Download Abstract Book - the ICAAM 2012 Conference in Gumushane
Download Abstract Book - the ICAAM 2012 Conference in Gumushane
Download Abstract Book - the ICAAM 2012 Conference in Gumushane
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
<strong>Abstract</strong><br />
FUZZY CONTINUOUS DYNAMICAL SYSTEM: A<br />
MULTIVARIATE OPTIMIZATION TECHNIQUE<br />
Abhirup Bandyopadhyay 1 and Samarjit Kar 1<br />
1 Department of Ma<strong>the</strong>matics, National Institute of Technology, Durgapur, India<br />
This paper presents a multivariate optimization technique for <strong>the</strong> numerical simulation of cont<strong>in</strong>-<br />
uous dynamical systems whose parameters, functional forms and/or <strong>in</strong>itial conditions are modeled by<br />
fuzzy distributions. Fuzzy differential equation (FDE) is <strong>in</strong>terpreted by us<strong>in</strong>g <strong>the</strong> strongly generalized<br />
differentiability concept and is shown that by this concept any FDE can be transformed to a system of<br />
ord<strong>in</strong>ary differential equations (ODEs). By solv<strong>in</strong>g <strong>the</strong> associate ODEs one can f<strong>in</strong>d solutions for FDE.<br />
This approach has an <strong>in</strong>herited drawback of <strong>in</strong>creas<strong>in</strong>g uncerta<strong>in</strong>ty at each <strong>in</strong>stance of time generally with<br />
nonl<strong>in</strong>ear functional forms. Here we present a methodology to numerically simulate <strong>in</strong>terval calculus and<br />
implements a new approach to <strong>the</strong> numerical <strong>in</strong>tegration of fuzzy dynamical systems, where <strong>the</strong> propa-<br />
gation of imprecision as a fuzzy distribution <strong>in</strong> <strong>the</strong> phase space is solved by a constra<strong>in</strong>ed multivariate<br />
optimization technique. Numerical simulations of some fuzzy dynamical systems (viz. Lotka Volterra<br />
model, Lorenz model) are also reported. F<strong>in</strong>ally ecological degradation <strong>in</strong> wetlands of India is modeled<br />
by fuzzy <strong>in</strong>itial value problem and some susta<strong>in</strong>able solution is proposed.<br />
References<br />
1. S. Abbasbandy, T. Allahv<strong>in</strong>loo, Numerical solutions of fuzzy differential equations by Taylor<br />
method, Journal of Computational Methods <strong>in</strong> applied Ma<strong>the</strong>matics 2, 113-124, 2002.<br />
2. S. Abbasbandy, T. Allahv<strong>in</strong>loo, O. Lopez-Pouso, J.J Nieto, Numerical methods for fuzzy differential<br />
<strong>in</strong>clusions, Journal of Computer and Ma<strong>the</strong>matics with Applications 48, 1633-1641, 2004.<br />
3. T. Allahv<strong>in</strong>loo, N. Ahmadi, E. Ahmadi, Numerical solution of fuzzy differential equations by<br />
predictor-corrector method, Information Sciences 177, 1633-1647, 2007.<br />
4. B. Bede, Note on “Numerical solutions of fuzzy differential equations by predictor-corrector<br />
method”, Information Sciences, 178, 1917-1922, 2008.<br />
5. B. Bede., S.G Gal, Almost periodic fuzzy-number-valued functions, Fuzzy Sets and Systems 147,<br />
385-403, 2004.<br />
6. B. Bede., S.G Gal S.G., Generalizations of <strong>the</strong> differentiability of fuzzy-number-valued functions<br />
with applications to fuzzy differential equations, Fuzzy Sets and Systems 151, 581-599, 2005.<br />
7. J.C. Butcher, Numerical methods for ord<strong>in</strong>ary differential equations, John Wiley & Sons, Great<br />
Brita<strong>in</strong>, 2003.<br />
8. Y. Chalco-Cano, H. Roman-Flores, On new solutions of fuzzy differential equations,Chaos, Solitons<br />
and Fractals 38, 112-119, 2008.<br />
9. S.L Chang, L.A. Zadeh, On fuzzy mapp<strong>in</strong>g and control, IEEE Trans, Systems Man Cybernet. 2,<br />
30-34, 1972.<br />
10. P. Diamond, Stability and periodicity <strong>in</strong> fuzzy differential equations, IEEE Trans. Fuzzy Systems<br />
8, 583-590, 2000.<br />
Page 21