Zarrinkamar, H. Panahi and F. Hosseini, Laplace transform approach for one-dimensional Fokker-Planck equation, U.P.B. Scientific Bulletin Series A 79(3) (2017), 213 – 220, URL: https://www.scientificbulletin.upb.ro/rev_docs_arhiva/fullcd7_405563.pdf.

J. Zhao, Y. Fan and Y. Xu, Delay-dependent stability of symmetric Runge-Kutta methods for second order delay differential equations with three parameters, Applied Numerical Mathematics 117 (2017), 103 – 114, DOI: 10.1016/j.apnum.2017.03.005.




DOI: http://dx.doi.org/10.26713%2Fcma.v11i3.1359

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