Advertisement

Fully Fuzzy Semi-linear Dynamical System Solved by Fuzzy Laplace Transform Under Modified Hukuhara Derivative

  • Purnima Pandit
  • Payal SinghEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1048)

Abstract

Semi-linear dynamical systems draw attention in many useful real world problems like population model, epidemic model, etc., they also occur in various applications involving parabolic equations. Now, when the modelling of such applications has inbuilt possibilistic uncertainty, it can be efficiently realized using fuzzy numbers. In this paper, we modify the existing Hukuhara derivative and give the pertaining results for it. We also redefine the Fuzzy Laplace Transform (FLT) and use it to solve such fully fuzzy semi-linear dynamical system.

Keywords

Fuzzy semi-linear dynamical system Fuzzy differential equation (FDE) Fuzzy Laplace transform (FLT) Modified Hukuhara derivative (mH-derivative) Fuzzy convolution theorem 

References

  1. 1.
    Zadeh L.A.: Fuzzy sets and systems. In: Proceedings of Syrup on Systems Theory. Polytechnic Institute Press, Brooklyn, NY (1965)Google Scholar
  2. 2.
    Chang, S.L., Zadeh, L.A.: On fuzzy mapping and control. IEEEE Trans. Syst., Man Cybern. 2, 30–34 (1972)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dubois, D., Prade, H.: Towards fuzzy differential calculus, part 3, differentiation. Fuzzy Sets Syst. 8, 225–233 (1982)CrossRefGoogle Scholar
  4. 4.
    Nazaroff, G.J.: Fuzzy topological polysystems. J. Math. Anal. Appl. 91, 478–485 (1973)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Puri M.L., Ralescu: Differential of fuzzy functions. J. Math. Anal. Appl. 91, 321–325 (1983)Google Scholar
  6. 6.
    Buckley, J.J., Feuring, T.: Fuzzy differential equations. Fuzzy Sets Syst. 110, 43–54 (2000)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Buckley, J.J., Jowers, L.: Simulating Continuous Fuzzy Systems. Springer, Berlin, Heidelberg (2006)zbMATHGoogle Scholar
  8. 8.
    Bede, B., Gal, S.G.: Generalization of the differentiability of fuzzy number valued function with application to fuzzy differential equation. Fuzzy Sets Syst. 151, 581–599 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kaleva, O.: Fuzzy differential equations. Fuzzy Sets Syst. 24, 301–317 (1987)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Seikala, S.: On the fuzzy initial value problem. Fuzzy Sets Syst. 24(3), 319–330 (1987)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Song, S., Wu, C.: Existence and uniqueness of solutions to Cauchy problem of fuzzy differential equations. Fuzzy Sets Syst. 110, 55–67 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lupulescu, V.: Initial value problem for fuzzy differential equations under dissipative conditions. Inf. Sci. 178, 4523–4533 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Nieto, J.J.: Rodríguez-Lopez, Euler polygonal method for metric dynamical systems. Inf. Sci. 177, 4256–4270 (2007)CrossRefGoogle Scholar
  14. 14.
    Hullermiere, E.: Numerical methods for fuzzy initial value problems. Int. J. Uncertain., Fuzziness Knowl.-Based Syst. 7(5), 439–461 (1999)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ma, M., Friedman, M., Kandel, A.: Numerical solutions of fuzzy differential equations. Fuzzy Sets Syst. 105, 133–138 (1999)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Nieto, J.J.: The cauchy problem for continuous fuzzy differential equations. Fuzzy Sets Syst. 102, 259–262 (1999)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Abbasbandy, S., Allahviranloo, T.: Numerical solutions of fuzzy differential equations by taylor method. Comput. Methods Appl. Math. 2, 113–124 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Abbasbandy, S., Allahviranloo, T.: Numerical solutions of fuzzy differential equations by runge-kutta method of order 2. Nonlinear Stud. 11(1), 117–129 (2004)MathSciNetGoogle Scholar
  19. 19.
    Abbasbandy, S., Allahviranloo, T., Darabi, P.: Numerical solutions of N-order fuzzy differential equations by runge-kutta method. Math. Comput. Appl. 16, 935–946 (2011)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Allahviranloo, T., Ehmady, N., Ehmady E.: Numerical solutions of fuzzy differential equations by predictor-corrector method. Inf. Sci. 177, 1633–1647 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Parandin, N.: Numerical solutions of fuzzy differential equations by runge-kutta method of 2nd order. J. Math. Ext. 7, 47–62 (2013)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Bede, B.: Note on numerical solutions of fuzzy differential equations by predictor corrector method. Inf. Sci. 178, 1917–1922 (2008)CrossRefGoogle Scholar
