Document Type : Research Paper
Authors
Department of Mathematics, Islamic Azad University, Arak Branch, Arak, Iran
Abstract
In this paper a linear Fuzzy Fredholm Integral Equation(FFIE)
with arbitrary Fuzzy Function input and symmetric triangular (Fuzzy Interval)
output is considered. For each variable, output is the nearest triangular fuzzy
number (fuzzy interval) to the exact fuzzy solution of (FFIE).
Keywords
[1] S. Abbasbandy and T. Allahviranloo, Numerical solution of fuzzy di erential equation by
runge-Kutta method, Nonlinear Stud, 11 (2004), 117-129.
[2] S. Abbasbandy and B. Asady, The nearest trapezoidal fuzzy number to a fuzzy quantity, Appl
Math Comput, 156 (2004), 381-386.
[3] S. Abbasbandy, E. Babolian and M. Alavi, Numerical method for solving linear Fredholm
fuzzy integral equations of the second kind, Chaos, Soliton and Fractals, 31 (2007), 138-146.
[4] T. Allahviranloo and M. Otadi, Gaussian quadratures for approximate of fuzzy integrals,
Applied Mathematics and Computation, 170 (2005), 874-885.
[5] K. Atkinson, A survey of numerical methods for solving nonlinear integral equations, J. of
Integral Equat. and Appl., 4 (1992), 15-46.
[6] E. Babolian, H. Sadeghi Goghary and S. Abbasbandy, Numerical solution of linear Fredholm
fuzzy integral equations of the second kind by Adomian method, Applied Mathematics and
Computation, 161 (2005), 733-744
[7] E. Babolian, H. Sadeghi and S. Javadi, Numerically solution of fuzzy di erential equations
by Adomian method, Appl. Math. Comput., 149 (2004), 547-557.
[8] K. Balachandran and K. Kanagarajan, Existence of solutions of general nonlinear fuzzy
Voltera-Feredholm integral equations, J. Appl. Math. Stochastic Anal, 3 (2005), 333-343.
[9] K. Balachandran and P. Prakash, Existence of solution of nonlinear fuzzy Voltera-Feredholm
integral equations, Indian J. Pure Appl. Math, 333 (2002), 329-343.
[10] B. Bede and S. G. Gal, Quadrature rules for integral of fuzzy-number-valued functions, Fuzzy
Sets and Systems, 145 (2004), 359-380.
[11] A. M. Bica, Error estimation in the Approximation of the solution of nonlinear fuzzy Fered-
holm integral equations, Information Sciences, 174 (2008), 1279-1292.
[12] J. J. Buckley and T. Furing, Fuzzy integral equations, J. Fuzzy Math, 10 (2002), 1011-1024.
[13] S. S. L. Chang and L. Zadeh, On fuzzy mapping and control, IEEE Trans System Man
Cybernet, 2 (1972), 30-34.
[14] W. Cingxin and M. Ming, On embedding problem of fuzzy number spaces: part I, Fuzzy Sets
and Systems, 44 (1991), 33-38.
[15] W. Cingxin and M. Ming, On embedding problem of fuzzy number spaces: part II, Fuzzy Sets
and Systems, 45 (1992), 189-202.
[16] W. Cingxin and M. Ming, On embedding problem of fuzzy number spaces: part III, Fuzzy
Sets and Systems, 44 (1992), 281-286.
[17] W. Congxin and M. Ming, On the integrals. series and integral equations of fuzzy set-valued
functions, J. Harbin Inst Technol, 21 (1990), 11-19.
[18] L. M. Delves and J. L. Mohemed, Computational methods for bntegral equations, Cambridge
University Press, Cambridge, 1985.
[19] K. Deimling, Multivalued di erential equations, Walter de Gruyter, New York, 1992.
[20] P. Diamond, Stability and periodicity in fuzzy di erential equations, IEEE Trans. Fuzzy Syst,
8 (2000), 583-590.
[21] D. Dubois and H. prade, Towards fuzzy di erential calculus, Fuzzy Sets and System, 8 (1982),
1-7.
[22] M. Friedman, M. Ma and A. Kandel, Numerical solutions of fuzzy di erential and integral
equations, Fuzzy Sets and Systems, 106 (1999), 35-48.
[23] M. Fridman, M. Ma and A. Kandel, On fuzzy integral equations, Fundam. Inform, 37 (1999),
89-99.
[24] M. Fridman, M. Ming and A. Kandel, Solution to fuzzy integral equations with arbitrary
kernels, Internat. J. Approx. Reason, 20 (1999), 249-262.
[25] D. N. Georgion and I. E. Kougias, Bounded solutions for fuzzy integral equations, Int. j.
Math.sci, 312 (2002), 109 114.
[26] D. N. Georgion and I. E. Kougias, On fuzzy fredholm and Voltera integral equations, J. Fuzzy
Math, 94 (2001), 943-951.
runge-Kutta method, Nonlinear Stud, 11 (2004), 117-129.
