Generalized Intuitionistic Fuzzy Laplace Transform and Its Application in Electrical Circuit

Abstract

In this paper we describe the generalized intuitionistic fuzzy laplace transform method for solving Şrst order generalized intutionistic fuzzy differential equation.The procedure is applied in imprecise electrical circuit theory problem. Here the initial condition of those applications is taken as Generalized Intuitionistic triangular fuzzynumbers GITFNs

Keywords

• Fuzzy differential equation,Generalized intuitionistic triangular fuzzy number,First order differential equation

References

1. Zadeh, L. A., (1965), Fuzzy sets, Information and Control, 8, pp. 338-353.
2. Dubois, D and Parade, H., (1978), Operation on Fuzzy Number. International Journal of Fuzzy
3. system, 9: pp. 613-626.
4. Atanassov, K.T., (1983), Intuitionistic fuzzy sets, VII ITKR’s Session, SoŞa, Bulgarian.
5. Atanassov, K.T., (1986), Intuitionistic fuzzy sets, Fuzzy Sets and Systems, vol.20, pp.87-96.
6. Deng-Feng-Li., (2008), A note on : Using intuitionistic fuzzy sets for fault tree analysis on printed
7. circuit board assembly, Micro Electronics Reliability, 48, pp. 1741.
8. Mahapatra, G. S. and Roy, T. K., (2009), Reliability Evaluation using Triangular Intuitionistic Fuzzy

