# Fuzzy collocation methods for second- order fuzzy Abel-Volterra integro-differential equations

Document Type : Research Paper

Authors

1 Department of Mathematics, Islamic Azad University, Qazvin Branch, Qazvin, Iran.

2 Department of Mathematics, Science and Research Branch, Is- lamic Azad University, Tehran, Iran.

3 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.

Abstract

In this paper we intend to offer new numerical methods to solve the second-order fuzzy Abel-Volterra
integro-differential equations under the generalized $H$-differentiability. The existence and uniqueness of the
solution and convergence of the proposed methods are proved in details and the efficiency of the methods is illustrated through a numerical example.

Keywords

#### References

\bibitem{7}
S. Abbasbandy and M. S. Hashemi, {\it Fuzzy integro-differential
equations: formulation and solution using the variational
iteration method}, Nonlinear Science Letters A, { \bf 1} (2010),
413-418.
\bibitem{5}
T. Allahviranloo, S. Abbasbandy, O. Sedaghgatfar and
P. Darabi, {\it A new method for solving fuzzy
integro-differential equation under generalized
differentiability}, Neural Computing and Applications, {\bf 21}
(2012), 191-196.
\bibitem{6}
T. Allahviranloo, M. Khezerloo, O. Sedaghatfar and S.
Salahshour, {\it Toward the existence and uniqueness of solutions
of second-order fuzzy volterra integro-differential equations with
fuzzy kernel}, Neural Computing and Applications, DOI
10.1007/s00521-012-0849-x, 2012.
\bibitem{9}
T. Allahviranloo, M. Khezerloo, M. Ghanbari and S. Khezerloo,{\it
The homotopy perturbation method for fuzzy Volterra integral
equations}, International Journal of Computational Cognition, {\bf
8} (2010), 31-37.
\bibitem{10}
R. Alikhani, F. Bahrami and A. Jabbari, {\it Existence of global
solutions to nonlinear fuzzy Volterra integro-differential
equations}, Nonlinear Analysis, {\bf 75} (2012), 1810-1821.
\bibitem{28}
T. Allahviranloo, S. Abbasbandy and S. Salahshour, {\it A new method
for solving fuzzy integro-differential equation under generalized
differentiability}, Neural Computing and Applications, {\bf
21} (2012), 191-196.
\bibitem{4}
E. Babolian, H. Sadeghi Goghary and S. Abbasbandy, {\it Numerical solution of linear fredholm fuzzy integral equations
of the second kind by Adomian method}, Applied Mathematics and
Computation, {\bf 161} (2005), 733-744.
\bibitem{17}
B. Bede and S. G. Gal, {\it Generalizations of differentiability of
fuzzy-number valued function with application to fuzzy
differential equations}, Fuzzy Sets and Systems, {\bf 151} (2005),
581-599.
\bibitem{22}
B. Bede, Note on {\it Numerical solutions of fuzzy differential
equations by predictor–corrector method}, Information Sciences,
{\bf 178} (2008), 1917-1922.
\bibitem{25}
A. M. Bijura, {\it Error bounded analysis and singularly perturbed
Abel-Volterra equations}, Journal of Applied Mathematics, {\bf 6}
(2004), 479-494.
\bibitem{18}
Y. Chalco-Cano and H. Román-Flores, {\it On new solutions of fuzzy
differential equations}, Chaos Soliton and Fractals, {\bf
38} (2006), 112-119.
\bibitem{16}
R. N. Desmarais and S. R. Bland, {\it Tables of properties of airfoil
polynomials}, Nasa Reference Publication, 1343, September 1995.
\bibitem{23}
\bibitem{2}
M. Friedman, M. Ma and A. Kandel, {\it Numerical methods for calculating the fuzzy integral}, Fuzzy Sets and Systems, {\bf83} (1996), 57–62.
\bibitem{12}
R. Goetschel and W. Voxman, {\it Elementary calculus}, Fuzzy
Sets Syst, {\bf 18} (1986), 31-43.
\bibitem{1}
M. Jahantigh, T. Allahviranloo and M. Otadi,{\it Numerical solution of fuzzy integral equation}, Applied Mathematical Sciences, {\bf2} (2008), 33-46.
\bibitem{14}
A. Kauffman and M. M. Gupta, {\it Introduction to fuzzy arithmetic: theory and application}, Van Nostrand Reinhold. New York, 1991.
\bibitem{24}
H. Kim and R. Sakthivel, {\it Numerical solution of hybrid fuzzy
differential equations using improved predictor-corrector method},
Communications in Nonlinear Science and Numerical Simulation, {\bf
17} (2012), 3788-3794.
\bibitem{20}
X. Li and T. TANG, {\it Convergence analysis of Jacobi spectral
collocation methods for Abel-Volterra integral equations of second
kind}, Frontiers of Mathematics in China, {\bf 7} (2012), 69–84.
\bibitem{8}
N. Mikaeilvand, S. Khakrangin and T. Allahviranloo, {\it Solving fuzzy
Volterra integro-differential equation by fuzzy differential
transform method}, EUSFLAT- LFA, (2011), 891-896.
\bibitem{11}
A. Molabahrami, A. Shidfar and A. Ghyasi, {\it An analytical method
for solving linear Fredholm fuzzy integral equations of the second
kind}, Computers and Mathematics with Applications, {\bf 61}
(2011), 2754-2761.
\bibitem{13}
M. L. Puri and D. Ralescu, { \it Fuzzy random variables}, Journal of Mathematical Analysis and Applications, { \bf 114} (1986), 409-422.
\bibitem{19}
Volterra-Fredholm integral equations by using homotopy analysis
and Adomian decomposition methods}, Journal of Fuzzy Set Valued
Analysis, { \bf 2011}(2011), 1-13.
\bibitem{21}
fuzzy second-order nonlinear Volterra–Fredholm
integro-differential equations by using Picard method}, Neural
Computing and Applications, DOI 10.1007/s00521- 012 - 0926 -1,
2012.
\bibitem{26}
use of fuzzy expansion method for Solving Fuzzy linear
Volterra-Fredholm Integral Equations}, Journal of Intelligent and
Fuzzy Systems, DOI:10.3233/IFS-130861, 2013.
\bibitem{27}
S. Salahshour and T. Allahviranloo, {\it Application of fuzzy
differential tansform method for solving fuzzy Volterra integral
equations}, Applied Mathematical Modelling, {\bf 37} (2013),
1016-1027.
\bibitem{29}
S. Salahshour, T. Allahviranloo and S. Abbasbandy, {\it Solving fuzzy
fractional differential equations by fuzzy Laplace transforms},
Communication in Nonlinear Science and Numerical Simulation, {\bf
17} (2012), 1372-1381.
\bibitem{3}
M. Sugeno, {\it Theory of fuzzy integrals and its application}, Ph.D. Thesis, Tokyo Institute of Technology, 1974.
\bibitem{15}
L. A. Zadeh, {\it Information and control}, Fuzzy Sets and Systems, {\bf 8} (1965), 338-353.