Applications of Fuzzy Differential Equations on Vibrating Spring Mass System

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-21-2023-120
Published: Oct 30, 2023
Cite as:
B. Divya, K. Ganesan, Applications of Fuzzy Differential Equations on Vibrating Spring Mass System, Int. J. Anal. Appl., 21 (2023), 120.

Main Article Content

B. Divya, K. Ganesan

Abstract

Modelling several real-world issues in the fuzzy world extensively uses ordinary differential equations. In this paper, a mechanical vibration system with the given mass, spring constant, damping and external force is modelled as a second-order ordinary differential equation. Due to measurement errors, the initial displacement of the string is approximate and assumed to be a fuzzy number. A fuzzy version of the Sumudu transform procedure is used to figure out this vibrating spring-mass system with fuzzy initial displacement. The output is displayed as a table at various computational stages. The consequences are visibly presented diagrammatically for different values of r and t. There is a good agreement between the computed results and the analytical solution.

Article Details

References

  1. N. Rahman, M. Ahmad, Applications of the Fuzzy Sumudu Transform for the Solution of First Order Fuzzy Differential Equations, Entropy. 17 (2015), 4582–4601. https://doi.org/10.3390/e17074582.
  2. N.A.A. Rahman, M.Z. Ahmad, Fuzzy Sumudu Transform for Solving Fuzzy Partial Differential Equations, J. Nonlinear Sci. Appl. 09 (2016), 3226–3239. https://doi.org/10.22436/jnsa.009.05.111.
  3. K.S. Aboodh, N.E. Mohamed, A.M. Ali, Solving Ordinary Differential Equations With Constant Coefficients Using Sumudu Transform Method, Int. J. Eng. Inf. Syst. 1 (2017), 70–76.
  4. B. Bede, S.G. Gal, Generalizations of the Differentiability of Fuzzy-Number-Valued Functions With Applications to Fuzzy Differential Equations, Fuzzy Sets Syst. 151 (2005), 581–599. https://doi.org/10.1016/j.fss.2004.08.001.
  5. J.J. Buckley, T. Feuring, Fuzzy Differential Equations, Fuzzy Sets Syst. 110 (2000), 43–54. https://doi.org/10.1016/s0165-0114(98)00141-9.
  6. J.J. Buckley, T. Feuring, Fuzzy Initial Value Problem for Nth-Order Linear Differential Equations, Fuzzy Sets Syst. 121 (2001), 247–255. https://doi.org/10.1016/s0165-0114(00)00028-2.
  7. Y. Chalco-Cano, H. Román-Flores, Comparation Between Some Approaches to Solve Fuzzy Differential Equations, Fuzzy Sets Syst. 160 (2009), 1517–1527. https://doi.org/10.1016/j.fss.2008.10.002.
  8. S.S.L. Chang, L.A. Zadeh, On Fuzzy Mapping and Control, IEEE Trans. Syst., Man, Cybern. SMC-2 (1972), 30–34. https://doi.org/10.1109/TSMC.1972.5408553.
  9. D. Dubois, H. Prade, Operations on Fuzzy Numbers, Int. J. Syst. Sci. 9 (1978), 613–626. https://doi.org/10.1080/00207727808941724.
  10. D. Dubois, H. Prade, Towards Fuzzy Differential Calculus Part 2: Integration on Fuzzy Intervals, Fuzzy Sets Syst. 8 (1982), 105–116. https://doi.org/10.1016/0165-0114(82)90001-x.
  11. D. Dubois, H. Prade, Towards Fuzzy Differential Calculus Part 3: Differentiation, Fuzzy Sets Syst. 8 (1982), 225–233. https://doi.org/10.1016/s0165-0114(82)80001-8.
