Skip to main content
SpringerLink
Log in

Fuzzy Initial Value Problem: A Short Survey

  • Conference paper
  • First Online:
Fuzzy Information Processing (NAFIPS 2018)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 831))

Included in the following conference series:

  • 774 Accesses

Abstract

This article provides a survey of the available literature on Fuzzy Initial Value Problem (FIVP) and various different interpretations. The fuzzy differential equations can be studied using the derivative concept or without it. The Malthusian population model with fuzzy initial condition is used to illustrate the different approaches, namely, Hukuhara derivative, gh-differentiability, \(\pi -\)derivative and Zadeh’s extension applied to derivative operator using the differentiability and differential inclusion theory, Zadeh’s extension principle applied in deterministic solution without derivative concept.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aubin, J.P.: Fuzzy differential inclusions. Probl. Control Inf. Sci. Theory 19(1), 55–67 (1990)

    MathSciNet  MATH  Google Scholar 

  2. Bassanezi, R.C., de Barros, L.C., Tonelli, P.A.: Attractors and asymptotic stability for fuzzy dynamical systems. Fuzzy Sets Syst. 113, 473–483 (2000)

    Article  MathSciNet  Google Scholar 

  3. Barros, L.C., Bassanezi, R.C., Lodwick, A.W.: A First Course in Fuzzy Logic, Fuzzy Dynamical Systems, and Biomathematics: Theory and Applications. Studies in Fuzziness and Soft Computing. Springer, New York (2017). https://doi.org/10.1007/978-3-662-53324-6

    Book  MATH  Google Scholar 

  4. Baidosov, V.A.: Fuzzy differential inclusions. Prikl. Matem. Mekhan. 54(1), 12–17 (1990)

    MathSciNet  Google Scholar 

  5. Barros, L.C., Bassanezi, R.C., Tonelli, P.A.: On the continuity of Zadeh’s extension. In: Proceedings of the 7th IFSA World Congress, Praga, pp. 3–8 (1997)

    Google Scholar 

  6. Bede, B., Gal, S.G.: Generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equation. Fuzzy Set Syst. 151, 581–599 (2005)

    Article  MathSciNet  Google Scholar 

  7. Bhaskar, T.G., Lakshmikantham, V., Devi, V.: Revisiting fuzzy differential equations. Nonlinear Anal. 58, 351–358 (2004)

    Article  MathSciNet  Google Scholar 

  8. Buckley, J.M., Eslami, E., Feuring, T.: Fuzzy Mathematics in Economics and Enginnering. Springer, New York (2002). https://doi.org/10.1007/978-3-7908-1795-9

    Book  MATH  Google Scholar 

  9. Chalco-Cano, Y., Román-Flores, H., Jiménez-Gamero, M.D.: Fuzzy differential equations with \(\pi -\)derivative. In: Proceedings of the Joint 2009 International Fuzzy Systems Association World Congress and 2009 European Society of Fuzzy Logic and Technology Conference, Lisbon, pp. 20–24 (2009)

    Google Scholar 

  10. Cecconello, M.S., Bassanezi, R.C., Brandão, A.V., Leite, J.: Periodic orbits for fuzzy flows. Fuzzy Sets Syst. 230, 21–38 (2013)

    Article  MathSciNet  Google Scholar 

  11. Chen, M., Li, D., Xue, X.: Periodic problems of first order uncertain dynamical systems. Fuzzy Sets Syst. 162, 67–78 (2011)

    Article  MathSciNet  Google Scholar 

  12. Corliss, G.F.: Survey of interval algorithms for ordinary differential equations. Appl. Math. Comput. 31, 112–120 (1989)

    MathSciNet  MATH  Google Scholar 

  13. Diamond, P.: Time-dependent differential inclusions, cocycle attractors and fuzzy differential equations. IEEE Trans. Fuzzy Syst. 7, 734–740 (1999)