  23. 23.
    Abbasbandy, S., Allahviranloo, T., Lopez, O., Nieto, J.J.: Numerical methods for fuzzy differential inclusions. Comput. Math Appl. 48, 1633–1641 (2004)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Jayakumar, T., Kanagarajan, K., Indrakumar, S.: Numerical solution of nth-order fuzzy differential equation by runge-kutta method of order five. Int. J. Math. Anal. 6, 2885–2896 (2012)zbMATHGoogle Scholar
  25. 25.
    Ghazanfari, B., Shakerami, A.: Numerical solution of fuzzy differential equations by extended Runge-Kutta like formula of order 4. Fuzzy sets and syst. 189, 74–91 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Bede, B., Rudas, I.J., Bencsik: First order linear fuzzy differential equation under generalized differentiability. Inform. Sci. 177, 1648–1662 (2007)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Georgion, D.N., Nieto, J.J., Rodrigue-Lopez, R.: Initial value problem for higher order fuzzy differential equations. Nonlinear Anal. 63(4), 587–600 (2005)Google Scholar
  28. 28.
    Buckley, J.J., Feuring, T.: Fuzzy initial value problem for nth-order linear differential equations. Fuzzy Sets Syst. 121(2), 247–255 (2001)CrossRefGoogle Scholar
  29. 29.
    Mosleh, M.: Fuzzy neural network for solving a system of fuzzy differential equations. Appl. Soft Comput. 13, 3597–3607 (2013)CrossRefGoogle Scholar
  30. 30.
    Mosleh, M., Otadi, M.: Simulation and evaluation of fuzzy differential equations by fuzzy neural network. Appl. Soft Comput. 12, 2817–2827 (2012)CrossRefGoogle Scholar
  31. 31.
    Purnima, P., Payal, S.: Prey-Predator model and fuzzy initial condition. Int. J. Eng. Innov. Technol. (IJEIT) 3(12) (2014)Google Scholar
  32. 32.
    Buckley, J.J., Feuring, T., Hayashi, Y.: Linear System of first order ordinary differential equations: fuzzy initial conditions. Soft. Comput. 6, 415–421 (2002)CrossRefGoogle Scholar
  33. 33.
    Oberguggenberger, M., Pittschmann: Differential equations with fuzzy parameters. Math. Comput. Model. Dyn. Syst. 5(3), 181–202 (1999)Google Scholar
  34. 34.
    Purnima, P., Payal, S.: Numerical technique to solve dynamical system involving fuzzy parameters. Int. J. Emerg. Trends Technol. Comput. Sci. (IJETTCS) 6(4), 051–057 (2017). ISSN 2278-6856Google Scholar
  35. 35.
    Xu, J., Zhigao, L., Neito, J.J.: A class of linear differential dynamical system with fuzzy matrices. J. Math. Anal. Appl. 368, 54–68 (2010)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Ghazanfari, B., Niazi, S., Ghazanfari, A.G.: Linear matrix differential dynamical system with fuzzy matrices. Appl. Math. Model. 36, 348–356 (2012)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Pandit, P., Payal, S.: Fuzzy Laplace transform technique to solve linear dynamical system with fuzzy parameters. In: Proceeding International Conference on “Research and Innovations in Science, Engineering and Technology” ICRISET-2017 (2017)Google Scholar
  38. 38.
    Allahviranloo, T., Ahmadi: Fuzzy Laplace transforms. Soft Comput. 14, 235–243 (2010)CrossRefGoogle Scholar
  39. 39.
    Salahshour, S., Allahviranloo, T.: Applications of fuzzy Laplace transforms. Soft. Comput. 17, 145–158 (2013)CrossRefGoogle Scholar
  40. 40.
    Eljaoui, E., Mellani, S., Saadia Chadli, L.: Solving second order fuzzy differential equation by the fuzzy Laplace transform method. Adv. Differ. Equ. (2015).  https://doi.org/10.1186/s13662-015-0414-x
  41. 41.
    Hayder, A.K., Ali, H.F.M.: Fuzzy Laplace transforms for derivatives of higher orders. Math. Theory Model. 4(2) (2014)Google Scholar
  42. 42.
    Sita, C.: On the solutions of first and second order nonlinear initial value problems. In: Proceedings of the World Congress on Engineering 2013, vol. I, WCE 2013, July 3–5, London, U.K. (2013)Google Scholar
  43. 43.
    Wu, H.C.: The improper fuzzy Riemann integral and its numerical integration. Infom. Sci. 111, 109–137 (1999)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Kaleva, A note on fuzzy differential equations. Nonlinear Anal. 64, 895–900 (2006)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Klir, G.J., Yuan, B.: Fuzzy Sets and fuzzy Logic: Theory and Applications. Prentice Hall, Englewood Cliffs, NJ (1995)zbMATHGoogle Scholar
  46. 46.
    Akin, O., Oruc, O.: A Prey Predator model with fuzzy initial values. Hacet. J. Math. Stat. 41(3), 387–395 (2012)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Technology & EngineeringThe Maharaja Sayajirao University of BarodaVadodaraIndia
  2. 2.Department of Applied Sciences, Faculty of Engineering and TechnologyParul UniversityVadodaraIndia

Personalised recommendations