[2] S. Abbasbandy and B. Asady, The nearest trapezoidal fuzzy number to a fuzzy quantity, Appl
Math Comput, 156 (2004), 381-386.
[3] S. Abbasbandy, E. Babolian and M. Alavi, Numerical method for solving linear Fredholm
fuzzy integral equations of the second kind, Chaos, Soliton and Fractals, 31 (2007), 138-146.
[4] T. Allahviranloo and M. Otadi, Gaussian quadratures for approximate of fuzzy integrals,
Applied Mathematics and Computation, 170 (2005), 874-885.
[5] K. Atkinson, A survey of numerical methods for solving nonlinear integral equations, J. of
Integral Equat. and Appl., 4 (1992), 15-46.
[6] E. Babolian, H. Sadeghi Goghary and S. Abbasbandy, Numerical solution of linear Fredholm
fuzzy integral equations of the second kind by Adomian method, Applied Mathematics and
Computation, 161 (2005), 733-744
[7] E. Babolian, H. Sadeghi and S. Javadi, Numerically solution of fuzzy di erential equations
by Adomian method, Appl. Math. Comput., 149 (2004), 547-557.
[8] K. Balachandran and K. Kanagarajan, Existence of solutions of general nonlinear fuzzy
Voltera-Feredholm integral equations, J. Appl. Math. Stochastic Anal, 3 (2005), 333-343.
[9] K. Balachandran and P. Prakash, Existence of solution of nonlinear fuzzy Voltera-Feredholm
integral equations, Indian J. Pure Appl. Math, 333 (2002), 329-343.
[10] B. Bede and S. G. Gal, Quadrature rules for integral of fuzzy-number-valued functions, Fuzzy
Sets and Systems, 145 (2004), 359-380.
[11] A. M. Bica, Error estimation in the Approximation of the solution of nonlinear fuzzy Fered-
holm integral equations, Information Sciences, 174 (2008), 1279-1292.
[12] J. J. Buckley and T. Furing, Fuzzy integral equations, J. Fuzzy Math, 10 (2002), 1011-1024.
[13] S. S. L. Chang and L. Zadeh, On fuzzy mapping and control, IEEE Trans System Man
Cybernet, 2 (1972), 30-34.
[14] W. Cingxin and M. Ming, On embedding problem of fuzzy number spaces: part I, Fuzzy Sets
and Systems, 44 (1991), 33-38.
[15] W. Cingxin and M. Ming, On embedding problem of fuzzy number spaces: part II, Fuzzy Sets
and Systems, 45 (1992), 189-202.
[16] W. Cingxin and M. Ming, On embedding problem of fuzzy number spaces: part III, Fuzzy
Sets and Systems, 44 (1992), 281-286.
[17] W. Congxin and M. Ming, On the integrals. series and integral equations of fuzzy set-valued
functions, J. Harbin Inst Technol, 21 (1990), 11-19.
[18] L. M. Delves and J. L. Mohemed, Computational methods for bntegral equations, Cambridge
University Press, Cambridge, 1985.
[19] K. Deimling, Multivalued di erential equations, Walter de Gruyter, New York, 1992.
[20] P. Diamond, Stability and periodicity in fuzzy di erential equations, IEEE Trans. Fuzzy Syst,
8 (2000), 583-590.
[21] D. Dubois and H. prade, Towards fuzzy di erential calculus, Fuzzy Sets and System, 8 (1982),
1-7.
[22] M. Friedman, M. Ma and A. Kandel, Numerical solutions of fuzzy di erential and integral
equations, Fuzzy Sets and Systems, 106 (1999), 35-48.
[23] M. Fridman, M. Ma and A. Kandel, On fuzzy integral equations, Fundam. Inform, 37 (1999),
89-99.
[24] M. Fridman, M. Ming and A. Kandel, Solution to fuzzy integral equations with arbitrary
kernels, Internat. J. Approx. Reason, 20 (1999), 249-262.
[25] D. N. Georgion and I. E. Kougias, Bounded solutions for fuzzy integral equations, Int. j.
Math.sci, 312 (2002), 109 114.
[26] D. N. Georgion and I. E. Kougias, On fuzzy fredholm and Voltera integral equations, J. Fuzzy
Math, 94 (2001), 943-951.
[27] R. Goetschel and W. Voxman, Elementary calculus, Fuzzy Sets and Systems, 18 (1986),
31-43.
[28] P. Grzegorzewski, Metricsand orders in space of fuzzy numbers, Fuzzy Sets and Systems, 97
(1987), 83-94
[29] P. Grzegorzewski, Nearst interval approximation of a fuzzy number, Fuzzy Sets ans Systems,
130 (2002), 321-330.
[30] P. Grzegorzewski, Trapezoidal approximations of fuzzy numbers preserving the expected in-
terval -algorithms and properties, Fuzzy Sets and Systems, 159 (2008), 1354-1364.
[31] H. Hochstadt, Integral equations, Wiley, New York, 1973.
