AUTHORS: Manoj Sahni, Ritu Sahni, Rajkumar Verma, Ashnil Mandaliya, Dhairya Shah
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In this paper, the solution for second order Cauchy Euler equation is derived using generalized trapezoidal intuitionistic fuzzy number. Further as an application we study the problem for finding the radial displacement of a solid disk with given boundary condition in the form of generalized trapezoidal intuitionistic fuzzy number. The obtained solutions are drawn graphically for different radii using (α,β)-cuts.
KEYWORDS: Fuzzy Sets; Cauchy-Euler differential equation; α,β–cuts; Trapezoidal Intuitionistic fuzzy number; Generalized Hukuhara Derivative, Solid Disk.
REFERENCES:
[1] L. A. Zadeh, Fuzzy Sets, Information and
Control, Vol. 8, 1965, pp. 338-353.
[2] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy
Sets and Systems, Vol. 20, 1986, pp. 87-96.
[3] K. T. Atanassov, More on Intuitionistic fuzzy
sets, Fuzzy Sets and Systems, Vol. 33, No. 1,
1989, pp. 37-46.
[4] K. Atanassov, and G. Gargov, Interval valued
intu-itionistic fuzzy sets, Fuzzy Sets and
Systems, Vol. 31, No. 3, 1989, pp.343-349.
[5] V. Khatibi, and G. A. Montazer, Intuitionistic
fuzzy set vs. fuzzy set application in medical
pattern recognition, Artificial Intelligence in
Medicine, Vol. 47, No. 1, 2009, pp. 43-52.
[6] M. Rencova, An Example of Applications of
Intuitionistic Fuzzy Sets to Sociometry,
Bulgarian Academy of Sciences, Cybernetics
and Information Technologies, Vol. 9, No. 2,
2009, pp. 43-45.
[7] B. Davvaz, and E. H. Sadrabadi, An application
of intuitionistic fuzzy sets in medicine,
International Journal of Biomathematics, Vol.
9, No. 3, 2016, pp. 1-15.
[8] F. Tugrul, M. Gezercan, and M. Citil,
Application of intuitionistic fuzzy sets in high
school determination via normalized Euclidean
distance method, 4th IFSCOM Notes on
Intuitionistic fuzzy sets, Vol. 23, No. 1, 2017,
pp. 42-47.
[9] T. Johnson, Applications of Intuitionistic fuzzy
sets in the academic career of the students,
Indian Journal of Science and Technology, Vol.
10, No. 34, 2017.
[10] M. Hukuhara, Integration des applications
measurables dont la valeur est un compact
convex, Funkcial. Ekvacioj, Vol. 10, 1967, pp.
205-229.
[11] S. L Chang, and L.A. Zadeh, On fuzzy
mapping and control, IEEE Trans, Systems Man
Cybernet,. Vol. 2, 1972, pp. 30-34.
[12] A. Kandel, and W. J. Byatt, Fuzzy differential
equations, Proceedings of the International
Conference on Cybernetics and Society, Tokyo,
Japan , 1978, pp. 1213–1216.
[13] D. Dubois, and H. Prade, Towards fuzzy
differential calculus: Part 3, differentiation,
Fuzzy Sets and Systems , Vol. 8, 1982, pp. 225-
233.
[14] M. L. Puri, and D. Ralescu, Differential for
fuzzy function, J. Math. Anal. Appl., Vol. 91,
1983, pp. 552-558.
[15] R. G. Voxman, Elementary fuzzy calculus.
Fuzzy sets and systems, Vol. 18, No. 1, 1986,
pp. 31-43.
[16] O. Kaleva, Fuzzy differential Equations, Fuzzy
Sets and Systems, Vol. 24, No. 3, 1987, pp. 301-
317.
[17] S. Seikkala, On the fuzzy initial value problem,
Fuzzy Sets Systems, Vol. 24, 1987, pp. 319–330.
[18] V. Lakshmikantham, and R. Mohapatra,
Theory of fuzzy differential equations and
inclusions, 1st Edition. London: Taylor and
Francis, 2003.
