Collocation method for fuzzy Volterra integral equations of the second kind
Abstract
In this paper we consider fuzzy Volterra integral equation of the second kind whose kernel may change sign. We give conditions for smoothness of the upper and lower functions of the solution. For numerical solution we propose the collocation method with two different basis function sets: triangular and rectangular basis. The smoothness results allow us to obtain the convergence rates of the methods. The proposed methods are illustrated by numerical examples, which confirm the theoretical convergence estimates.
Keyword : fuzzy Volterra integral equation, smoothness of solution, triangular basis, rectangular basis, collocation method, convergence rate
How to Cite
Alijani, Z., & Kangro, U. (2020). Collocation method for fuzzy Volterra integral equations of the second kind. Mathematical Modelling and Analysis, 25(1), 146-166. https://doi.org/10.3846/mma.2020.9695
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
S. Abbasbandy, E. Babolian and M. Alavi. Numerical method for solving linear Fredholm fuzzy integral equations of the second kind. Chaos, Solitons and Fractals, 31(1):138–146, 2007. https://doi.org/10.1016/j.chaos.2005.09.036
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 Computation, 161(3):733–744, 2005. https://doi.org/10.1016/j.amc.2003.12.071
B. Bede. Mathematics of Fuzzy Sets and Fuzzy Logic, volume 295 of Studies in Fuzziness and Soft Computing. Springer, 2013. https://doi.org/10.1007/978-3642-35221-8
A.M. Bica. Error estimation in the approximation of the solution of nonlinear fuzzy Fredholm integral equations. Information Sciences, 178(5):1279–1292, 2008. https://doi.org/10.1016/j.ins.2007.10.021
H. Brunner. Collocation methods for Volterra integral and related functional differential equations, volume 15 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, 2004. https://doi.org/10.1017/CBO9780511543234
H. Brunner. Volterra integral equations: an introduction to theory and applications, volume 30 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, 2017.
S.S.L. Chang and L.A. Zadeh. On fuzzy mapping and control. IEEE Transactions on Systems, Man and Cybernetics, 2:30–34, 1972. https://doi.org/10.1109/TSMC.1972.5408553
D. Dubois and H. Prade. Towards fuzzy differential calculus part 3: Differentiation. Fuzzy sets and systems, 8(3):225–233, 1982. https://doi.org/10.1016/S0165-0114(82)80001-8
M. Friedman, Ma Ming and A. Kandel. Numerical solutions of fuzzy differential and integral equations. Fuzzy sets and Systems, 106(1):35–48, 1999. https://doi.org/10.1016/S0165-0114(98)00355-8
J.R. Goetschel and W. Voxman. Elementary fuzzy calculus. Fuzzy sets and systems, 18(1):31–43, 1986. https://doi.org/10.1016/0165-0114(86)90026-6
O. Kaleva. Fuzzy differential equations. Fuzzy sets and systems, 24(3):301–317, 1987. https://doi.org/10.1016/0165-0114(87)90029-7
R. Kress. Linear integral equations, volume 82 of Applied Mathematical Sciences. Springer, 2014. https://doi.org/10.1007/978-1-4614-9593-2
M. Mosleh and M. Otadi. Solution of fuzzy Volterra integral equations in a Bernstein polynomial basis. Journal of Advances in Information Technology, 4(3):148–155, 2013. https://doi.org/10.4304/jait.4.3.148-155
J.Y. Park and H.K. Han. Existence and uniqueness theorem for a solution of fuzzy Volterra integral equations. Fuzzy Sets and Systems, 105(3):481–488, 1999. https://doi.org/10.1016/S0165-0114(97)00238-8
F. Saberidad, S.M. Karbassi and M. Heydari. Numerical solution of nonlinear fuzzy Volterra integral equations of the second kind for changing sign kernels. Soft Computing, 2018. https://doi.org/10.1007/s00500-018-3668-x