9. numbers Arithmetic operations, World Academy of science, Engineering and Technology, 50, pp. 574-581.
10. Nagoorgani, A., Ponnalagu, K., (2012), A New Approach on Solving Intuitionistic Fuzzy Linear
11. Programming Problem, Applied Mathematical Sciences, Vol.6, no.70, pp. 3467-3474.
12. Atanassov, K.T., and Gargov, G., (1989), Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and
13. Systems, vol.31, no.3, pp.343-349.
14. Atanassov, K.T., (1989), More on intuitionistic fuzzy sets, Fuzzy Sets and Systems, vol.33, no.1, pp.37-46.
15. Atanassov, K.T., (1999), Intuitionistic Fuzzy Sets, Physica-Verlag, Heidelberg, New York.
16. Atanassov, K.T., (2000), Two theorems for Intuitionistic fuzzy sets, Fuzzy Sets and Systems, vol.110, pp.267-269.
17. Atanassov, K.T., and Gargov, G., (1998), Elements of intuitionistic fuzzy logic, Part I, Fuzzy Sets and Systems, vol.95, no.1, pp.39-52.
18. Szmidt, E., and Kacprzyk, J., (2000), Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systems, vol.114, no.3, pp.505-518.
19. Buhaesku, T., (1989), Some observations on intuitionistic fuzzy relations, Itinerant Seminar of Func- tional Equations, Approximation and Convexity, pp.111-118.
20. Ban, A.I., (2006), Nearest interval approximation of an intuitionistic fuzzy number, Computational Intelligence, Theory and Applications, Springer-Verlag, Berlin, Heidelberg, pp.229-240.
21. Deschrijver, G., and Kerre, EE.,(2002), On the relationship between intuitionistic fuzzy sets and some other extensions of fuzzy set theory, Journal of Fuzzy Mathematics, vol.10, no.3, pp.711-724.
22. Stoyanova, D., (1993), More on Cartesian product over intuitionistic fuzzy sets, BUSEFAL, vol.54, pp.9-13.
23. Cornelis, C., Deschrijver, G. and Kerre, E.E., (2004) ,Implication in intuitionistic fuzzy and interval- valued fuzzy set theory: construction, application, International Journal of Approximate Reasoning, Vol. 35, pp.55-95.
24. Buhaesku, T., (1988), On the convexity of intuitionistic fuzzy sets, Itinerant Seminar on Functional Equations, Approximation and Convexity, Cluj-Napoca, pp.137-144.
25. Gerstenkorn, T. and Manko, J., (1991), Correlation of intuitionistic fuzzy sets, Fuzzy Sets and Systems, Vol. 44, pp.39-43.
26. Stoyanova, D. and Atanassov, K.T., (1990), Relation between operators, deŞned over intuitionistic fuzzy sets’, IM-MFAIS, Vol. 1, SoŞa, Bulgaria, pp.46-49.
27. Stoyanova, D., (1993), More on cartesian product over intuitionistic fuzzy sets, BUSEFAL, Vol. 54, pp.9-13.
28. Mahapatra, G.S. and Roy, T.K., (2009) ,Reliability evaluation using triangular intuitionistic fuzzy numbers arithmetic operations, Proceedings of World Academy of Science, Engineering and Technol- ogy, Vol. 38, pp.587-595.
29. Hajeeh, M.A., (2011) ,Reliability and availability of a standby system with common cause fail- ure,International Journal of Operational Research, Vol. 11, No. 3, pp.343-363.
30. Persona, A., Sqarbossa, F. and Pham, H., (2009), Systemability function to optimization reliability in random environment, International Journal of Mathematics in Operational Research,Vol. 1, No. 3, pp.397-417.
31. Prabha, B., Sujatha, R. and Srikrishna, S., (2011) ,Posfust reliability of a uniŞed fuzzy markov model, International Journal of Reliability and Safety, Vol. 5, No. 1, pp.83-94.
32. Nikolaidis, E. and Mourelatos, Z.P., (2011), Imprecise reliability assessment when the type of the probability distribution of the random variables is unknown’, International Journal of Reliability and Safety, Vol. 5, No. 2, pp.140-157.
33. Kumar, M., Yadav, S.P. and Kumar, S., ( 2011), A new approach for analyzing the fuzzy system relia- bility using intuitionistic fuzzy number’, International Journal of Industrial and Systems Engineering, Vol. 8, No. 2, pp.135-156.
34. Wang, Y., (2010), Imprecise probabilities based on generalized intervals for system reliability assess- ment’, International Journal of Reliability and Safety, Vol. 4, No. 4, pp.319-342.
35. Shaw A. K. and Roy T. K., (2013), Trapezoidal Intuitionistic Fuzzy Number with some arithmetic operations and its application on reliability evaluation, Int. J. Mathematics in Operational Research, Vol. 5, No. 1.
36. Adak, A. K., Bhowmik, M. and Pal, M., (2012), Intuitionistic Fuzzy Block Matrix and its Some Properties, Annals of Pure and Applied Mathematics, Vol. 1, No. 1, pp. 13-31.
37. Varghese, A. and Kuriakose, S., (2012), Centroid of an intuitionistic fuzzy number, Notes on Intu- itionistic Fuzzy Sets Vol. 18, No. 1, pp. 19-24.
38. Kandel, A., Byatt, J. W., (1978), Fuzzy differential equations, in: Proceedings of International Con- ference Cybernetics and Society, Tokyo, pp. 1213- 1216.
39. Chang, S. L., Zadeh, L. A., (1972), On fuzzy mapping and control, IEEE Transactions on Systems Man Cybernetics 2, pp. 330-340.
40. Dubois, D., Prade, H., (1982), Towards fuzzy differential calculus: Part 3, Differentiation. Fuzzy Sets and Systems 8, pp. 225-233.
41. Puri, M. L., Ralescu, D., (1983), Differential for fuzzy function, J. Math. Anal. Appl. 91, pp. 552-558.
42. Goetschel, R., Voxman, W., (1986), Elementary calculus, Fuzzy Sets and Systems 18, pp. 31-43.
43. Seikkala, S., (1987), On the fuzzy initial value problem, Fuzzy Sets and Systems 24, pp. 319-330.
44. Friedmen, M., Ming, M. and Kandel, A., (1996), Fuzzy derivatives and fuzzy Cauchy problems us- ing LP metric, in: Da Ruan (ED.), Fuzzy Logic Foundations and Industrial Applications, Kluwer Dordrecht, pp. 57-72.
45. Kandel A., Friedmen, M. and Ming, M., (1996), On Fuzzy dynamical processes, Proc. FUZZIEEE’ 96, New Orleans, pp. 1813-1818.
46. Chalco-Cano, Y. and Romn-Flores, H., (2009), Comparation between some approaches to solve fuzzy differential equations, Fuzzy Sets and Systems 160, pp. 1517-1527.
