In this paper, we use double fuzzy Sumudu transform method (DSTM) to solve two dimensional fuzzy convolution Volterra integral equations (2D-FCVIE). By using double fuzzy Sumudu transform method the problem reducing to algebraic problem. The convolution, its properties and convolution theorem with a proof are discussed in some detail. We give some preliminary results of the double fuzzy Sumudu transform method and describe the method of this paper. Finally, illustrative example with fuzzy and crisp convolution kernels is given to show the ability of the proposed method.
REFERENCES
1.
A.
Alidema
, A.
Georgieva
. Adomian decomposition method for solving two-dimensional nonlinear Volterra fuzzy integral equations
. AIP Conference Proceedings.
2048
:050009
, 2018
.2.
S.
Chang
, L.
Zadeh
. On fuzzy mapping and control
. IEEE Transactions on Systems, Man, and Cybernetics
, 1
:30
–34
, 1972
.3.
D.
Dubois
, and H.
Prade
. Towards fuzzy differential calculus I,II,III
. Fuzzy Sets and Systems
, 8
:105-116
, 225
–233
, 1982
.4.
S.
Enkov
, A.
Georgieva
, R.
Nikolla
. Numerical solution of nonlinear Hammerstein fuzzy functional integral equations
. AIP Conference Proceedings
, 1789
:030006-1
–030006-8
, 2016
.5.
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
.6.
A.
Georgieva
Solving two-dimensional nonlinear Volterra-Fredholm fuzzy integral equations by using Adomian decomposition method
. Dynamic Systems and Applications.
27
:819
–837
, 2018
.7.
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
.8.
A.
Georgieva
, I.
Naydenova
. Numerical solution of nonlinear Urisohn-Volterra fuzzy functional integral equations
. AIP Conference Proceedings
, 1910
:050010-1
–050010-8
, 2017
.9.
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
.10.
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
)11.
O.
Kaleva
, Fuzzy differential equations
, Fuzzy Sets and Systems
, 24
:301
–317
, 1987
.12.
N. A.
Khan
, O. A.
Razzaq
and M.
Ayyaz
. On the solution of fuzzy differential equations by Fuzzy Sumudu Transform
. Nonlinear Engineering
, 4
:4960
, 2015
.13.
A.
Kilicman
, H.
Eltaybed
and K. A. M.
Atan
. A note on the comparison between Laplace and Sumudu transforms
. Bulletin of the Iranian Mathematical Society
, 37
:131
–141
, 2011
.14.
A.
Kilicman
, M.
Omran
. On double Natural transform and its applications
. Joural of Nonlinear science and Applications
, 10
:1744
–1754
, 2017
.15.
J. W.
Layman
. The Hankel transform and some of its properties
. Journal of Integer Sequences
, 4
:111
, 2001
.16.
J.
Mordeson
, W.
Newman
. Fuzzy integral equations
. Information Sciences
, 87
:215
–229
, 1995
.17.
V.
Namias
. The fractional order Fourier transform and its application to quantum mechanics
. IMA Journal Appllied Mathemathics
, 25
:241265
, 1980
.18.
S.
Seikkala
. On the fuzzy initial value problem
. Fuzzy Sets and Systems
, 24
:319
–330
, 1987
.19.
S.
Saitoh
. The Weierstrass transform and an isometry in the heat equation
. Applicable Analysis
, 16
:16
, 1983
.20.
R.
Spinelli
. Numerical inversion of a Laplace transform
. SIAM Journal Numerical Analysis
, 3
:636649
, 1966
.21.
C.
Tranter
. The use of the Mellin transform in finding the stress distribution in an infinite wedge
, Q. J. Mech. Appl. Math.
, 1
:125130
, 1948
.22.
J. M.
Tchuenche
and N. S.
Mbare
. An application of the double Sumudu transform
. Applied Mathematical Sciences
, 1
: 31
–39
, 2007
.23.
G. K.
Watugala
. Sumudu transform a new integral transform to solve differential equations and control engineering problems
. Internat. J. Math. Ed. Sci. Tech.
, 24
:35
–43
, 1993
.24.
G. K.
Watugala
. The Sumudu transform for functions of two variables
. Mathematical Engineering in Industry
, 8
:293
–302
, 2002
.25.
R.
Goetschel
, W.
Voxman
. Elementary fuzzy calculus
, Fuzzy Sets and Systems
, 18
:31
–43
, 1986
.26.
C.
Wu
, Z.
Gong
. On Henstock integral of fuzzy-number-valued functions(I)
. Fuzzy Sets and Systems
, 120
:523
–532
, 2001
.27.
C.
Wu
, C.
Wu
. The supremum and infimum of these to fuzzy-numbers and its applications
, J. Math. Anal. Appl.
, 210
:499
–511
, 1997
.
This content is only available via PDF.
© 2019 Author(s).
2019
Author(s)
