Document Type : Research Paper
Author
Department of Mathematics, Aliabad Katoul Branch, Islamic Azad University, Aliabad Katoul, Iran
Abstract
In this paper, a fuzzy numerical procedure for solving fuzzy linear Volterra integro-differential
equations of the second kind under strong generalized differentiability is designed. Unlike the existing numerical methods, we do not replace the original fuzzy equation by a $2\times 2$ system of
crisp equations, that is the main difference between our method and other numerical methods.
Error analysis and numerical examples are given to show the convergency and efficiency of the
proposed method, respectively.
Keywords
[1] R. Alikhani and F. Bahrami, Global solutions of fuzzy integro-di erential equations under
generalized di erentiability by the method of upper and lower solutions, Information Sciences,
295 (2015), 600{608.
[2] R. Alikhani, F. Bahrami and A. Jabbari, Existence of global solutions to nonlinear fuzzy
Volterra integro-di erential equations, Nonlinear Analysis: Theory, Methods & Applications,
75(4) (2012), 1810{1821.
[3] G. A. Anastassiou and S. G. Gal, On a fuzzy trigonometric approximation theorem of
Weierstrass-type, J. Fuzzy Math., 9(3) (2001), 701{708.
[4] K. Balachandran and P. Prakash, On fuzzy volterra integral equations with deviting argu-
ments, Journal of Applied Mathematics and Stochastic Analysis, 2 (2004), 169{176.
[5] B. Bede and S. G. Gal, Almost Periodic fuzzy-number valued functions, Fuzzy Sets and
Systems, 147 (2004), 385{403.
[6] B. Bede and S. G. Gal, Generalizations of di erentiablity of fuzzy number valued function
with application to fuzzy di erential equations, Fuzzy Sets and Systems, 151 (2005), 581{599.
[7] B. Bede and S. G. Gal, Quadrature rules for integrals of fuzzy-number-valued functions, Fuzzy
Sets and Systems, 145 (2004), 359{380.
[8] Y. Chalco-Cano and H. Roman-Flores, On new solutions of fuzzy di erential equations,
Chaos, Solitons and Fractals, 38 (2008), 112{119.
[9] D. Dubois and H. Prade, Towards fuzzy di erential calculus. I. Integration of fuzzy mappings,
Fuzzy Sets and Systems, 8(1) (1982), 1{17.
[10] D. Dubois and H. Prade, Towards fuzzy di erential calculus. II. Integration on fuzzy intervals,
Fuzzy Sets and Systems, 8(2) (1982), 105{116.
[11] D. Dubois and H. Prade, Towards fuzzy di erential calculus: III, di erentiation, Fuzzy Sets
and Systems, 8 (1982), 225{233.
[12] S. G. Gal, Approximation theory in fuzzy setting, in: G.A. Anastassiou (Ed.), Handbook
of Analytic-Computational Methods in Applied Mathematics, Chapman Hall, CRC Press,
(2000), 617{666.
[13] M. Ghatee and S. M. Hashemi, Ranking function-based solutions of fully fuzzi ed minimal
coast
ow problem, Information Sciences, 177 (2007), 4271{4294.
[14] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy sets and Systems, 18 (1986),
31{43.
[15] S. Hajighasemi, T. Allahviranloo, M. Khezerloo, M. Khorasany and S. Salahshour, Existence
and uniqueness of solutions of fuzzy volterra integro-di erential equations, IPMU, Part II,
CCIS 81, (2010), 491{500.
[16] O. Kaleva, Fuzzy di erential equations, Fuzzy Sets and Systems, 24 (1987), 301{317.
[17] J. Y. Park and H. K. Han, Existence and uniqueness theorem for a solution of fuzzy Volterra
integral equations, Fuzzy Sets and Systems, 105 (1999), 481{488.
[18] J. Y. Park and J. U. Jeong, A note on fuzzy integral equations, Fuzzy Sets and Systems, 108
(1999), 193{200.
[19] M. L. Puri and D. Ralescu, Di erentials of fuzzy functions, J. Math. Anal. Appl., 91 (1983),
552{558.
[20] M. L. Puri and D. Ralescu, Fuzzy random variables, J. Math. Anal. Appl., 114 (1986),
409{422.
[21] A. Ralston and P. Rabinowitz, First course in numerical analysis, McGraw-Hill, 1978.
[22] S. Salahshour and T. Allahviranloo, Application of fuzzy di erential transform method for
solving fuzzy Volterra integral equations, Applied Mathematical Modelling, 37(3) (2013),
1016{1027.
[23] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems, 24(3) (1987), 319{
330.
