On interactive fuzzy boundary value problems

Abstract In this paper we use the concept of interactivity between fuzzy numbers for the solution to a linear fuzzy boundary value problem (FBVP). We show that a solution of a FBVP, with non-interactive fuzzy numbers as boundary values, can be obtained by the Zadeh's Extension Principle. In addition, we show it is possible to obtain a fuzzy solution by means of the extension principle based on joint possibility distributions for the case where the boundary values are given by interactive fuzzy numbers. Examples of FBVPs with both cases, interactive and non-interactive, are presented. Also from arithmetic operations for linearly correlated fuzzy numbers, we compare our solutions to the one proposed by Gasilov et al. We conclude that the fuzzy solution in the interactive case (when the boundary values are linearly correlated fuzzy numbers) is contained in the fuzzy solution for the non-interactive case. Finally, we present the fuzzy solution for a nonlinear FBVP with Gaussian fuzzy numbers as boundary values.

[1]  B. Bede,et al.  Fuzzy Differential Equations in Various Approaches , 2015 .

[2]  Hung T. Nguyen,et al.  A note on the extension principle for fuzzy sets , 1978 .

[3]  Xiaoping Xue,et al.  Two-point boundary value problems of uncertain dynamical systems , 2011, Fuzzy Sets Syst..

[4]  Witold Pedrycz,et al.  Fuzzy Systems Engineering - Toward Human-Centric Computing , 2007 .

[5]  David L Powers,et al.  Boundary Value Problems and Partial Differential Equations Ed. 6 , 2009 .

[6]  Sahin Emrah Amrahov,et al.  Solution of linear differential equations with fuzzy boundary values , 2014, Fuzzy Sets Syst..

[7]  X. Guo,et al.  Fuzzy approximate solutions of second-order fuzzy linear boundary value problems , 2013 .

[8]  Lotfi A. Zadeh,et al.  The Concepts of a Linguistic Variable and its Application to Approximate Reasoning , 1975 .

[9]  Laécio C. Barros,et al.  Fuzzy differential equations with interactive derivative , 2017, Fuzzy Sets Syst..

[10]  Christer Carlsson,et al.  Additions of completely correlated fuzzy numbers , 2004, 2004 IEEE International Conference on Fuzzy Systems (IEEE Cat. No.04CH37542).

[11]  Barnabás Bede,et al.  A note on "two-point boundary value problems associated with non-linear fuzzy differential equations" , 2006, Fuzzy Sets Syst..

[12]  J. Nieto,et al.  A boundary value problem for second order fuzzy differential equations , 2010 .

[13]  V. Lakshmikantham,et al.  Initial and boundary value problems for fuzzy differential equations , 2003 .

[14]  T. Allahviranloo,et al.  A Numerical Method for Two-Point Fuzzy Boundary Value Problems , 2011 .

[15]  Laécio C. Barros,et al.  A First Course in Fuzzy Logic, Fuzzy Dynamical Systems, and Biomathematics - Theory and Applications , 2016, Studies in Fuzziness and Soft Computing.

[16]  Peter Sussner,et al.  A parametrized sum of fuzzy numbers with applications to fuzzy initial value problems , 2018, Fuzzy Sets Syst..

[17]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[18]  Laécio Carvalho de Barros,et al.  Fuzzy differential equation with completely correlated parameters , 2015, Fuzzy Sets Syst..