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Differential equation with interval valued fuzzy number and its applications

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Abstract

The paper presents an adaptation of solution of first order differential equation with initial value as interval valued triangular fuzzy number. The arithmetic operation of interval-valued triangular fuzzy number is re-established and studied with the help of fuzzy extension principle method. Demonstration of fuzzy solutions of the governing differential equation is carried out using the approaches namely, generalized Hukuhara derivative. Additionally, different illustratively examples and applications are also undertaken with the useful table and graph for usefulness for attained to the proposed approaches.

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References

  • Abirami D, Dinagar DS (2015) On L–R type interval valued fuzzy numbers in critical path analysis. Int J Fuzzy Math Arch 6(1):77–83

    Google Scholar 

  • Allahviranloo T, Ahmadi MB (2010) Fuzzy Laplace transforms. Soft Comput 14:235–243

    Article  MATH  Google Scholar 

  • Allahviranloo T, Abbasbandy S, Salahshour S, Hakimzadeh A (2011) A new method for solving fuzzy linear differential equations. Computing 92:181–197

    Article  MathSciNet  MATH  Google Scholar 

  • Araghi MAF, Fallahzadeh A (2016) Inherited collocation method for fuzzy differential equations. J Fuzzy Set Valued Anal 2016:8–18

    Google Scholar 

  • Bede B (2006) A note on two-point boundary value problems associated with non-linear fuzzy differential equations. Fuzzy Sets Syst 157:986–989

    Article  MathSciNet  MATH  Google Scholar 

  • Bede B, Gal SG (2005a) Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets Syst 151:581–599

    Article  MathSciNet  MATH  Google Scholar 

  • Bede B, Gal SG (2005b) Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Set Syst 151:581–599

    Article  MathSciNet  MATH  Google Scholar 

  • Bede B, Rudas IJ, Bencsik AL (2007) First order linear fuzzy differential equations under generalized differentiability. Inf Sci 177:1648–1662

    Article  MathSciNet  MATH  Google Scholar 

  • Bhatia N, Kumar A (2012) Sensitivity analysis for interval valued fully fuzzy linear programming problems. J Appl Res Technol 10(6):871–884

    MathSciNet  Google Scholar 

  • Buckley JJ, Feuring T (2000) Fuzzy differential equations. Fuzzy Sets Syst 110:43–54

    Article  MathSciNet  MATH  Google Scholar 

  • Buckley JJ, Feuring T (2001) Fuzzy initial value problem for Nth-order linear differential equations. Fuzzy Sets Syst 121:247–255

    Article  MathSciNet  MATH  Google Scholar 

  • Chalco-Cano Y, Romn-Flores H (2008) On the new solution of fuzzy differential equations. Chaos Solitons Fractals 38:112–119

    Article  MathSciNet  MATH  Google Scholar 

  • Chalco-Cano Y, Rojas-Medar MA, Romn-Flores H (2007) Sobre ecuaciones diferencial esdifusas. Bol Soc Esp Mat Apl 41:91–99

    Google Scholar 

  • Chalco-Cano Y, Romn-Flores H, Rojas-Medar MA (2008) Fuzzy differential equations with generalized derivative. In: Proceedings of the 27th North American fuzzy information processing society international conference. IEEE

  • Chang SSL, Zadeh LA (1972) On fuzzy mappings and control. IEEE Trans Syst Man Cybern SMC–2:30–34

    Article  MathSciNet  MATH  Google Scholar 

  • Chang SL, Zadeh LA (1972b) On fuzzy mapping and control. IEEE Trans Syst Man Cybern 2:30–34

    Article  MathSciNet  MATH  Google Scholar 

  • Chen M, Wu C, Xue X, Liu G (2008) On fuzzy boundary value problems. Inf Sci 178:1877–1892

    Article  MathSciNet  MATH  Google Scholar 

  • Diamond P, Kloeden P (1994) Metric space of fuzzy sets. World Scientific, Singapore

    MATH  Google Scholar 

  • Dirbaz M, Dirbaz F (2016) Numerical Solution of impulsive fuzzy initial value problem by modified Euler’s method. J Fuzzy Set Valued Anal 2016:50–57

    Article  Google Scholar 

  • Dizicheh AK, Salahshour S, Ismail F, Hosseini AA (2016) On new solutions of linear system of first-order fuzzy differential equations with fuzzy coefficient. J Fuzzy Set Valued Anal 2016:110–117

    Article  Google Scholar 

  • Dubois D, Prade H (1978) Operations on fuzzy numbers. Int J Syst Sci 9:613–626

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois D, Prade H (1982) Towards fuzzy differential calculus: part 3. Differ Fuzzy Sets Syst 8:225–233

    Article  MathSciNet  MATH  Google Scholar 

  • Firouz E, Keshavarz M, Khakrangin S (2016) Advantage of gH-difference on the Nth-fuzzy linear differential equations with constant coefficients. J Fuzzy Set Valued Anal 2016:85–93

