حل عددی معادلات دیفرانسیل فازی مرتبه n با استفاده از روش آدامز- بشفورث
الموضوعات :
1 - گروه ریاضی, دانشکده علوم, دانشگاه آزاد کرمانشاه, کرمانشاه, ایران.
الکلمات المفتاحية: Fuzzy differential equations, Adams-Bashforth Fuzzy differen, Adams-Bashforth,
ملخص المقالة :
در این مقاله، روشی عددی برای حل معادلات دیفرانسیل مرتبه پیشنهاد شده است. تاکنون روشهای زیادی برای حل معادلات دیفرانسیل فازی مرتبه اول، توسط محققین ارائه شده است. اما روشهای عددی کمتری نسبت به روشهای مرتبه اول، برای حل معادلات دیفرانسیل فازی مرتبه بالا پیشنهاد شده است. در این تحقیق، ابتداء معادله دیفرانسیل مرتبه n به دستگاهی از معادلات دیفرانسیل فازی مرتبه اول تبدیل میشود، سپس از روش آدامز- بشفورث برای حل این دستگاه معادلات استفاده میشود. نهایتاً با ارائه مثالهایی، دقت روش سنجیده میشود.
[1] S. Abbasbandy and T. Allahviranloo, Numerical solution of fuzzy differential equation by Runge-Kutta method, J. Sci. Teacher Training University,1(3), 2002.
[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] S. Abbasbandy, T. Allahviranloo, O. Lopez-Pouso, and J. J. Nieto, Numerical methods for fuzzy differential inclusions, Journal of Computer and Mathematics with Applications, 48 (2004), 1633-1641.
[4] T. Allahviranloo, N. Ahmady, and E. Ahmady, Numerical solution of fuzzy differential equations by predictor-corrector method, Information Sciences, 177 (2007), 1633-1647.
[5] T. Allahviranloo, E. Ahmady, and N. Ahmady, Nth-order fuzzy linear differential equations, Information Sciences, 178 (2008), 1309-1324.
[6] T. Allahviranloo, N. A. Kiani, and N. Motamedi, Solving fuzzy differential equations by differential transformation method, Information Sciences, 179 (2009), 956-966.
[7] M. Barkhordari Ahmadi, N. A. Kiani, N. Mikaeilvand, Extended predictor-corrector methods for solving fuzzy differential equations under generalized differentiability, International Journal of Mathematical Modeling and Computations, 5(2015), 149-171
[8] B. Bede, A note on two-point boundary value problems associated with non-linear fuzzy differential equations, Fuzzy Sets and Systems, 157(2006), 986-989.
[9] B. Bede and S. G. Gal, Generalizations of the differntiability of fuzzy number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems, 151 (2005), 581-599.
[10] J. J. Buckley and T. Feuring, Fuzzy differential equations, Fuzzy Sets and Systems, 110 (2000), 43-54.
[11] C.J. Budd, A. Iserles, Geometric integration: numerical solution of differential equations on manifolds, philosophical transactions: mathematical, Physical and Engineering Sciences: Mathematical 357 (1999) 945–956.
[12] J. Behlke, O. Ristau, A new approximate whole boundary solution of the Lamm differential equation for the analysis of sedimentation velocity experiments, Biophysical Chemistry 95 (2002) 59–68.
[13] R.L. Burden, J.D. Faires, Numerical Analysis.
[14] G.A. Bocharov, F.A. Rihan, Numerical modelling in biosciences using delay differential equations, Journal of Computational and Applied Mathematics 125 (2000) 183–199.
[15] R.V. Culshaw, S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Mathematical Biosciences 165 (2000) 27–39.
[16] P. Diamond, Stability and periodicity in fuzzy differential equations, IEEE Trans, Fuzzy Systems, 8 (2000), 583-590.
[17] P. Diamond, Brief note on the variation of constants formula for fuzzy differential equations, Fuzzy Sets and Systems, 129 (2002), 65-71.
