Numerical solutions of nonlinear fuzzy Fredholm integro-differential equations of\ the second kind

Document Type: Research Paper

Authors

Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran

Abstract

In this paper, we use parametric form of fuzzy number, then an
iterative approach for obtaining approximate solution for a class
of nonlinear fuzzy Fredholm
integro-differential equation of the second kind
is proposed. This paper presents a method based on Newton-Cotes
methods with positive coefficient. Then we obtain approximate
solution of the nonlinear fuzzy integro-differential equations by an iterative
approach.

Keywords


[1] S. Abbasbandy and T. Allahviranloo,Numerical solution of fuzzy di erential equation by
Runge-Kutta method, Nonlinear studies, 11(1) (2004), 117-129.
[2] S. Abbasbandy, T. Allaviranloo, O. Lopez-Pouso and J. J. Nieto, Numerical methods for
fuzzy di erential inclusions, Computers & mathematics with applications, 48(10-11) (2004),
1633-1641.
[3] S. Abbasbandy and B. Asady, Newtons method for solving fuzzy nonlinear equations, Applied
Mathematics and Computation, 159(2) (2004), 349-356.
[4] S. Abbasbandy, E. Babolian and M. Alavi, Numerical method for solving linear Fredholm
fuzzy integral equations of the second kind, Chaos Solitons & Fractals, 31(1) (2007), 138-
146.
[5] S. Abbasbandy and A. Jafarian, Steepest descent method for solving fuzzy nonlinear equa-
tions, Applied Mathematics and Computation, 175(1) (2006), 581-589.
[6] S. Abbasbandy, J. J. Nieto and M. Alavi, Tuning of reachable set in one dimensional fuzzy
di erential inclusions, Chaos, Solitons & Fractals, 26(5) (2005), 1337-1341.
[7] T. Allahviranloo, S. Abbasbandy, N. Ahmady and E. Ahmady, Improved predictorcorrector
method for solving fuzzy initial value problems, Information Sciences, 179(7) (2009), 945-955.
[8] T. Allahviranloo, N. Ahmady and E. Ahmady, Numerical solution of fuzzy di erential equa-
tions by predictorcorrector method, Information Sciences, 177(7) (2007), 1633-1647.
[9] T. Allahviranloo, N. A. Kiani and M. Barkhordari,Toward the existence and uniqueness of
solutions of second-order fuzzy di erential equations, Information Sciences, 179(8) (2009),
1207-1215.
[10] T. Allahviranloo, N. A. Kiani and N. Motamedi, Solving fuzzy di erential equations by dif-
ferential transformation method, Information Sciences, 179(7) (2009), 956-966.
[11] K. E. Atkinson,An introduction to numerical analysis, New York: Wiley, 1987.
[12] E. Babolian, H. S. Goghary and S. Abbasbandy,Numerical solution of linear Fredholm fuzzy
integral equations of the second kind by Adomian method, Applied Mathematics and Com-
putation, 161(3) (2005), 733-744.
[13] C. T. H. Baker, A perspective on the numerical treatment of Volterra equations, J. Comput.
Appl. Math., 125(1-2) (2000), 217-249.
[14] P. Balasubramaniam and S. Muralisankar, Existence and uniqueness of fuzzy solution for the
nonlinear fuzzy integro-di erential equations, Applied mathematics letters, 14(4) (2001),
455-462.
[15] M. I. Berenguer, D. Gamez, A. I. Garralda-Guillem, M. Ruiz Galan and M. C. Serrano Perez,
Biorthogonal systems for solving Volterra integral equation systems of the second kind, J.
Comput. Appl. Math., 235(7) (2011), 1875-1883.
[16] A. H. Borzabadi and O. S. Fard, A numerical scheme for a class of nonlinear Fredholm
integral equations of the second kind, Journal of Computational and Applied Mathematics,
232(2) (2009), 449-454.
[17] S. S. L. Chang and L. Zadeh,On fuzzy mapping and control, IEEE Trans. System Man
Cybernet, 2(1) (1972), 30-34.
[18] Y. Chen and T. Tang, Spectral methods for weakly singular Volterra integral equations with
smooth solutions, J. Comput. Appl. Math., 233(4) (2009), 938-950.
[19] D. Dubois and H. Prade, Operations on fuzzy numbers, J. Systems Sci., 9(6) (1978), 613-626.
[20] D. Dubois and H. Prade, Towards fuzzy di erential calculus, Fuzzy Sets and Systems, 8(1-3)
(1982), 1-7.
[21] M. Friedman, M. Ma and A. Kandel, Numerical solutions of fuzzy di erential and integral
equations, Fuzzy Sets and Systems, 106(1) (1999), 35-48.
[22] R. Goetschel and W. Vaxman, Elementary calculus, Fuzzy sets and Systems, 18(1) (1986),
31-43.
[23] H. Hochstadt, Integral equations, New York: Wiley, 1973.
[24] A. Kaufmann and M. M. Gupta, Introduction Fuzzy Arithmetic, Van Nostrand Reinhold,
New York, 1985.
[25] O. Kaleva, Fuzzy di erential equations, Fuzzy Sets and Systems, 24(3) (1987), 301-317.
[26] J. P. Kauthen, Continuous time collocation method for Volterra-Fredholm integral equations,
Numer. Math., 56(1) (1989), 409-424.
[27] G. J. Klir, U. S. Clair and B. Yuan, Fuzzy set theory: foundations and applications, Prentice-
Hall Inc., 1997.
[28] H. Kwakernaak, Fuzzy random variables. Part I: de nitions and theorems, Information Sci-
ences, 15(1) (1978), 129.
[29] P. Linz, Analytical and numerical methods for Volterra equations, SIAM, Philadelphia, PA,
1985.
[30] M. T. Malinowski,On random fuzzy di erential equations, Fuzzy Sets and Systems, 160(21)
(2009), 3152-3165.
[31] M. T. Malinowski, Existence theorems for solutions to random fuzzy di erential equations,
Nonlinear Analysis: Theory, Methods & Applications, 73(6) (2010), 1515-1532.
[32] M. T. Malinowski, Random fuzzy di erential equations under generalized Lipschitz condition,
Nonlinear Analysis: Real World Applications, 13(2) (2012), 860-881.
[33] M. Mosleh and M. Otadi, Simulation and evaluation of fuzzy di erential equations by fuzzy
neural network, Applied Soft Computing, 12(9) (2012), 2817-2827.
[34] M. Mosleh and M. Otadi, Minimal solution of fuzzy linear system of di erential equations,
Neural Computing and Applications, 21(1) (2012), 329-336.
[35] M. L. Puri and D. Ralescu, Fuzzy random variables, Journal of Mathematical Analysis and
Applications, 114(2) (1986), 409-422.
[36] M. L. Puri and D. Ralescu, Di erentials of fuzzy functions, Journal of Mathematical Analysis
and Applications, 91(2) (1983), 552-558.
[37] H. H. Sorkun and S. Yalcinbas, Approximate solutions of linear Volterra integral equation
systems with variable coecients, Applied Mathematical Modelling, 34(11) (2010), 3451-
3464.
[38] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer-Verlag, New York,
1993.
[39] W. Congxin and M. Ming, Embedding problem of fuzzy number space, Fuzzy Sets and Systems,
45(2) (1992), 189-202.
[40] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning,
Information Sciences, 8(3) (1975), 199-249.