  12. H. Eltayeb, A. Kilicman, A Note on the Sumudu Transforms and Differential Equations, Appl. Math. Sci. 4 (2010), 1089–1098.
  13. R. Goetschel Jr., W. Voxman, Elementary Fuzzy Calculus, Fuzzy Sets Syst. 18 (1986), 31–43. https://doi.org/10.1016/0165-0114(86)90026-6.
  14. H.Z. Mjthap, S.N. Al-Azzawi, Some Properties of Sumudu Transformation, J. Interdiscip. Math. 22 (2019), 1543– 1547. https://doi.org/10.1080/09720502.2019.1706853.
  15. H.Z. Mjthap, S.N. Al-Azzawi, Mixing Sumudu Transform and Adomain Decomposition Method for Solving Riccati Equation of Variable Fractional Order, J. Interdiscip. Math. 22 (2019), 1559–1563. https://doi.org/10.1080/09720502.2019.1706858.
  16. R. Jafari, S. Razvarz, Solution of Fuzzy Differential Equations Using Fuzzy Sumudu Transforms, Math. Comput. Appl. 23 (2018), 5. https://doi.org/10.3390/mca23010005.
  17. O. Kaleva, Fuzzy Differential Equations, Fuzzy Sets Syst. 24 (1987), 301–317. https://doi.org/10.1016/0165-0114(87)90029-7.
  18. O. Kaleva, The Cauchy Problem for Fuzzy Differential Equations, Fuzzy Sets Syst. 35 (1990), 389–396. https://doi.org/10.1016/0165-0114(90)90010-4.
  19. N.A. Khan, O.A. Razzaq, M. Ayyaz, On the Solution of Fuzzy Differential Equations by Fuzzy Sumudu Transform, Nonlinear Eng. 4 (2015), 49–60. https://doi.org/10.1515/nleng-2015-0003.
  20. A. Kilicman, H. Eltayeb, M.R. Ismail, A Note on Integral Transforms and Differential Equations, Malays. J. Math. Sci. 6 (2012), 1–18.
  21. V. Laksmikantham, Set Differential Equations Versus Fuzzy Differential Equations, Appl. Math. Comput. 164 (2005), 277–294. https://doi.org/10.1016/j.amc.2004.06.068.
  22. M. Sahni, M. Parikh, R. Sahni, Sumudu Transform for Solving Ordinary Differential Equation in a Fuzzy Environment, J. Interdiscip. Math. 24 (2021), 1565–1577. https://doi.org/10.1080/09720502.2020.1845468.
  23. M. Ma, M. Friedman, A. Kandel, A New Fuzzy Arithmetic, Fuzzy Sets Syst. 108 (1999), 83–90. https://doi.org/10.1016/s0165-0114(97)00310-2.
  24. S.P. Mondal, T.K. Roy, First Order Linear Homogeneous Ordinary Differential Equation in Fuzzy Environment Based On Laplace Transform, J. Fuzzy Set Valued Anal. 2013 (2013), 1-18. https://doi.org/10.5899/2013/jfsva-00174.
  25. M.L. Puri, D.A. Ralescu, Differentials of Fuzzy Functions, J. Math. Anal. Appl. 91 (1983), 552–558. https://doi.org/10.1016/0022-247x(83)90169-5.
  26. G.K. Watugala, Sumudu Transform: A New Integral Transform to Solve Differential Equations and Control Engineering Problems, Int. J. Math. Educ. Sci. Technol. 24 (1993), 35–43. https://doi.org/10.1080/0020739930240105.
  27. S. Weerakoon, Application of Sumudu Transform to Partial Differential Equations, Int. J. Math. Educ. Sci. Technol. 25 (1994), 277–283. https://doi.org/10.1080/0020739940250214.
  28. S. Weerakoon, Complex Inversion Formula for Sumudu Transform, Int. J. Math. Educ. Sci. Technol. 29 (1998), 618–621.
  29. L.A. Zadeh, Fuzzy Sets, Inf. Control. 8 (1965), 338–353. https://doi.org/10.1016/s0019-9958(65)90241-x.