A numerical method for solving two-dimensional nonlinear Volterra fuzzy integral equation (2D-NVFIE) of the second kind will is introduced. We convert a nonlinear Volterra fuzzy integral equation to a nonlinear system of Volterra integral equation in crisp case. We use Adomian Decomposition Method (ADM) to find the approximate solution of this system and hence obtain an approximation for fuzzy solution of the nonlinear Volterra fuzzy integral equation. Also, the existence and uniqueness of the solution and convergence of the proposed method are proved. Some numerical examples are included to demonstrate the validity and applicability of the proposed technique.
REFERENCES
1.
G.
Adomian
. A review of the decomposition method in applied mathematics
. Journal Mathematics Anal App.
, 135
:501
–544
, 1988
.2.
G.
Adomian
. Solving Frontier Problems of Physics. The Decompositon Method. Kluver Academic Publish-ers
, Boston
, 12
, 1994
.3.
T.
Allahvirnaloo
, S.
Abbasbandy
. The Adomian decompositon method applied to fuzzy system of Fredholm integral equations of the second kind
. Int. J. Uncertainty Fuzziness Knowl-Based Systems
, 10
:101
–110
, 2006
.4.
E.
Babolian
, H. S. Goghary ans S.
Abbasbandy
. Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomain method
. Applied Math.Comput.
, 161
:733
–744
, 2005
.5.
A. M.
Barkhordari
, M.
Khezerloo
. Fuzzy bivariate Chebyshev method for solving fuzzy Volterra-Fredholm integral equations
. International Journal of Industrial Mathematics
, 3
:67
–77
, 2011
.6.
S.S.
Behzadi
. Solving Fuzzy Nonlinear Volterra-Fredholm Integral Equations by Using Homotopy Analysis and Adomian Decomposition Methods
. International of Fuzzy Set Valued Analysis
, 35
:13
pages, 2011
.7.
S. S.
Behzadi
, T.
Allahviranloo
, S.
Abbasbandy
. Solving fuzzy second-order nonlinear Volterra-Fredholm integro-differential equations by using Picard method
. Neural Computing and Applications
, 21
:337
–346
, 2012
.8.
D.
Dubois
, and H.
Prade
. Towards fuzzy differential calculus I,II,III
. Fuzzy Sets and Systems
, 8
:105-116
, 225
–233
, 1982
.9.
I. L.
El-Kalla
. Convergence of the Adomian method applied to a class of nonlinear integral equations
. Journal Applied Mathematics and Computating
, 21
:327
–376
, 2008
.10.
S.
Enkov
, A.
Georgieva
, R.
Nikolla
. Numerical solution of nonlinear Hammerstein fuzzy functional integral equations
. AIP Conference Proceedings
, 1789
:030006-1
–030006-8
, 2016
.11.
S.
Enkov
, A.
Georgieva
. Numerical solution of two-dimensional nonlinear Hammerstein fuzzy functional integral equations based on fuzzy Haar wavelets
. AIP Conference Proceedings
, 1910
:050004-1
–050004-8
, 2017
.12.
S.
Enkov
, A.
Georgieva
, A.
Pavlova
. Quadrature rules and iterative numerical method for two-dimensional nonlinear Fredholm fuzzy integral equations
. Communications in Applied Analysis
, 21
:479
–498
, 2017
.13.
A.
Georgieva
, A.
Alidema
. Convergence of homotopy perturbation method for solving of two-dimensional fuzzy Volterra functional integral equations
, Advanced Computing in Industrial Mathematics, Studies in Computational Intelligence
, 793
, 2019
.14.
A.
Georgieva
, I.
Naydenova
. Numerical solution of nonlinear Urisohn-Volterra fuzzy functional integral equations
. AIP Conference Proceedings
, 1910
:050010-1
–050010-8
, 2017
.15.
A.
Georgieva
, A.
Pavlova
, I.
Naydenova
. Error estimate in the iterative numerical method for two-dimensional nonlinear Hammerstein-Fredholm fuzzy functional integral equations
. Advanced Computing in Industrial Mathematics, Studies in Computational Intelligence
, 728
:41
–45
, 2018
.16.
R.
Goetschel
, W.
Voxman
. Elementary fuzzy calculus
, Fuzzy Sets and Systems
, 18
:31
–43
, 1986
.17.
A.
Jafarian
, N. S.
Measoomy
, S.
Tavan
. A numerical scheme to solve fuzzy linear Volterra integral equations system
. J. Appl. Math.
: article ID 216923, (2012
)18.
O.
Kaleva
, Fuzzy differential equations
, Fuzzy Sets and Systems
, 24
:301
–317
, 1987
.19.
J.
Mordeson
, W.
Newman
. Fuzzy integral equations
. Information Sciences
, 87
:215
–229
, 1995
.20.
M.
Mosleh
, M.
Otadi
. Solution of fuzzy Volterra integral equations in a Bernstein polynomial basis
, Journal of Advances in Information Technology
, 4
:148
–155
, 2013
.21.
S.
Sadatrasoul
, R.
Ezzati
. Numerical solution of two-dimensional nonlinear Hammerstein fuzzy integral equations based on optimal fuzzy quadrature formula
, Journal of Computational and Applied Mathematics
, 292
:430
–446
, 2016
.22.
S.
Salahshour
, T.
Allahviranloo
. Applicationm of fuzzy differential transform method for solving fuzzyVolterra integral equations
. Applied Mathematical Modelling
, 37
:1016
–1027
, 2013
.23.
P.
Salehi
, M.
Nejatiyan
. Numerical method for nonlinear fuzzy Volterra integral equations of the second kind
. International Journal of Industrial Mathematics
, 3
:169
–179
, 2011
.24.
S.
Seikkala
On the fuzzy initial value problem
, Fuzzy Sets and Systems
, 24
:319
–330
, 1987
.25.
C.
Wu
, Z.
Gong
. On Henstock integral of fuzzy-number-valued functions(I)
. Fuzzy Sets and Systems
, 120
:523
–532
, 2001
.26.
C.
Wu
, C.
Wu
. The supremum and infimum of these to fuzzy-numbers and its applications
, J. Math. Anal. Appl.
, 210
:499
–511
, 1997
.
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