A Genetic Programming-based Scheme for Solving Fuzzy Differential Equations

Authors

Department of Mathematics, Gorgan Branch, Gomishan Center, Islamic Azad University, Gomishan, Iran.

Abstract

This paper deals with a new approach for solving fuzzy differential equations based on genetic programming. This method produces some trial solutions and seeks the best of them. If the solution cannot be expressed in a closed analytical form then our method produces an approximation with a controlled level of accuracy. Furthermore, the numerical results reveal the potential of the proposed approach.

Keywords

References

1. S. Abbasbandy, Numerical methods for fuzzy differential inclusions, Computers and Mathematics with Applications, 48 (2004) 1633-1641.
2. S. Abbasbandy and T. AllahViranloo, Numerical solution of fuzzy differential equation by RungeKutta method, Nonlinear Studies, 11 (1) (2004) 117-129.
3. E. Babolian, H. Sadeghi, and Sh. Javadi, Numerically solution of fuzzy differential equations by Adomian method, Applied Mathematics and Computation, 149 (2004) 547-557.
4. B. Bede and S.G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems, 151(2005) 581-599.
5. B. Bede, I.J. Rudas, and A.L. Bencsik, First order linear fuzzy differential equations under generalized differentiability, Information Sciences, 177 (2007) 1648-1662.
6. Y.C. Cano and H.R. Flores, On new solutions of fuzzy differential equations, Chaos, Solitons and Fractals, 38 (1) (2008) 112-119.
7. S.S.L. Chang and L. Zadeh, On fuzzy mapping and control, IEEE Transactions on System, Man and Cybernetics, 2 (1972) 30-34.
8. D. Dubois and H. Prade, Towards fuzzy differential calculus, Fuzzy Sets and Systems, 8(1982) 225-233.
9. M. Friedman, M. Ma, and A. Kandel, Numerical solutions of fuzzy differential and integral equations, Fuzzy Sets and Systems, 106 (1999) 35-48.
10. R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18 (1986) 31-43.
11. E. Hullermeier, Numerical methods for fuzzy initial value problems, International Journal of Uncertainty Fuzziness Knowledge-Based Systems, 7 (1999) 439{61.
12. O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987) 301-317.
13. J.R. Koza, Genetic Programming, On the Programming of Computer by Means of Natural Selection, MIT Press: Cambridge, MA, 1992.
14. M.T. Mizukoshi, L.C. Barros, Y. Chalco-Cano, H. Roman-Flores, and R.C. Bassanezi, Fuzzy differential equations and the extension principle, Information Sciences, 177 (2007) 3627-3635.
15. M. O'Neill, Automatic Programming in an Arbitrary Language: Evolving Programs with Grammatical Evolution, PhD thesis, University Of Limerick, Ireland, August, 2001.
16. M. O'Neill and C. Ryan, Grammatical Evolution: Evolutionary Automatic Programming in a Arbitrary Language, volume 4 of Genetic programming. Kluwer Academic Publishers, 2003.
17. M. O'Neill and C. Ryan, Grammatical evolution, IEEE Trans. Evolutionary Computation, 5 (2001) 349{358.
18. M.L. Puri and D. Ralescu, Differential for fuzzy function, Journal of Mathematical Analysis and Applications, 91 (1983) 552-558.
19. C. Ryan, J. J. Collins, and M.O. Neill, Evolving programs for an arbitrary language, in Proceedings of the First European Workshop on Genetic Programming, Volume 1391 of LNCS, W. Banzhaf, Ri. Poli, M. Schoenauer and T.C. Fogarty, (eds.), Springer-Verlag, 8395, Paris, 1415 April 1998.
20. I.G. Tsoulos and I.E. Lagaris, Solving differential equations with genetic programming,7 (2006) 33-54.
21. L.A. Zadeh, Toward a generalized theory of uncertainty (GTU) an outline, Information Sciences, 172 (2005) 1-40.
22. L.A. Zadeh, Fuzzy sets, Information and Control, 8 (1965) 338-353.
Global Analysis and Discrete Mathematics
Volume 2, Issue 1
May 2017
Pages 75-89

Files

History

  • Receive Date: 10 October 2016
  • Revise Date: 04 July 2017
  • Accept Date: 14 March 2017

Share

How to cite

Statistics