    Article  Google Scholar 

  14. Diamond, P.: Stability and periodicity in fuzzy differential equations. IEEE Trans. Fuzzy Syst. 8, 583–590 (2000)

    Article  Google Scholar 

  15. Diamond, P.: Brief note on the variation of constants formula for fuzzy differential equations. Fuzzy Sets Syst. 129, 65–71 (2002)

    Article  MathSciNet  Google Scholar 

  16. Gomes, L.T., de Barros, L.C., Bede, B.: Fuzzy differential equations. Fuzzy Differential Equations in Various Approaches. SM, pp. 69–113. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-22575-3_4

    Chapter  MATH  Google Scholar 

  17. Gomes, L.T.: On fuzzy differential equations. Ph.D. thesis, IMECC, UNICAMP (2014)

    Google Scholar 

  18. Hullermeier, E.: An approach to modeling and simulation of uncertain dynamical systems. Int. J. Uncertain. Fuzziness Knowl.-Bases Syst. 5, 117–137 (1997)

    Article  Google Scholar 

  19. de Silva, J.D.M., Leite, J., Bassanezi, R.C., Cecconello, M.S.: Stationary points-I: one-dimensional p-fuzzy dynamical systems. J. Appl. Math. 2013, 1–11 (2013)

    Article  MathSciNet  Google Scholar 

  20. Jafelice, R.S.M., de Barros, L.C., Bassanezi, R.C., Gomide, F.: Methodology to determine the evolution of asymptomatic HIV population using fuzzy set theory. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 13(1), 39–58 (2005)

    Article  Google Scholar 

  21. Kaleva, O.: Fuzzy differential equations. Fuzzy Sets Syst. 24, 301–317 (1987)

    Article  MathSciNet  Google Scholar 

  22. Kandel, A., Byatt, W.J.: Fuzzy differential equations. In: Proceedings of the International Conference on Cybernetics and Society, Tokyo, pp. 1213–1216 (1978)

    Google Scholar 

  23. Kandel, A., Byatt, W.J.: Fuzzy processes. Fuzzy Sets Syst. 4(2), 117–152 (1980)

    Article  MathSciNet  Google Scholar 

  24. Khastan, A., Nieto, J.J., Rodríguez-López, R.: Periodic boundary value problems for the first-order linear differential equations with uncertainty under generalized differentiability. Inf. Sci. 222, 544–558 (2013)

    Article  MathSciNet  Google Scholar 

  25. Kloeden, P.E., Schmalfuss, B.: Asymptotic behaviour of nonautonomous difference inclusions. Syst. Control Lett. 33, 275–280 (1998)

    Article  MathSciNet  Google Scholar 

  26. Krivan, V., Colombo, G.: A non-stochastic approach for modelling uncertainty in population dinamics. Bull. Math. Biol. 60, 721–751 (1998)

    Article  Google Scholar 

  27. Kupka, J.: On devaney chaotic induced fuzzy and set-valued dynamical systems. Fuzzy Sets Syst. 177, 34–44 (2011)

    Article  MathSciNet  Google Scholar 

  28. Lakshmikantham, V., Vatsala, A.S.: Basic theory of fuzzy difference equations. J. Differ. Equ. Appl. 8, 957–968 (2002)

    Article  MathSciNet  Google Scholar 

  29. Lodwick, W.A.: Constrained interval arithmetic. Technical report, University Of Colorado at Denver, Denver (1999)

    Google Scholar 

  30. Lodwick, W.A., Dubois, D.: Interval linear systems as a necessary step in fuzzy linear systems. Fuzzy Sets Syst. 281, 227–251 (2015)

    Article  MathSciNet  Google Scholar 

  31. Marchaud, A.: Sur les champs de demicônes et les équations differentielles du premier ordre. Bull. Soc. Math. France 62, 1–38 (1934)

    MathSciNet  MATH  Google Scholar 

  32. Mizukoshi, M.T., de Barros, L.C., Bassanezi, R.C.: Stability of fuzzy dynamics systems. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 17, 69–83 (2009)

