A Runge–Kutta method with reduced number of function evaluations to solve hybrid fuzzy differential equations

Soft Computing

Volumn 19, Issue 4, 2015, Pages 1051-1062

  Ahmadian, Ali ^a   Salahshour, Soheil ^b   Chan, Chee Seng ^a  

^a UNIVERSITY OF MALAYA   (Malaysia)

^b ISLAMIC AZAD UNIVERSITY   (Iran)

Author keywords

Bede s characterization theorem; Fuzzy ordinary differential equation; High order Runge Kutta method; Hybrid system; Seikkala derivative

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; HYBRID SYSTEMS; ORDINARY DIFFERENTIAL EQUATIONS;

CHARACTERIZATION THEOREMS; FIRST ORDER; FUZZY DIFFERENTIAL EQUATIONS; HIGH-ORDER; NUMERICAL ALGORITHMS;

RUNGE KUTTA METHODS;

EID: 84925290496     PISSN: 14327643     EISSN: 14337479     Source Type: Journal    
DOI: 10.1007/s00500-014-1314-9     Document Type: Article

References (35)

1. Abbod MF, Von Keyserlingk DG, Linkens DA, Mahfouf M (2001) Survey of utilisation of fuzzy technology in medicine and healthcare. Fuzzy Set Syst 120:33117349
2. Ahmadian A, Suleiman M, Salahshour S, Baleanu D (2013) A Jacobi operational matrix for solving a fuzzy linear fractional differential equation. Adv Differ Equ 1:1–29
3. Allahviranloo T, Ahmady N, Ahmady E (2007) Numerical solution of fuzzy differential equations by predictor-corrector method. Inform Sci 177:1633–1647
4. Allahviranloo T, Salahshour S (2011) Euler method for solving hybrid fuzzy differential equation. Soft Comput J 15:1247171253
5. Allahviranloo T, Abbasbandy S, Salahshour S, Hakimzadeh A (2011) A new method for solving fuzzy linear differential equations. Computing 92:181–197
6. Barro S, Marn R (2002) Fuzzy logic in medicine. Physica-Verlag, Heidelberg
7. Bede B (2008) Numerical solutions of fuzzy differential equations by predictor corrector method. Inform Sci 178:1917–1922
8. Casasnovas J, Rossell F (2005) Averaging fuzzy biopolymers. Fuzzy Set Syst 152:13917158
9. Chang BC, Halgamuge SK (2002) Protein motif extraction with neuro-fuzzy optimization. Bioinformatics 18:10841790
10. Dubios D, Prade H (1982) Towards fuzzy differential calculus-part3. Fuzzy Sets Syst 8:225–234
11. Enright WH (1974) Second derivative multi-step methods for stiff ordinary differential equations. SIAM J Numer Anal 11:32117331
12. Feng G, Chen G (2005) Adaptative control of discrete-time chaotic systems: a fuzzy control approach. Chaos, Solitons Fractals 23:4591767
13. Goeken D, Johnson O (2000) Runge-Kutta with higher derivative approximations. Appl Numer Math 39:24917257
14. Guang-Quan Z (1991) Fuzzy continuous function and its properties. Fuzzy Sets Syst 43:159–171
15. Jackiewicz Z, Tracogna S (1995) A general class of two-step Runge-Kutta methods for ordinary differential equations. SIAM J Numer Anal 32(5):1390171427
16. Jiang W, Guo-Dong Q, Bin D (2005) $$H_{\infty }$$H∞ variable universe adaptative fuzzy control for chaotic systems. Chaos, Solitons Fractals 24:10751786
17. Kima H, Sakthivel R (2012) Numerical solution of hybrid fuzzy differential equations using improved predictorcorrector method. Commun Nonlinear Sci Numer Simulat 17:3788173794
18. Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets Syst 24:301–317
19. Kloeden P (1991) Remarks on Peano-like theorems for fuzzy differential equations. Fuzzy Set Syst 44:161174
20. Ma M, Friedman M, Kandel A (1999) Numerical solution of fuzzy differential equations. Fuzzy Sets Syst 105:13317138
21. Palligkinis SCh, Papageorgiou G, Famelis ITh (2009) Runge-Kutta methods for fuzzy differential equations. Appl Math Comput 209:9717105
22. Pederson S, Sambandham M (2009) Numerical solution of hybrid fuzzy differential equation IVPs by a characterization theorem. Inform Sci 179:31917328
23. Pederson S, Sambandham M (2008) The Runge-Kutta method for hybrid fuzzy differential equations. Nonlinear Anal Hybrid Syst 2:62617634
24. Pederson S, Sambandham M (2007) Numerical solution to hybrid fuzzy systems. Math Comput Model 45:1133171144
25. Prakash P, Kalaiselvi V (2009) Numerical solution of hybrid fuzzy differential equations by predictor-corrector method. Int J Comput Math 86:1211734
26. Puri ML, Ralescu D (1983) Differential for fuzzy function. J Math Anal Appl 91:552–558
27. Rosenbrock HH (1963) Some general implicit processes for the numerical solution of differential equations. Comput J 5:32917330
28. Salahshour S, Allahviranloo T, Abbasbandy S (2012) Solving fuzzy fractional differential equations by fuzzy Laplace transforms. Commun Nonlinear Sci Numer Simulat 17:1372–1381
29. Salahshour S, Allahviranloo T, Abbasbandy S, Baleanu D (2012) Existence and uniqueness results for fractional differential equations with uncertainty. Adv Differ Equ 2012:112
30. Salahshour S, Allahviranloo T (2013) Applications of fuzzy Laplace transforms. Soft Comput 17:145–158
31. Salahshour S, Allahviranloo T (2013) Application of fuzzy differential transform method for solving fuzzy Volterra integral equations. Appl Math Model 37:1016–1027
32. Seikkala S (1987) On the fuzzy initial value problem. Fuzzy Sets Syst 24:319–330
33. Wu C, Song S, Stanley Lee E (1996) Approximate solution, existence and uniqueness of the Cauchy problem of fuzzy differential equations. J Math Anal Appl 202:629–644
34. Xu J, Liao Z, Hu Z (2007) A class of linear differential dynamical systems with fuzzy initial condition. Fuzzy Sets Syst 158:2339–2358
35. Zhang H, Liao X, Yu J (2005) Fuzzy modeling and synchronization of hyperchaotic systems. Chaos, Solitons Fractals 26:8351743