The main goal of the paper is to present an approximate method for solving of a nonlinear Volterra-Fredholm fuzzy integro-differential equation (NVFFIDE). It is applied the homotopy analysis method (HAM). The studied equation is converted to a nonlinear system of Volterra-Fredholm integro-differential equations in a crisp case. Approximate solutions of this system are obtained by the help with HAM and hence an approximation for the fuzzy solution of the NVFFIDE is presented. The convergence of the proposed method is proved. A numerical example is given to demonstrate the validity and applicability of the proposed technique.
REFERENCES
1.
S.
Abbasbandy
, E.
Magyari
, E.
Shivanian
, The homotopy analysis method for multiple solutions of nonlinear boundary value problems
, Commun. Nonlinear Sci. Numer. Simul.
, 14
, 3530
–3536
(2009
).2.
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
).3.
A. S.
Bataineh
, M.S.M.
Noorani
, I.
Hashim
, Approximate analytical solutions of systems of PDEs by homo- topy analysis method
, Comput. Math. Appl.
, 55
, 2913
–2923
(2008
).4.
Y.
Cherruault
Convergence of Adomians method
, Kybernetes
, 18
, 31
–38
(1989
).5.
Y.
Chalco-Cano
, H.
Roman-Flores
On New Solutions of Fuzzy Differential Equations
, Chaos, Solitons and Fractals
, 38
, 112
–119
(2008
).6.
M.
Friedman
, M.
Ming
, A.
Kandel
Numerical Solution of fuzzy Differential and Integral Equations
, Fuzzy Sets and System
, 106
, 35
–48
(1999
).7.
M.
Ghanbari
Numerical Solution of Fuzzy Linear Volterra Integral Equations of the Second Kind by Homo- topy Analysis Method
, Intern. J. Industrial Math.
, 2
, 73
–87
(2010
).8.
A.
Georgieva
Solving two-dimensional nonlinear Volterra-Fredholm fuzzy integral equations by using Ado-mian decomposition method
, Dynamic Systems and Applications
, 27
, 819
–837
(2018
).9.
A.
Georgieva
Double Fuzzy Sumudu Transform to Solve Partial Volterra Fuzzy Integro-Differential Equations
, Mathematics
8
, 692
, (2020
).10.
A.
Georgieva
Solving two-dimensional nonlinear fuzzy Volterra integral equations by homotopy analysis method
, Demonstratio Mathematica
, 54
, 1124
(2021
).11.
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
, 129
–145
(2019
).12.
A.
Georgieva
, S.
Hristova
Homotopy Analysis Method to Solve Two-Dimensional Nonlinear Volterra- Fredholm Fuzzy Integral Equations
, Fractal Fract.
, 4
, 9
(2020
).13.
A.
Georgieva
, I.
Naydenova
Numerical solution of nonlinear Urisohn-Volterra fuzzy functional integral equa- tions
, AIP Conference Proceedings
, 1910
, 050010
-1
–050010-8 (2017
).14.
A.
Georgieva
, I.
Naydenova
. Application of homotopy analysis method for solving of two-dimensional linear Volterra fuzzy integral equations
. AIP Conference Proceedings
, 2159
, 030012
, 2019
.15.
R.
Goetschel
, W.
Voxman
. Elementary fuzzy calculus
, Fuzzy Sets and Systems
, 18
:31
–43
, 1986
.16.
R. V.
Gorder
, K.
Vajravelu
, On the selection of auxiliary functions, operators, and convergence control pa-rameters in the application of the homotopy analysis method to nonlinear differential equations: a general approach
, Commun. Nonlinear Sci. Numer. Simul.
, 14
, 4078
–4089
, (2009
).17.
L.
Hooshangian
. Nonlinear Fuzzy Volterra Integro-differential Equation of N-th Order: Analytic Solution and Existence and Uniqueness of Solution
, Int. J. Industrial Mathematics
, 11
, 1
(2019
).18.
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
)19.
S.
Liao
Homotopy Analysis Method in Nonlinear Differential Equations
, Springer, Higher Education Press
, Berlin Beijing
, (2012
).20.
S. J.
Liao
On the homotopy analysis method for nonlinear problem
, Appl. Math. Comput.
, 147
, 499
–513
(2004
).21.
F.
Mirzaee
, M. K.
Yari
, M.
Paripour
Solving linear and nonlinear Abel fuzzy integral equations by homotopy analysis method
, J. Taibah University for Science
, 9
, 104
–115
, (2015
).22.
Z.
Odibat
A study on the convergence of homotopy analysis method
, Appl. Math. Comput.
, 217
, 782
–789
(2010
).23.
M.
Puri
, D.
Ralescu
. Differential and fuzzy functions
. Mathematics Analysis and Applications
91
, 552
–558
(1983
).24.
M.
Russo
, R. V.
Gorder
Control of error in the homotopy analysis of nonlinear Klein-Gordon initial value problems
, Appl. Math. Comput.
, 219
, 6494
–6509
(2013
).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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