    Article  MathSciNet  Google Scholar 

  33. Mizukoshi, M.T., Barros, L.C., Chalco-Cano, Y., Roman-Flores, H., Bassanezi, R.C.: Fuzzy differential equation and the extension principle. Inf. Sci. 177, 3627–3635 (2007)

    Article  MathSciNet  Google Scholar 

  34. Moore, R.E.: Automatic error analysis in digital computation, Technical Report LMSD-48421, Lockheed Missile and Space Division, Sunnyvale, CA (1959). http://interval.lousiana.edu/Moores_early_papers/bibliography.html

  35. Nieto, J.J., Rodríguez-López, R.: Bounded solutions for fuzzy differential and integral equations. Chaos, Solitons Fractals 27, 1376–1386 (2006)

    Article  MathSciNet  Google Scholar 

  36. Oberguggenberger, M., Pittschmann, S.: Differential equations with fuzzy parameters. Math. Comput. Model. Dyn. Syst. 5, 181–202 (1999)

    Article  Google Scholar 

  37. Ouyang, H., Wu, Y.: On fuzzy differential equations. Fuzzy Sets Syst. 32, 321–325 (1989)

    Article  MathSciNet  Google Scholar 

  38. Peixoto, M.S., Barros, L.C., Bassanezi, R.C.: Predator-prey fuzzy model. Ecol. Model. 214, 39–44 (2008)

    Article  Google Scholar 

  39. Puri, M.L., Ralescu, D.A.: Differentials for fuzzy functions. J. Math. Anal. Appl. 91, 552–558 (1983)

    Article  MathSciNet  Google Scholar 

  40. Puri, M.L., Ralescu, D.A.: Fuzzy random variables. J. Math. Anal. Appl. 114, 409–422 (1986)

    Article  MathSciNet  Google Scholar 

  41. Rihm, R.: Interval methods for initial value problems in ODEs. In: Topics in Validated Computations, pp. 173–206 (1994)

    Google Scholar 

  42. Román-Flores, H., Barros, L.C., Bassanezi, R.C.: A note on Zadeh’s extension. Fuzzy Sets Syst. 117, 327–331 (2001)

    Article  MathSciNet  Google Scholar 

  43. Román-Flores, H.: A note on transitivity in set-valued discrete systems. Chaos, Solitons Fractals 17(1), 99–104 (2003)

    Article  MathSciNet  Google Scholar 

  44. Rouvray, D.H.: The treatment of uncertainty in the sciences. Endevour 21(4), 154–158 (1997)

    Article  Google Scholar 

  45. Seikkala, S.: On the fuzzy initial value problem. Fuzzy Sets Syst. 24, 309–330 (1987)

    Article  MathSciNet  Google Scholar 

  46. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  Google Scholar 

  47. Zaremba, S.K.: Sur les équations au paratingent. Bull. Sci. Math. 60, 139–160 (1936)

    MATH  Google Scholar 

  48. Wasewski, T.: Sur une genéralisation de la notion des solutions d’une équation au contingent. Bull. Acad. Pol. Sci. Ser. Math. Astronom. Phys. 10(1), 1–15 (1962)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade Federal de Goiás, Campus II - Samambaia, Goiânia, GO, 74690-900, Brazil

    Marina Tuyako Mizukoshi

Authors
  1. Marina Tuyako Mizukoshi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marina Tuyako Mizukoshi .

Editor information

Editors and Affiliations

  1. Department of Teleinformatics Engineering, Federal University of Ceará, Fortaleza, Ceará, Brazil

    Guilherme A. Barreto

  2. Department of Statistics & Applied Mathematics, Federal University of Ceará, Fortaleza, Ceará, Brazil

    Ricardo Coelho

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Mizukoshi, M.T. (2018). Fuzzy Initial Value Problem: A Short Survey. In: Barreto, G., Coelho, R. (eds) Fuzzy Information Processing. NAFIPS 2018. Communications in Computer and Information Science, vol 831. Springer, Cham. https://doi.org/10.1007/978-3-319-95312-0_40

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-95312-0_40

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-95311-3

  • Online ISBN: 978-3-319-95312-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation