Numerical methods for m-polar fuzzy initial value problems

Document Type : Research Paper

Authors

1 Department of Mathematics, University of Gujrat, Gujrat, Pakistan.

2 Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan.

3 Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey.

Abstract

Some problems in science and technology are modeled using ambiguous, imprecise, or lacking contextual data. In the modeling of some real-world problems, differential equations often involve multi-agent, multi-index, multi-objective, multi-attribute, multi-polar information or uncertainty, rather than single bits. These types of differentials are not well represented by fuzzy differential equations or bipolar fuzzy differential equations. Therefore, m-pole fuzzy set theory can be applied to differential equations to deal with problems with multi-polar information. In this paper, we study differential equations in m extremely fuzzy environments. We introduce the concept gH-derivative of m-polar fuzzy valued function. By considering different types of differentiability, we propose some properties of the gH-differentiability of m-polar fuzzy-valued functions. We consider the m-polar fuzzy Taylor expansion. Using Taylor expansion, Euler's method and modified Euler's method for solving the m-polar fuzzy initial value problem are proposed. We discuss the convergence analysis of these methods. Some numerical examples are described to see the convergence and stability of the proposed method. We compare these methods by computing the global truncation error. From the numerical results, it can be seen that the modified Euler method converges to the exact solution faster than the Euler method.

