Advertisement

\(\mathrm {L}^p\)-calculus Approach to the Random Autonomous Linear Differential Equation with Discrete Delay

  • J. Calatayud
  • J.-C. Cortés
  • M. JornetEmail author
Article
  • 32 Downloads

Abstract

In this paper, we provide a full probabilistic study of the random autonomous linear differential equation with discrete delay \(\tau >0\): \(x'(t)=ax(t)+bx(t-\tau )\), \(t\ge 0\), with initial condition \(x(t)=g(t)\), \(-\tau \le t\le 0\). The coefficients a and b are assumed to be random variables, while the initial condition g(t) is taken as a stochastic process. Using \(\mathrm {L}^p\)-calculus, we prove that, under certain conditions, the deterministic solution constructed with the method of steps that involves the delayed exponential function is an \(\mathrm {L}^p\)-solution too. An analysis of \(\mathrm {L}^p\)-convergence when the delay \(\tau \) tends to 0 is also performed in detail.

Keywords

Random autonomous linear differential equation with discrete delay \(\mathrm {L}^p\) random calculus method of steps uncertainty quantification 

Mathematics Subject Classification

34F05 34K50 60H10 65C30 

Notes

Acknowledgements

This work has been supported by the Spanish Ministerio de Economía y Competitividad Grant MTM2017–89664–P. The author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València.

Compliance with Ethical Standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this article.