    Article  Google Scholar 

  • Goetschel R, Voxman W (1986) Elementary calculus. Fuzzy Sets Syst 18:31–43

    Article  MathSciNet  MATH  Google Scholar 

  • Guijun W, Xiaoping L (1998) The applications of interval-valued fuzzy numbers and intervaldistribution numbers. Fuzzy Sets Syst 98(3):331–335

    Article  MATH  Google Scholar 

  • Hllermeier E (1997) An approach to modeling and simulation of uncertain dynamical systems. Int J Uncertain Fuzziness Knowl-Based Syst 5:117–137

    Article  Google Scholar 

  • Kalaichelvi A, Haritha Malini P, Janofer K (2012) Application of interval fuzzy matrices and interval valued fuzzy soft sets in the analysis of the factors influencing high scores in higher secondary examinations. Int J Math Sci Appl 2(2):777–780

    Google Scholar 

  • Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets Syst 24:301–317

    Article  MathSciNet  MATH  Google Scholar 

  • Khan NA, Razzaq OA (2016) An efficient computer based wavelets approximation method to solve Fuzzy boundary value differential equations. Nonlinear Eng 5(1):1–6

    Article  Google Scholar 

  • Khastan A, Nieto JJ (2010) A boundary value problem for second-order fuzzy differential equations. Nonlinear Anal 72:3583–3593

    Article  MathSciNet  MATH  Google Scholar 

  • Kumar P, Singh SB (2015) Fuzzy fault tree analysis using level (λ, π) interval-valued fuzzy numbers. Ind Eng Lett 5(2):36–42

    Google Scholar 

  • Lin FT (2002) Fuzzy job-shop scheduling based on ranking level (λ, 1) interval-valued fuzzy numbers. IEEE Trans Fuzzy Syst 10(4):510–522

    Article  Google Scholar 

  • Mondal SP, Roy TK (2013) First order linear homogeneous fuzzy ordinary differential equation based on lagrange multiplier method. J Soft Comput Appl 2013:1–17

    Google Scholar 

  • Mondal SP, Roy S, Das B (2016) Numerical solution of first order linear differential equations in fuzzy environment by Runge–Kutta–Fehlberg method and its application. Int J Differ Equ 2016:8150497

    MathSciNet  Google Scholar 

  • ORegan D, Lakshmikantham V, Nieto J (2003) Initial and boundary value problems for fuzzy differential equations. Nonlinear Anal 54:405–415

    Article  MathSciNet  MATH  Google Scholar 

  • Otadi M, Mosleh M (2016) Solution of fuzzy differential equations. Int J Ind Math 8(1):73–80

    MathSciNet  MATH  Google Scholar 

  • Paul S, Mondal SP, Bhattacharya P (2016) Discussion on fuzzy quota harvesting model in fuzzy environment: fuzzy differential equation approach. Model Earth Syst Environ 2:70

    Article  Google Scholar 

  • Puri ML, Ralescu DA (1983) Differentials of fuzzy functions. J Math Anal Appl 91:552–558

    Article  MathSciNet  MATH  Google Scholar 

  • Shen Y, Wang F (2016) A fixed point approach to the Ulam stability of fuzzy differential equation under generalized differentiability. J Intell fuzzy Syst 30(6):3253–3260

    Article  Google Scholar 

  • Stefanini L (2008) A generalization of Hukuhara difference. In: Dubois D, Lubiano MA, Prade H, Gil MA, Grzegorzewski P, Hryniewicz O (eds) Soft methods for handling variability and imprecision, Series on advances in soft computing, Vol 48. Springer Berlin, Heidelberg pp 203–210

  • Stefanini L, Bede B (2009) Generalized Hukuhara differentiability of interval-valued functions and interval differential equations. Nonlinear Anal 71:1311–1328

    Article  MathSciNet  MATH  Google Scholar 

  • Su J-S (2007) Fuzzy programming based on interval-valued fuzzy numbers and ranking. Int J Contemp Math Sci 2(8):393–410

    Article  MathSciNet  MATH  Google Scholar 

  • Tolouti SJR, Ahmadi MB (2010) Fuzzy Laplace transform on two order derivative and solving fuzzy two order differential equation. Int J Ind Math 2(4):279–293

    Google Scholar 

  • Wang G, Li X (2001) Correlation and information energy of interval-valued fuzzy number. Fuzzy Sets Syst 103(1):169–175

    Article  MATH  Google Scholar 

  • Wei SH, Chen SM (2008) A new approach for fuzzy risk analysis based on similarity measures of generalized fuzzy numbers. Expert Syst Appl 36(1):589–598

    Article  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh L (2005) Toward a generalized theory of uncertainty (GTU) an outline. Inf Sci 172:140

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology, Agartala, Jirania, Tripura, 799046, India

    Sankar Prasad Mondal

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  1. Sankar Prasad Mondal
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Correspondence to Sankar Prasad Mondal.

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Mondal, S.P. Differential equation with interval valued fuzzy number and its applications. Int J Syst Assur Eng Manag 7, 370–386 (2016). https://doi.org/10.1007/s13198-016-0474-7

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  • DOI: https://doi.org/10.1007/s13198-016-0474-7

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