[19] M. Friedman, M. Ming, and A. Kandel.,
Solutions to fuzzy integral equations with
arbitrary kernels, International Journal of
Approxiamte Reasoning, Vol. 20, No. 3, 1999,
pp. 249-262.
[20] P. Diamond, Stability and periodicity in fuzzy
differential equations, IEEE Transactions in
Fuzzy Systems, Vol. 8, 2000, pp. 583-590.
[21] S. Melliani, and L. S. Chadli, Introduction to
intuitionistic fuzzy differential equations, Notes
on IFS, Vol. 6, No. 2, 2000, pp. 31-41.
[22] S. Melliani, and L.S. Chadli, Introduction to
intuitionistic fuzzy partial differential equations,
Notes on IFS, Vol. 7, No. 3, 2001, pp. 39-42,.
[23] S. Abbasbandy, and T. A.Viranloo, Numerical
solution of Fuzzy differential equations by
Runge-kutta method, Mathematical and
Computational Applications, Vol. 11, No. 1,
2004, pp. 117-129.
[24] T. Allahviranloo, and M. Barkhordari Ahmadi,
Fuzzy Laplace transforms, Soft Computing,
Vol.14, 2010, pp. 235-243.
[25] S. J. Ramazannia Tolouti, and M. Barkhordary
Ahmadi, Fuzzy Laplace Transform on Two
Order Derivative and Solving Fuzzy Two Order
Differential Equation, Int. J. Industrial
Mathematics, Vol. 2, No. 4, 2010, pp. 279-293.
[26] S.P. Mondal, and T. K. Roy, Generalized
intuitionistic fuzzy Laplace transform and its
application in electrical circuit, TWMS J.Appl.
Eng. Math., Vol. 5, No. 1, 2015, pp. 30-45.
[27] S. P. Mondal, and T. K. Roy, Second order
linear differential equations with generalized
trapezoidal intuitionistic fuzzy boundary value,
Journal of Linear and Topological Algebra,
Vol. 4, No. 2, 2015, pp. 115-129.
[28] N. Ahmad, M. Mamat, J. K. Kumar, and N. S.
Amir Hamzah, Solving Fuzzy Duffing’s
Equation by the Laplace Transform
Decomposition, Applied Mathematical Sciences,
Vol. 6, No. 59, 2012, pp. 2935-2944.
[29] B. Bede, I. J. Rudas, and A. L.Bencsik, First
order linear fuzzy differential equations under
generalized differentiability, Information
Sciences, Vol. 177, 2007, pp. 1648-1662.
[30] C. Vasavi, G. Suresh Kumar, T. Srinivasa Rao,
and B. V. Appa Rao, Application of fuzzy
differential equations for cooling problems,
International Journal of Mechanical
Engineering and Technology, Vol. 8, Issue 12,
2017, pp. 712-72.
[31] S. P. Mondal, and T. K. Roy, First order
homogeneous ordinary differential equation
with initial value as triangular intuitionistic
fuzzy number, Journal of Uncertainty in
Mathematics Science, Vol. 2014, 2014, pp. 1-
17,.
[32] S. Biswas, S. Banerjee and T. K. Roy, Solving
intuitionistic fuzzy differential equations with
linear differential operator by Adomian
decomposition method, 3rd Int. IFS Conf., ,
Mersin, Turkey, Vol. 22, No. 4, 29th Aug. 1
Sep. 2016, pp. 25-41.
[33] M. Shapique and C. Jesuraj, Solutions to Fuzzy
differential equations using pentagonal
intuitionistic fuzzy numbers, MAYFEB Journal
of Mathematics, Vol. 2, 2017, pp. 8-20.
[34] S. S. Saade, Mapping convex and normal
fuzzy sets, Fuzzy Sets and Systems, Vol. 81,
1996, pp. 251-256,.
[35] H .Nasseri, Fuzzy Numbers: Positive and
Nonnegative, International Mathematical
Forum, Vol. 3, No. 36, 2008, pp. 1777-1780.
[36] S. P. Mondal, and T. K. Roy, First order linear
homogeneous fuzzy ordinary differential
equation based on Lagrange multiplier method,
Journal of soft computing and applications,
Vol. 2013, 2013, pp. 1-17.