P.K. Sahu and S. Saha Ray. A new Bernoulli wavelet method for accurate solutions of nonlinear fuzzy Hammerstein-Volterra delay integral equations. Fuzzy Sets and Systems, 309:131–144, 2017. https://doi.org/10.1016/j.fss.2016.04.004
S. Salahshour and T. Allahviranloo. Application of fuzzy differential transform method for solving fuzzy Volterra integral equations. Applied Mathematical Modelling, 37(3):1016–1027, 2013. https://doi.org/10.1016/j.apm.2012.03.031
L.A. Zadeh. Fuzzy sets. Information and control, 8(3):338–353, 1965. https://doi.org/10.1016/S0019-9958(65)90241-X
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 Computation, 161(3):733–744, 2005. https://doi.org/10.1016/j.amc.2003.12.071
B. Bede. Mathematics of Fuzzy Sets and Fuzzy Logic, volume 295 of Studies in Fuzziness and Soft Computing. Springer, 2013. https://doi.org/10.1007/978-3642-35221-8
A.M. Bica. Error estimation in the approximation of the solution of nonlinear fuzzy Fredholm integral equations. Information Sciences, 178(5):1279–1292, 2008. https://doi.org/10.1016/j.ins.2007.10.021
H. Brunner. Collocation methods for Volterra integral and related functional differential equations, volume 15 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, 2004. https://doi.org/10.1017/CBO9780511543234
H. Brunner. Volterra integral equations: an introduction to theory and applications, volume 30 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, 2017.
S.S.L. Chang and L.A. Zadeh. On fuzzy mapping and control. IEEE Transactions on Systems, Man and Cybernetics, 2:30–34, 1972. https://doi.org/10.1109/TSMC.1972.5408553
D. Dubois and H. Prade. Towards fuzzy differential calculus part 3: Differentiation. Fuzzy sets and systems, 8(3):225–233, 1982. https://doi.org/10.1016/S0165-0114(82)80001-8
M. Friedman, Ma Ming and A. Kandel. Numerical solutions of fuzzy differential and integral equations. Fuzzy sets and Systems, 106(1):35–48, 1999. https://doi.org/10.1016/S0165-0114(98)00355-8
J.R. Goetschel and W. Voxman. Elementary fuzzy calculus. Fuzzy sets and systems, 18(1):31–43, 1986. https://doi.org/10.1016/0165-0114(86)90026-6
O. Kaleva. Fuzzy differential equations. Fuzzy sets and systems, 24(3):301–317, 1987. https://doi.org/10.1016/0165-0114(87)90029-7
R. Kress. Linear integral equations, volume 82 of Applied Mathematical Sciences. Springer, 2014. https://doi.org/10.1007/978-1-4614-9593-2
M. Mosleh and M. Otadi. Solution of fuzzy Volterra integral equations in a Bernstein polynomial basis. Journal of Advances in Information Technology, 4(3):148–155, 2013. https://doi.org/10.4304/jait.4.3.148-155
J.Y. Park and H.K. Han. Existence and uniqueness theorem for a solution of fuzzy Volterra integral equations. Fuzzy Sets and Systems, 105(3):481–488, 1999. https://doi.org/10.1016/S0165-0114(97)00238-8
F. Saberidad, S.M. Karbassi and M. Heydari. Numerical solution of nonlinear fuzzy Volterra integral equations of the second kind for changing sign kernels. Soft Computing, 2018. https://doi.org/10.1007/s00500-018-3668-x
P.K. Sahu and S. Saha Ray. A new Bernoulli wavelet method for accurate solutions of nonlinear fuzzy Hammerstein-Volterra delay integral equations. Fuzzy Sets and Systems, 309:131–144, 2017. https://doi.org/10.1016/j.fss.2016.04.004
S. Salahshour and T. Allahviranloo. Application of fuzzy differential transform method for solving fuzzy Volterra integral equations. Applied Mathematical Modelling, 37(3):1016–1027, 2013. https://doi.org/10.1016/j.apm.2012.03.031
L.A. Zadeh. Fuzzy sets. Information and control, 8(3):338–353, 1965. https://doi.org/10.1016/S0019-9958(65)90241-X