[32] O. Kaleva, Fuzzy di erential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.
[33] W. V. Lovitt, Linear integral equation, Dover, New York, 1950.
[34] M. Ma, M. Friedman and A. Kandel, Numerical solution of fuzzy di erential equations, Fuzzy
Sets and Systems, 105 (1999), 133-138.
[35] M. Matloka, On fuzzy integrals Proc, 2nd Polish Symp. on Interval and Fuzzy Mathematics,
Politechnika Poznnsk, (1987), 167-170.
[36] A. Maturo, On some structure of fuzzy numbers, Iranian Journal of Fuzzy Systems, 6 (2009),
49-59.
[37] A. Molabahrami, A. Shidfar and A. Ghyasi, An analytical method for solving linear Feredholm
fuzzy integral equations of the second kind, Computers and Mathematics with Applications,
61 (2011), 2754-2761.
[38] J. Mordeson and W. Newman, Fuzzy integral equations, Information Sciences, 814 (1995),
215-229.
[39] J. J. Nieto and R. Rodriguez-Lopaz, Bounded solution for fuzzy di erential and integral
equations, Choas Solitons and Fractals, 275 (2006), 1376-1386.
[40] N. Parandin and M. A. Fariborzi Araghi, The approximate solution of linear fuzzy Fered-
holm integral equations of the second kind by using iterative interpolation, Word Academy
of science, Engineering and Technology, 49 (2009), 425-431.
[41] E. Pasha, A. saiedifar and B. Asady, The presentation on fuzzy numbers and their applica-
tions, Iranian Journal of Fuzzy Systems, 6 (2009), 27-44.
[42] J. Y. Perk and J. U. Jeong, On the existence and uniquenes of solutions of fuzzy Voltera-
Feredholm, integral equations, Fuzzy Sets and Systems, 115 (2000), 425-431.
[43] O. Solaymani and A. Vahidian kamyad, Modi ed K-step method for solving fuzzy initial value
problems, Iranian Journal of Fuzzy Systems, 8 (2011), 49-59.
[44] J. Vrba, A note on inverse in arithwith fuzzy numbers, Fuzzy Sets and Systems, 50 (1992),
267-278.
31-43.
[28] P. Grzegorzewski, Metricsand orders in space of fuzzy numbers, Fuzzy Sets and Systems, 97
(1987), 83-94
[29] P. Grzegorzewski, Nearst interval approximation of a fuzzy number, Fuzzy Sets ans Systems,
130 (2002), 321-330.
[30] P. Grzegorzewski, Trapezoidal approximations of fuzzy numbers preserving the expected in-
terval -algorithms and properties, Fuzzy Sets and Systems, 159 (2008), 1354-1364.
[31] H. Hochstadt, Integral equations, Wiley, New York, 1973.
[32] O. Kaleva, Fuzzy di erential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.
[33] W. V. Lovitt, Linear integral equation, Dover, New York, 1950.
[34] M. Ma, M. Friedman and A. Kandel, Numerical solution of fuzzy di erential equations, Fuzzy
Sets and Systems, 105 (1999), 133-138.
[35] M. Matloka, On fuzzy integrals Proc, 2nd Polish Symp. on Interval and Fuzzy Mathematics,
Politechnika Poznnsk, (1987), 167-170.
[36] A. Maturo, On some structure of fuzzy numbers, Iranian Journal of Fuzzy Systems, 6 (2009),
49-59.
[37] A. Molabahrami, A. Shidfar and A. Ghyasi, An analytical method for solving linear Feredholm
fuzzy integral equations of the second kind, Computers and Mathematics with Applications,
61 (2011), 2754-2761.
[38] J. Mordeson and W. Newman, Fuzzy integral equations, Information Sciences, 814 (1995),
215-229.
[39] J. J. Nieto and R. Rodriguez-Lopaz, Bounded solution for fuzzy di erential and integral
equations, Choas Solitons and Fractals, 275 (2006), 1376-1386.
[40] N. Parandin and M. A. Fariborzi Araghi, The approximate solution of linear fuzzy Fered-
holm integral equations of the second kind by using iterative interpolation, Word Academy
of science, Engineering and Technology, 49 (2009), 425-431.
[41] E. Pasha, A. saiedifar and B. Asady, The presentation on fuzzy numbers and their applica-
tions, Iranian Journal of Fuzzy Systems, 6 (2009), 27-44.
[42] J. Y. Perk and J. U. Jeong, On the existence and uniquenes of solutions of fuzzy Voltera-
Feredholm, integral equations, Fuzzy Sets and Systems, 115 (2000), 425-431.
[43] O. Solaymani and A. Vahidian kamyad, Modi ed K-step method for solving fuzzy initial value
problems, Iranian Journal of Fuzzy Systems, 8 (2011), 49-59.
[44] J. Vrba, A note on inverse in arithwith fuzzy numbers, Fuzzy Sets and Systems, 50 (1992),
267-278.
)