Korean J. Math. Vol. 29 No. 3 (2021) pp.527-545
DOI: https://doi.org/10.11568/kjm.2021.29.3.527

Numerical solution of Abel's general fuzzy linear integral equations by fractional calculus method

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Himanshu Kumar

Abstract

The aim of this article is to give a numerical method for solving Abel's general fuzzy linear integral equations with arbitrary kernel. The method is based on approximations of fractional integrals and Caputo derivatives. The convergence analysis for the proposed method is also given and the applicability of the proposed method is illustrated by solving some numerical examples. The results show the utility and the greater potential of the fractional calculus method to solve fuzzy integral equations.



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References

[1] S. Jahanshahi, E. Babolian, D.F.M. Torres and A. Vahidi, Solving Abel integral equations of first kind via fractional calculus, Journal of King Saud University 27 (2015), 161–167. Google Scholar

[2] S.S. Chang and L.A. Zadeh, On fuzzy mappings and control transportation systems, Man and Cybernetics, SMC-2(1972), 30–34. Google Scholar

[3] D. Dubois and H. Prade, Towards fuzzy differential Calculus part 1, Fuzzy Sets and Systems 8 (1982), 1–17. Google Scholar

[4] R. Goetchel and W. Voxman, Topological Properties of Fuzzy Numbers, Fuzzy Sets and Systems 10 (1983), 87–99. Google Scholar

[5] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987), 301–317. Google Scholar

[6] S. Nanda, On Integration of Fuzzy Mappings, Fuzzy Sets and Systems 32 (1989), 95–101. Google Scholar

[7] D. Ralescu and G. Adams, The Fuzzy Integrals, Journal of Mathematical Analysis and Applications 75 (1980), 562–570. Google Scholar

[8] Z. Wang, The Autocontinuity of Set-Function and the Fuzzy Integral, Journal of Mathematical Analysis and Applications 99 (1984), 195–218. Google Scholar

[9] B. Bede and S.G. Gal, Generalizations of the differentiability of fuzzy number valued function with applications to fuzzy differential equations, Fuzzy Sets and Systems 151 (2005), 581–599. Google Scholar

[10] H. Wu, The fuzzy Riemann integral and its numerical integration, Fuzzy Sets and Systems 110 (2000), 1–25. Google Scholar

[11] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999. Google Scholar

[12] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, Inc., New York, 1993. Google Scholar

[13] K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. Google Scholar

[14] M. Caputo, Linear models of dissipation whose Q is almost frequency independent, Part II, Geophysical Journal International 13 (5)(1967), 529–539. Google Scholar

[15] K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Electronic Transactions on Numerical Analysis 5 (1997), 1–6. Google Scholar

[16] H. Kumar and P.K.Parida, Solving Abel’s general fuzzy linear integral equations by homotopy analysis method, International Journal of Fuzzy Computation and Modelling 1 (4) 2015, 382–396. Google Scholar

[17] K. Diethelm, N.J. Ford, A.D. Freed and Yu. Luchko, Algorithms for the fractional calculus: a selection of numerical methods, Computer Methods in Appled Mechanics and Engineering 194 (6-8) (2005), 743–773. Google Scholar

[18] K. Diethelm and A.D. Freed, The FracPECE subroutine for the numerical solution of differential equations of fractional order, In: Proc. of Forschung und Wwissenschaftliches Rechnen: Beitrage Zum Heinz-Billing-Preis, 1998, 57–71. Google Scholar

[19] H. Brass, Quadraturverfahren, Vandenhoeck and Ruprecht, Gottingen, 1977. Google Scholar

[20] G. Walz, Asymptotics and Extrapolation, Akademie-Verlag, Berlin, 1996. Google Scholar