[18] Ming Ma, M. Friedman, and A. Kandel, Numerical solutions of fuzzy differential equations, Fuzzy Sets and Systems, 105 (1999), 133-138.
[19] M. Friedman, M.Ming, and A. Kandel, Numerical solutions of fuzzy differential and integral equations, Fuzzy Sets and Systems, 106 (1999), 35-48.
[20] 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 (2011), 74-91.
[21] D. N. Georgiou, J. J. Nieto, and R. Rodriguez-Lopez, Initial value problems for higher-order fuzzy differential equations, Nonlinear Anal., 63
(2005), 587-600.
[22] H. Gang, H. Kaifen, Controlling chaos in systems described by partial differential equations, Phys. Rev. Lett. 71 (1993) 3794–3797.
[23] A.R. Its, A.G. Izergin, V.E. Korepin, N.A. Slavnov, Differential equations for quantum correlation functions, International Journal of Modern Physics B 4 (1990) 1003–1037.
[24] A.V. Kotikov, Differential equations method: the calculation of vertex-type Feynman diagrams, Physics Letters B 259 (1991) 314–322.
[25] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24
(1987), 301-317.
[26] O. Kaleva, The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Systems, 35 (1990), 389-396.
[27] A. Khastan and K. Ivaz, Numerical solution of fuzzy differential equations by Nystrom method. Chaos, Solitons and Fractals, 41 (2009), 859-868.
[28] H. Kim and R. Sakthivel, Numerical solution of hybrid fuzzy differential equations using improved predictor-corrector method, Commun Nonlinear Sci Numer Simulat, 17 (2012), 3788-3794.
[29] M. Mosleh and M. Otadi, Simulation and evalution of fuzzy differential equations by fuzzy neural network, Applied Soft Computing, 12 (2012), 2817-2827.
[30] J. J. Nieto and R. Rodriguez-Lopez, Bounded solutions for fuzzy differential and integral equations, Chaos, Solitons and Fractals, 27 (2006), 1376-1386.
[31] J. J. Nieto, The Cauchy problem for continuous fuzzy differential equations, Fuzzy Sets and Systems, 102 (1999), 259-262.
[32] R. Norberg, Differential equations for moments of present values in life insurance, Insurance: Mathematics and Economics 17 (1995) 171–180.
[33] H. Ouyang and Y. Wu, On fuzzy differential equations, Fuzzy Sets and Systems, 32 (1989), 321-325.
[34] S. Ch. Palligkinis, G. Papageorgiou, and I. Th. Famelis, Runge-Kutta methods for fuzzy differential equations, Applied Mathematics and Computation, 209 (2009), 97-105.
[35] S. Pederson and M. Sambandham, The Runge-Kutta method for hybrid fuzzy differential equations, Nonlinear Analysis: Hybrid Systems, 2 (2008), 626-634.
[36] Y.Z. Peng, Exact solutions for some nonlinear partial differential equations, Physics Letters A 314 (2003) 401–408.
[37] H. Roman-Flores and M. Rojas-Medar, Embedding of level-continuous fuzzy sets on Banach spaces, Information Sciences, 144 (2002), 227-247.
[38] S. Sekar, A. Sakthivel, Numerical investigation of the n-th order fuzzy differential equations using he’s homotopy perturbation method, International Journal of Pure and Applied Mathematics, 116(2017) , 899-911.
[39] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems, 24 (1987), 319-330.
[40] U. Salzner, P. Otto, J. Ladik, Numerical solution of a partial differential equation system describing chemical kinetics and diffusion in a cell with the aid of compartmentalization, Journal of Computational Chemistry 11 (1990) 194–204.
[41] J.G. Verwer, J.G. Blom, M. van Loon, E.J. Spee, A comparison of stiff ODE solvers for atmospheric chemistry problems, Atmospheric Environment 30 (1996) 49–58.
