Numerical solution of linear inhomogeneous fuzzy delay differential equations

  • A. G. Fatullayev
  • Nizami A. Gasilov
  • Şahin Emrah AmrahovEmail author


We investigate inhomogeneous fuzzy delay differential equation (FDDE) in which initial function and source function are fuzzy. We assume these functions be in a special form, which we call triangular fuzzy function. We define solution as a fuzzy bunch of real functions such that each real function satisfies the equation with certain membership degree. We develop an algorithm to find the solution, and we provide the existence and uniqueness results for the considered FDDE. We also present an example to show the applicability of the proposed algorithm.


Fuzzy differential equation Fuzzy delay differential equation Fuzzy set 



We are grateful to the Editor-in-Chief, the Associate Editor and the anonymous reviewers for their careful reading of the paper and their valuable comments and suggestions.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Amrahov, Ş. E., Khastan, A., Gasilov, N., & Fatullayev, A. G. (2016). Relationship between Bede–Gal differentiable set-valued functions and their associated support functions. Fuzzy Sets and Systems, 295, 57–71.MathSciNetCrossRefGoogle Scholar
  2. Azbelev, N. V., Maksimov, V. P., & Rakhmatullina, L. F. (2007). Introduction to the theory of functional differential equations: Methods and applications. Cairo: Hindawi Pub. Co.CrossRefGoogle Scholar
  3. Balasubramaniam, P., & Muralisankar, S. (2001). Existence and uniqueness of a fuzzy solution for the nonlinear fuzzy neutral functional differential equation. Computers & Mathematics with Applications, 42(6–7), 961–967.MathSciNetCrossRefGoogle Scholar
  4. Barzinji, K., Maan, N., & Aris, N. (2014). Linear fuzzy delay differential system: Analysis on stability of steady state. Matematika, 30(1a), 1–7.MathSciNetGoogle Scholar
  5. Donchev, T., & Nosheen, A. (2013). Fuzzy functional differential equations under dissipative-type conditions. Ukrainian Mathematical Journal, 65(6), 873–883.MathSciNetCrossRefGoogle Scholar
  6. Epstein, I. R., & Luo, Y. (1991). Differential delay equations in chemical kinetics. Nonlinear models: The cross-shaped phase diagram and the Oregonator. The Journal of Chemical Physics, 95(1), 244–254.CrossRefGoogle Scholar
  7. Gasilov, N. A., & Amrahov, Ş. E. (2018). Solving a nonhomogeneous linear system of interval differential equations. Soft Computing, 22(12), 3817–3828.CrossRefGoogle Scholar
  8. Gasilov, N. A., Amrahov, Ş. E., Fatullayev, A. G., & Hashimoglu, I. F. (2015). Solution method for a boundary value problem with fuzzy forcing function. Information Sciences, 317, 349–368.MathSciNetCrossRefGoogle Scholar
  9. Gasilov, N. A., Hashimoglu, I. F., Amrahov, Ş. E., & Fatullayev, A. G. (2012). A new approach to non-homogeneous fuzzy initial value problem. CMES: Computer Modeling in Engineering & Sciences, 85(4), 367–378.MathSciNetzbMATHGoogle Scholar
  10. Guo, M., Peng, X., & Xu, Y. (2012). Oscillation property for fuzzy delay differential equations. Fuzzy Sets and Systems, 200, 25–35.MathSciNetCrossRefGoogle Scholar
  11. Hale, J. K. (1997). Theory of Functional Differential Equations. New York: Springer.Google Scholar
  12. Hoa, N. V., Tri, P. V., Dao, T. T., & Zelinka, I. (2015). Some global existence results and stability theorem for fuzzy functional differential equations. Journal of Intelligent & Fuzzy Systems, 28(1), 393–409.MathSciNetzbMATHGoogle Scholar
  13. Hüllermeier, E. (1997). An approach to modeling and simulation of uncertain dynamical systems. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 5, 117–137.MathSciNetCrossRefGoogle Scholar
  14. Jafelice, R. M., Barros, L. C., & Bassanezi, R. C. (2009). A fuzzy delay differential equation model for HIV dynamics. In Proceedings of the IFSA/EUSFLAT conference (pp. 265–270).Google Scholar
  15. Jiang, C., & Zhou, D. H. (2005). Fault detection and identification for uncertain linear time-delay systems. Computers & Chemical Engineering, 30(2), 228–242.CrossRefGoogle Scholar
  16. Khastan, A., Nieto, J. J., & Rodríguez-López, R. (2014). Fuzzy delay differential equations under generalized differentiability. Information Sciences, 275, 145–167.MathSciNetCrossRefGoogle Scholar
  17. Kuang, Y. (1993). Delay differential equations with applications in population dynamics. Boston: Academic Press.zbMATHGoogle Scholar
  18. Lupulescu, V., & Abbas, U. (2012). Fuzzy delay differential equations. Fuzzy Optimization and Decision Making, 11(1), 99–111.MathSciNetCrossRefGoogle Scholar
  19. Malinowski, M. T. (2012). Itô type stochastic fuzzy differential equations with delay. Systems & Control Letters, 61(6), 692–701.MathSciNetCrossRefGoogle Scholar
  20. Min, C., Huang, N.-J., & Zhang, L.-H. (2014). Existence of local and global solutions of fuzzy delay differential inclusions. Advances in Difference Equations.
  21. Park, J. Y., & Jeong, J. U. (2013). On random fuzzy functional differential equations. Fuzzy Sets and Systems, 223, 89–99.MathSciNetCrossRefGoogle Scholar
  22. Roussel, M. R. (1996). The use of delay differential equations in chemical kinetics. The Journal of Physical Chemistry, 100(20), 8323–8330.CrossRefGoogle Scholar
  23. Shao, Y. B., Zhang, H. H., & Xue, G. L. (2014). Existence of solutions for the fuzzy functional differential equations. In B. -Y. Cao, & H. Nasseri (Eds.), Fuzzy information and engineering and operations research and management (pp. 215–227). Heidelberg: Springer.Google Scholar
  24. Tri, P. V., Hoa, N. V., & Phu, N. D. (2014). Sheaf fuzzy problems for functional differential equations. Advances in Difference Equations.
  25. Vu, H., & Hoa, N. V. (2016). On impulsive fuzzy functional differential equations. Iranian Journal of Fuzzy Systems, 13(4), 79–94.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • A. G. Fatullayev
    • 1
  • Nizami A. Gasilov
    • 1
  • Şahin Emrah Amrahov
    • 2
    Email author
  1. 1.Baskent UniversityAnkaraTurkey
  2. 2.Computer Engineering DepartmentAnkara UniversityAnkaraTurkey

Personalised recommendations