47. Hllermeier, E., (1997) , An approach to modelling and simulation of uncertain systems, International Journal of Uncertainty, Fuzziness and Knowledge- Based Systems 5, pp. 117-137.
48. Lan, H. Y. and Nieto, J. J., (2009), On initial value problems for Şrst-order implicit impulsive fuzzy differential equations,Dynamics Systems and Applications 18, pp. 677-686.
49. Nieto J. J., Rodrguez-Lpez, R. and Georgiou, D. N., (2008), Fuzzy differential systems under gener- alized metric spaces approach, Dynamic Systems & Applications 17, pp. 1-24.
50. Buckley J. J. and Feuring, T., (2000), Fuzzy differential equations, Fuzzy Sets and Systems 110, pp. 43-54.
51. Diamond, P. and Kloeden, P., (1994), Metric Spaces of Fuzzy Sets, World ScientiŞc, Singapore.
52. Kaleva, O., (2006), A note on fuzzy differential equations, Nonlinear Analysis 64, pp. 895-900.
53. Nieto, J. J., Rodrguez-Lpez, R. and Franco, D., (2006), Linear Şrst order fuzzy differential equations, International Journal of Uncertainty, Fuzziness, and Knowledge-Based Systems 14, pp. 687-709.
54. Khastan, A., Nieto, J. J. and Rosana Rodriguez-Lopez., (2011),Variation of constant formula for Şrst order fuzzy differential equations, Fuzzy Sets and Systems 177, pp. 20-33.
55. Lata, S. and Kumar, A., (2012), A new method to solve time-dependent intuitionstic fuzzy differential equations and its application to analyze the intuitionistic fuzzy reliability systems, SAGA ,Concurrent Engineering published online.
56. Melliani, S. and Chadli, L. S., (2001), Introduction to intuitionistic fuzzy partial differential equations, Notes on IFS,Vol 7, Number 3, pp. 39-42.
57. Abbasbandy, S. and Allah Viranloo, T., (2002), Numerical solution of fuzzy differential equation by Runge-Kutta method and the intutionistic treatment, Notes on IFS,Vol 8, Number 3, pp. 45-53.
58. Melliani .S and Chadli, L. S., (2000), Intuitionistic fuzzy differential equation, Notes on IFS, Vol 6, Number 2, pp. 37-41.
59. Buckley, J. J., Feuring, T. and Hayashi, Y., (2002), Linear systems of Şrst order ordinary differential equations: Fuzzy initial conditions, Soft Computing, 6, pp. 415-421.
60. Oberguggenberger, M. and Pittschmann, S., (1999), Differential equations with fuzzy parameters, Math. Mod. Syst. 5, pp. 181-202.
61. Casasnovas, J. and Rossell, F., (2005), Averaging fuzzy biopolymers, Fuzzy Sets and Systems 152, pp. 139-158.
62. El Naschie, M. S., (2005), From experimental quantum optics to quantum gravity via a fuzzy Khler manifold, Chaos, Solitons and Fractals 25, pp. 969-977.
63. Bencsik, A., Bede, B., Tar, J. and Fodor, J., (2006), Fuzzy differential equations in modeling hydraulic differential servo cylinders, In: Third Romanian-Hungarian joint symposium on applied computational intelligence (SACI), Timisoara, Romania.
64. Zarei, H., Vahidian Kamyad, A. and Heydari, A. A., Fuzzy Modeling and Control of HIV Infection, Computational and Mathematical Methods in Medicine Volume 2012, Article ID 893474, pp. 17.
65. Diniz, G. L., Fernandes, J.F.R., Meyer,J.F.C.A. and Barros, L. C., (2001), A fuzzy Cauchy problem modeling the decay of the biochemical oxygen demand in water, IEEE.
66. Ahmad, M. Z. and De Baets, B., (2009), A Predator-Prey Model with Fuzzy Initial Populations, IFSA-EUSFLAT.
67. Barros, L. C., Bassanezi, R. C. and Tonelli, P. A., (2000), Fuzzy modelling in population dynamics, Ecol. Model. 128, pp. 27-33.
68. Bede, B., Rudas, I. J. and Fodor, J., (2007), Friction Model by Fuzzy Differential Equations, IFSA 2007, LNAI 4529, Springer-Verlag Berlin Heidelberg, pp.23-32.
69. Mondal, S. P., Banerjee, S. and Roy, T. K., (2013), First Order Linear Homogeneous Ordinary Dif- ferential Equation in Fuzzy Environment, Int. J. Pure Appl. Sci. Technol. 14(1), pp. 16-26.
70. Mondal, S. P. and Roy, T. K., (2013), First Order Linear Non Homogeneous Ordinary Differential Equation in Fuzzy Environment, Mathematical theory and Modeling, Vol.3, No.1, pp. 85-95.
71. Mondal, S. P. and Roy, T. K., First Order Linear Homogeneous Ordinary Differential Equation in Fuzzy Environment Based On Laplace Transform, Accepted.
72. Mondal, S. P. and Roy, T. K., First Order Linear Homogeneous Fuzzy Ordinary Differential Equation Based on Lagrange Multiplier Method,Accepted.
73. Bede,B. and Gal, S. G., (2004), Almost periodic fuzzy-number-value functions, Fuzzy Sets and Systems 147, pp. 385-403.
74. Chalco-Cano, Y. and Roman-Flores, H., On new solutions of fuzzy differential equations, Chaos, Solitons and Fractals 38, pp. 112-119.
75. Mondal, S. P. and Roy, T. K., (2008), First Order Linear Homogeneous Fuzzy Ordinary Differential Equation with initial value as triangular intuitionistic fuzzy number, Accepted.
76. Mondal, S. P. and Roy, T. K., Non linear arithmetic operation on Generalized triangular intutionistic fuzzy numbers, Communicated.
77. Allahviranloo, T. and Barkhordari Ahmadi, M., (2010), Fuzzy Laplace transforms, Soft Computing 14, pp. 235-243.
78. Ramazannia Tolouti, S. J. and Barkhordary Ahmadi, M., (2010), Fuzzy Laplace Transform on Two Order Derivative and Solving Fuzzy Two Order Differential Equation, Int. J. Industrial Mathematics Vol. 2, No. 4, pp. 279-293.
79. Salahshour, S. and Haghi, E., Solving Fuzzy Heat Equation by Fuzzy Laplace Transforms.
80. Ahmad, N., Mamat, M., Kumar, J. K. and Amir Hamzah, N. S., (2012), Solving Fuzzy Duffing’s Equation by the Laplace Transform Decomposition, Applied Mathematical Sciences Vol. 6, no. 59, pp. 2935-2944.
81. Omar Abu-Arqub et al. (2013), Analytical Solutions of Fuzzy Initial Value Problems by HAM, Appl. Math. Inf. Sci.7, No. 5, pp. 1903-1919.