Keywords

References

  • [1] F. Abbasi and T. Allahviranloo, Computational procedure for solving fuzzy equations, Soft Computing, 25(2021), 2703-2717.
  • [2] S. Abbasbandy and T. Allahviranloo, Numerical solutions of fuzzy differential equations by Taylor method, Journal  of Computational Methods in Applied mathematics, 2(2002), 113-124.
  • [3] M. Akram, m-Polar fuzzy graphs, Studies in Fuzziness and Soft Computing, 371 (2019), Springer.
  • [4] M. Akram, D. Saleem, and T. Allahviranloo, Linear system of equations in  m-polar fuzzy environment, Journal       of Intelligent and Fuzzy Systems, 37(6) (2019), 8251-8266.
  • [5] M. Akram and G. Shahzadi, Certain characterization of m-polar fuzzy graphs by level graphs, Punjab University Journal of Mathematics, 49(1) (2017), 1-12.
  • [6] T. Allahviranloo, N. Ahmady, and E. Ahmady, Numerical solution of fuzzy differential equations by predictor- corrector method, Information Sciences, 177 (2007), 1633-1646.
  • [7] T. Allahviranloo, Z. Gouyandeh, and A. Armand, A full fuzzy method for solving differential equation  based  on taylor expansion, Journal of Intelligent and Fuzzy Systems, 29 (2015), 1039-1055.
  • [8] G. A. Anastassiou, Numerical initial value problems in ordinary differential equations, Prentice Hall, Englewood Clifs, (1971).
  • [9] A. Armand, T. Allahviranloo, and Z. Gouyandeh, Some fundamental results of fuzzy calculus, Iranian Journal of Fuzzy Systems, 15(3) (2018), 27-46.
  • [10] A. Arman, T. Allahviranloo, S. Abbasbandy, and Z. Gouyandeh, The fuzzy generalized Taylor’s expansion with application in fractional differential equations, Iranian Journal of Fuzzy Systems, 16(2) (2019), 55-72.
  • [11] B. Bede, Note on ”Numerical solutions of fuzzy differential equations by predictor-corrector method”, Information Sciences, 178 (2008), 1917-1922.
  • [12] B. Bede and S. G. Gal, Generalization of the differentiabilty of fuzzy number valued function with application to fuzzy differential equations, Fuzzy Sets and Systems, 151 (2005), 581-599.
  • [13] B. Bede, I. J. Rudas, and A. L. Bencsik, First order linear fuzzy differential equations under generalized differen- tiability, Information Sciences, 177 (2007), 1648-1662.
  • [14] B. Bede and L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, 230 (2013), 119-141.
  • [15] S. S. Behzadi and T. Allahviranloo, Solving fuzzy differential  equations  by  using  Picard  method, Iranian Journal of Fuzzy System, 13(3) (2016), 71-81.
  • [16] Y. C. Cano and M. S. Flores, On new solutions of fuzzy differential equations, IEEE Transaction on System Man and Cybernetics, Part-B Cybernetics, 38(1) (2008), 112-119.
  • [17] Y. Chalco-Cano, H. Roman-Flores, and M. D. Jimenez-Gamero, Generalized derivative and π-derivative for set- valued functions, Information Sciences, 181 (2011), 2177-2188.
  • [18] S. S. L. Chang and L. A. Zadeh, On fuzzy mapping and control, IEEE Transaction on Systems, Man and Cyber- netics, 2 (1972), 30-34.
  • [19] J. Chen, S. Li, S. Ma, and X. Wang, m-polar fuzzy sets: An extension of bipolar fuzzy sets, The Scientific World Journal, (2014).
  • [20] D. Dubois and H. Prade, Towards fuzzy differential calculus III, Fuzzy Sets and Systems, 8(3) (1982), 225-233.
  • [21] S. Effati and M. Pakdaman, Artificial neural network approach for solving fuzzy differential equations, Information Science, 180 (2010), 1434-1457.
  • [22] J. Elmi and M. Eftekhari, Dynamic ensemble selection based on hesitant fuzzy  multiple  criteria  decision  making,  Soft Computing, 24 (2020), 12241-12253.
  • [23] J. F. Epperson, An introduction to numerical methods and analysis, John Wiley and Sons, 2013.
  • [24] O. S. Fard and T. Feuring, Numerical solutions for linear system of first order fuzzy differential equations, Infor- mation Science, 181 (2011), 4765-4779.
  • [25] N. A. Gasilov, A. G. Fatullayev, S. E. Amrahov, and A. Khastan, A new approach to fuzzy  initial  value problem,  Soft Computing, 18 (2014), 217-225.
  • [26] B. Ghazanfari and A. Shakerami, Numerical solution of fuzzy differential equations by extended Runge-kutta like formulae of order 4, Fuzzy Sets and Systems, 189 (2012), 74-91.
  • [27] B. Ghazanfari and A. Shakerami, Numerical solution of second order fuzzy differential eqaution using improved Runge-Kutta nystrom method, Mathematical Problems in Engineering, 2013.
  • [28] Z. Guang-Quan, Fuzzy continuous function and its properties, Fuzzy sets and Systems, 43(2) (1991), 159- 171.
  • [29] K. Ivaz, A. Khastan, and J. J. Neito, A numerical method for fuzzy differential equations and hybrid fuzzy differential equations, Abstract and Applied Analysis, Article ID 735128 10 (2013).
  • [30] T. Jayakumar, K. Kanagarajan, and S. Indrakumar, Numerical  solution  of  nth-order  fuzzy  differential  equation by runge-kutta method of order five, International Journal of Mathematical Analysis, 6 (2012), 2885- 2896.
  • [31] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.
  • [32] M. Ma, M. Friedman, and M. Kandel, Numerical solutions of fuzzy differential equations, Fuzzy Sets and Systems, 105 (1999), 133-138.
  • [33] J. J. Nieto, The Cauchy problem for continuous differential equations, Fuzzy Sets  and  Systems,  102  (1999), 259-262.
  • [34] S. Palligkinis, G. Papageorgious, and I. Famelis, Runge-Kutta method for fuzzy differential equations, Applied Mathematics and Computations, 209 (2009), 97-105.
  • [35] N. Parandin, Numerical solutionof fuzzy differential equations of nth order by Runge-Kutta method, Neural Com- puting Applications, 181 (2011), 4765-4779.
  • [36] S. Pederson and M. Sambandham, The Runge-Kutta method for hybrid fuzzy differential equations, Nonlinear Analysis: Hybrid Systems, 2 (2008), 626-634.
  • [37] S. Pederson and M. Sambandham, Numerical solution of hybrid fuzzy differential equation IVPs by characteriza-  tion, Information Science, 179 (2009), 319-328.
  • [38] M. Saqib, M. Akram, S. Bashir, and T. Allahviranloo, Numerical solution of bipolar fuzzy initial value problem, Journal of Intelligent and Fuzzy System, 40(1) (2021), 1309-1341.
  • [39] L. Stefanini and B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval  differ-  ential equations, Nonlinear Analysis, 71 (2009), 1311-1328.
  • [40] S. Tapaswini and S. Chakraverty, A new approach to fuzzy initial value problem by improved Euler method, Fuzzy Information and Engineering, 4(3) (2012), 293-312.
  • [41] S. Tapaswini and S. Chakraverty, Euler based new solution method for fuzzy initial value problems, International Journal of Artificial Intelligence and Soft Computing, 4 (2014), 58-79.
  • [42] S. Tapaswini,  S. Chakraverty,  and T. Allahviranloo,  A new approach  to nth order fuzzy differential equations,  Fuzzy Sets and Systems, 28(2) (2017), 278-300.
  • [43] C. Wu and Z. Gong, On henstock integral of fuzzy number valued functions, Fuzzy Sets and Systems, 120 (2001), 523-532.
  • [44] L. A. Zadeh, Fuzzy sets, Information and control, 8 (1965), 338-353.
Computational Methods for Differential Equations
Volume 11, Issue 3
June 2023
Pages 412-439

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History

  • Receive Date: 27 January 2022
  • Revise Date: 31 December 2022
  • Accept Date: 18 January 2023

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