References

  1. 1.
    Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics. Springer, New York (2011)CrossRefGoogle Scholar
  2. 2.
    Driver, Y.: Ordinary and Delay Differential Equations. Applied Mathematical Science Series. Springer, New York (1977)CrossRefGoogle Scholar
  3. 3.
    Kuang, Y.: Delay Differential Equations: with Applications in Population Dynamics. Academic Press, Cambridge (2012)Google Scholar
  4. 4.
    Bocharov, G.A., Rihan, F.A.: Numerical modelling in biosciences using delay differential equations. J. Comput. Appl. Math. 125, 183–199 (2000).  https://doi.org/10.1016/S0377-0427(00)00468-4 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Jackson, M., Chen-Charpentier, B.M.: Modeling plant virus propagation with delays. J. Comput. Appl. Math. 309, 611–621 (2017).  https://doi.org/10.1016/j.cam.2016.04.024 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen-Charpentier, B.M., Diakite, I.: A mathematical model of bone remodeling with delays. J. Comput. Appl. Math. 291, 76–84 (2016).  https://doi.org/10.1016/j.cam.2017.01.005 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Erneux, T.: Applied Delay Differential Equations, Surveys and Tutorials in the Applied Mathematical Sciences Series. Springer, New York (2009)Google Scholar
  8. 8.
    Kyrychko, Y.N., Hogan, S.J.: On the Use of delay equations in engineering applications. J. Vib. Control 16(7–8), 943–960 (2017).  https://doi.org/10.1177/1077546309341100 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Matsumoto, A., Szidarovszky, F.: Delay Differential Nonlinear Economic Models (in Nonlinear Dynamics in Economics, Finance and the Social Sciences), 195–214. Springer-Verlag, Berlin Heidelberg (2010)Google Scholar
  10. 10.
    Harding, L., Neamtu, M.: A dynamic model of unemployment with migration and delayed policy intervention. Comput. Econ. 51(3), 427–462 (2018).  https://doi.org/10.1007/s10614-016-9610-3 CrossRefGoogle Scholar
  11. 11.
    Oksendal, B.: Stochastic Differential Equations. Springer, New York (1998)CrossRefGoogle Scholar
  12. 12.
    Shaikhet, L.: Lyapunov Functionals and Stability of Stochastic Functional Differential Equations. Springer, New York (2013)CrossRefGoogle Scholar
  13. 13.
    Hartung, F., Pituk, M.: Recent Advances in Delay Differential and Differences Equations. Springer-Verlag, Berlin Heidelberg (2014)CrossRefGoogle Scholar
  14. 14.
    Shaikhet, L.: Stability of equilibrium states of a nonlinear delay differential equation with stochastic perturbations. Int. J. Robust Nonlinear Control 27(6), 915–924 (2016).  https://doi.org/10.1002/rnc.3605 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shaikhet, L.: About some asymptotic properties of solution of stochastic delay differential equation with a logarithmic nonlinearity. Funct. Differ. Equ. 4(1–2), 57–67 (2017)MathSciNetGoogle Scholar
  16. 16.
    Fridman, E., Shaikhet, L.: Delay-induced stability of vector second-order systems via simple Lyapunov functionals. Automatica 74, 288–296 (2016).  https://doi.org/10.1016/j.automatica.2016.07.034 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Benhadri, M., Zeghdoudi, H.: Mean square asymptotic stability in nonlinear stochastic neutral Volterra-Levin equations with Poisson jumps and variable delays. Functiones et Approximatio Commentarii Mathematici 58(2), 157–176 (2018).  https://doi.org/10.7169/facm/1657 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Nouri, K., Ranjbar, H.: Improved Euler-Maruyama method for numerical solution of the Itô stochastic differential systems by composite previous-current-step idea. Mediterr. J. Math. 15, 140 (2018).  https://doi.org/10.1007/s00009-018-1187-8 CrossRefzbMATHGoogle Scholar
  19. 19.
    Santonja, F., Shaikhet, L.: Probabilistic stability analysis of social obesity epidemic by a delayed stochastic model. Nonlinear Anal. Real World Appl. 17, 114–125 (2014).  https://doi.org/10.1016/j.nonrwa.2013.10.010 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Santonja, F., Shaikhet, L.: Analysing social epidemics by delayed stochastic models. Discret. Dyn. Nat. Soc. 2012, 13 (2012).  https://doi.org/10.1155/2012/530472. (Article ID 530472)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Liu, L., Caraballo, T.: Analysis of a stochastic 2D-Navier-Stokes model with infinite delay. J. Dyn. Differ. Equ. pp 1–26 (2018).  https://doi.org/10.1007/s10884-018-9703-x
  22. 22.
    Caraballo, T., Colucci, R., Guerrini, L.: On a predator prey model with nonlinear harvesting and distributed delay. Commun. Pure Appl. Anal. 17(6), 2703–2727 (2018).  https://doi.org/10.3934/cpaa.2018128 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Smith, R.C.: Uncertainty Quantification. Theory, Implementation and Applications. SIAM, Philadelphia (2014)zbMATHGoogle Scholar
  24. 24.
    Soong, T.T.: Random Differential Equations in Science and Engineering. Academic Press, New York (1973)zbMATHGoogle Scholar
  25. 25.
    Nouri, K., Ranjbar, H.: Mean square convergence of the numerical solution of random differential equations. Mediterr. J. Math. 12(3), 1123–1140 (2015).  https://doi.org/10.1007/s00009-014-0452-8 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhou, T.: A stochastic collocation method for delay differential equations with random input. Adv. Appl. Math. Mech. 6(4), 403–418 (2014).  https://doi.org/10.4208/aamm.2012.m38 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Shi, W., Zhang, C.: Generalized polynomial chaos for nonlinear random delay differential equations. Appl. Numer. Math. 115, 16–31 (2017).  https://doi.org/10.1016/j.apnum.2016.12.004 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lupulescu, V., Abbas, U.: Fuzzy delay differential equations. Fuzzy Optim. Decis. Mak. 11(1), 91–111 (2012).  https://doi.org/10.1007/s10700-011-9112-7 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Liu, S., Debbouche, A., Wang, J.R.: Fuzzy delay differential equations. On the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths. J. Comput. Appl. Math. 312, 47–57 (2017).  https://doi.org/10.1016/j.cam.2015.10.028 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Krapivsky, P.L., Luck, J.L., Mallick, K.: On stochastic differential equations with random delay. J. Stat. Mech. Theory Exp. (2011).  https://doi.org/10.1088/1742-5468/2011/10/P10008 CrossRefGoogle Scholar
  31. 31.
    Garrido-Atienza, M.J., Ogrowsky, A., Schmalfuss, B.: Random differential equations with random delays. Stoch. Dyn. 11(2–3), 369–388 (2011).  https://doi.org/10.1142/S0219493711003358 MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Khusainov, D.Y., Ivanov, A.F., Kovarzh, I.V.: Solution of one heat equation with delay. Nonlinear Oscil. 12, 260–282 (2009).  https://doi.org/10.1007/s11072-009-0075-3 MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Asl, F.M., Ulsoy, A.G.: Analysis of a system of linear delay differential equations. J. Dyn. Syst. Meas. Control 125, 215–223 (2003).  https://doi.org/10.1115/1.1568121 CrossRefGoogle Scholar
  34. 34.
    Kyrychko, Y.N., Hogan, S.J.: On the use of delay equations in engineering applications. J. Vib. Control 16(7–8), 943–960 (2010).  https://doi.org/10.1177/1077546309341100 MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Villafuerte, L., Braumann, C.A., Cortés, J.C., Jódar, L.: Random differential operational calculus: theory and applications. Comput. Math. Appl. 59(1), 115–125 (2010).  https://doi.org/10.1016/j.camwa.2009.08.061 MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Strand, J.L.: Random ordinary differential equations. J. Diff. Equ. 7(3), 538–553 (1970).  https://doi.org/10.1016/0022-0396(70)90100-2 MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Khusainov, D.Y., Pokojovy, M.: Solving the linear 1d thermoelasticity equations with pure delay. Int. J. Math. Math. Sci. 2015, 1–11 (2015).  https://doi.org/10.1155/2015/479267 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Instituto Universitario de Matemática Multidisciplinar Building 8G, access C, 2nd floorUniversitat Politècnica de ValènciaValenciaSpain

Personalised recommendations