[24] S. J. Song, Q.Y. Liu and Q. C Xu, Existence and comparison theorems to Volterra fuzzy
integral equation in (En;D), Fuzzy Sets and Systems, 104 (1999), 315{321.
generalized di erentiability by the method of upper and lower solutions, Information Sciences,
295 (2015), 600{608.
[2] R. Alikhani, F. Bahrami and A. Jabbari, Existence of global solutions to nonlinear fuzzy
Volterra integro-di erential equations, Nonlinear Analysis: Theory, Methods & Applications,
75(4) (2012), 1810{1821.
[3] G. A. Anastassiou and S. G. Gal, On a fuzzy trigonometric approximation theorem of
Weierstrass-type, J. Fuzzy Math., 9(3) (2001), 701{708.
[4] K. Balachandran and P. Prakash, On fuzzy volterra integral equations with deviting argu-
ments, Journal of Applied Mathematics and Stochastic Analysis, 2 (2004), 169{176.
[5] B. Bede and S. G. Gal, Almost Periodic fuzzy-number valued functions, Fuzzy Sets and
Systems, 147 (2004), 385{403.
[6] B. Bede and S. G. Gal, Generalizations of di erentiablity of fuzzy number valued function
with application to fuzzy di erential equations, Fuzzy Sets and Systems, 151 (2005), 581{599.
[7] B. Bede and S. G. Gal, Quadrature rules for integrals of fuzzy-number-valued functions, Fuzzy
Sets and Systems, 145 (2004), 359{380.
[8] Y. Chalco-Cano and H. Roman-Flores, On new solutions of fuzzy di erential equations,
Chaos, Solitons and Fractals, 38 (2008), 112{119.
[9] D. Dubois and H. Prade, Towards fuzzy di erential calculus. I. Integration of fuzzy mappings,
Fuzzy Sets and Systems, 8(1) (1982), 1{17.
[10] D. Dubois and H. Prade, Towards fuzzy di erential calculus. II. Integration on fuzzy intervals,
Fuzzy Sets and Systems, 8(2) (1982), 105{116.
[11] D. Dubois and H. Prade, Towards fuzzy di erential calculus: III, di erentiation, Fuzzy Sets
and Systems, 8 (1982), 225{233.
[12] S. G. Gal, Approximation theory in fuzzy setting, in: G.A. Anastassiou (Ed.), Handbook
of Analytic-Computational Methods in Applied Mathematics, Chapman Hall, CRC Press,
(2000), 617{666.
[13] M. Ghatee and S. M. Hashemi, Ranking function-based solutions of fully fuzzi ed minimal
coast
ow problem, Information Sciences, 177 (2007), 4271{4294.
[14] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy sets and Systems, 18 (1986),
31{43.
[15] S. Hajighasemi, T. Allahviranloo, M. Khezerloo, M. Khorasany and S. Salahshour, Existence
and uniqueness of solutions of fuzzy volterra integro-di erential equations, IPMU, Part II,
CCIS 81, (2010), 491{500.
[16] O. Kaleva, Fuzzy di erential equations, Fuzzy Sets and Systems, 24 (1987), 301{317.
[17] J. Y. Park and H. K. Han, Existence and uniqueness theorem for a solution of fuzzy Volterra
integral equations, Fuzzy Sets and Systems, 105 (1999), 481{488.
[18] J. Y. Park and J. U. Jeong, A note on fuzzy integral equations, Fuzzy Sets and Systems, 108
(1999), 193{200.
[19] M. L. Puri and D. Ralescu, Di erentials of fuzzy functions, J. Math. Anal. Appl., 91 (1983),
552{558.
[20] M. L. Puri and D. Ralescu, Fuzzy random variables, J. Math. Anal. Appl., 114 (1986),
409{422.
[21] A. Ralston and P. Rabinowitz, First course in numerical analysis, McGraw-Hill, 1978.
[22] S. Salahshour and T. Allahviranloo, Application of fuzzy di erential transform method for
solving fuzzy Volterra integral equations, Applied Mathematical Modelling, 37(3) (2013),
1016{1027.
[23] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems, 24(3) (1987), 319{
330.
[24] S. J. Song, Q.Y. Liu and Q. C Xu, Existence and comparison theorems to Volterra fuzzy
integral equation in (En;D), Fuzzy Sets and Systems, 104 (1999), 315{321.
[25] C. Wu and Z. Gong, On Henstock integral of fuzzy-number-valued functions I, Fuzzy Sets
and Systems, 120 (2001), 523{532.
and Systems, 120 (2001), 523{532.
)