ON INTERACTIVE SOLUTION FOR TWO POINT FUZZY BOUNDARY VALUE PROBLEM

Year 2024, Volume: 7 Issue: 2, 85 - 98, 31.07.2024

Tahir Ceylan

https://doi.org/10.33773/jum.1375017

Abstract

In this manuscript, the eigenvalues and eigenfunctions of the twopoint
fuzzy boundary value problem (FBVP) are analyzed under the concept of
interactivity between the fuzzy numbers found in the boundary conditions. A
fuzzy solution is provided for this problem via sup-J extension, which can be
seen as a generalization of Zadeh’s extension principle. Finally, an example is
presented in order to compare the given features.

Keywords

Fuzzy eigenfunction, Sup-J extension prensible, Interactive fuzzy numbers, Zadeh

References

• L.C. Barros, R.C. Bassanezi and P.A. Tonelli, On the continuity of the Zadeh's extension, Seventh IFSA World Congress, Prague, pp. 22-26 (1997).
• L.C. Barros, R.C. Bassanezi and W.A. Lodwick, A first Course in Fuzzy Logic, Fuzzy Dynamical Systems, and Biomathematics: Theory and applications, Springer Cham., London, (2017).
• L.C. Barros, F.S. Pedro, Fuzzy differential equations with interactive derivative, Fuzzy Sets and Systems, Vol. 309, pp. 64-80 (2017).
• B. Bede and L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, Vol. 230, pp. 119-141 (2013).
• C. Carlsson, R. Fuller and P. Majlender, Additions of Completely Correlated Fuzzy Numbers, IEEE International Conference on Fuzzy Systems, (2004).
• T. Ceylan and N. Altınışık, Eigenvalue problem with fuzzy coeffcients of boundary conditions, Scholars Journal of Physics, Mathematics and Statistics, Vol. 5, N. 2, pp. 187-193 (2018).
• P. Diamond and P. Kloeden, Metric spaces of fuzzy sets World Scientific, World Scientific, Singapore, (1994).
• E. Esmi, D.E. Sanchez, V.F. Wasques and L.C. Barros, Solutions of higher order linear fuzzy differential equations with interactive fuzzy values, Fuzzy Sets and Systems, Vol. 419, N. 1, pp. 122-140 (2021).
• N. Gasilov, S.E. Amrahov and A.G. Fatullayev, Linear differential equations with fuzzy boundary values, CoRR, pp. 696-700 (2011).
• L.T. Gomes, L.C. Barros and B. Bede, Fuzzy Differential Equations in Various Approaches, Springer Cham, London, pp. 120 (2015).
• H.G. Çitil and N. Altınışık, On the eigenvalues and the eigenfunctions of the Sturm-Liouville fuzzy boundary value problem, J. Math. Comput. Sci., Vol. 7, N. 4, pp. 786-805 (2017).
• D.S Ibanez, E. Esmi and L.C. Barros, Linear Ordinary Differential Equations with Linearly Correlated Boundary Values, Proceedings of 2018 IEEE international conference on fuzzy systems (FUZZ-IEEE), (2018).
• A. Kandel and W.J. Byatt, Fuzzy differential equations, Proceedings of the International Conference on Cybernetics and Society, (1978).
• O. Kaleva , Fuzzy differential equations, Fuzzy sets and systems, Vol. 24, N. 3, pp. 301-317 (1987).
• A. Khastan and J.J. Nieto, A boundary value problem for second order fuzzy differential equations, Fuzzy sets and systems, Vol. 72, N. 9-11, pp. 3583-3593 (2010).
• G. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic,Prentice Hall Inc., Upper Saddle River, (1995).
• M.T. Mizukoshi, L.C. Barros, Y. Chalco-Cano, H. Roman-Flores and R.C. Bassanezi, Fuzzy differential equations and the extension principle, Information Sciences, Vol. 177, pp. 3627-3635 (2007).
• M. L. Puri and D. A. Ralescu , Differentials of fuzzy functions, Journal of Math. Analysis and App., Vol. 91, N. 2, pp. 552-558 (1983).
• D.E. Sanchez, V.F. Wasques, E. Esmi and L.C. Barros, Solution to the Bessel differential equation with interactive fuzzy boundary conditions, Computational and Applied Mathematics, Vol. 4, N. 1, pp. 1-12 (2022).
• L. Stefanini and B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Analysis: Theory, Methods and Applications, Vol. 71, N. 3-4, pp. 1311-1328 (2009).
• I. Sadeqi, M. Moradlou and M. Salehi, On approximate cauchy equation in Felbin's type fuzzy normed linear spaces, Iranian Journal of Fuzzy Systems, Vol. 10, N. 3, pp. 51-63 (2013).
• T. Allahviranloo and K. Khalilpour, A numerical method for two-point fuzzy boundary value problems, World Applied Sciences Journal, Vol. 13, N. 10, pp. 2137-2147 (2011).
• K. Sabzi, T. Allahviranloo and S. Abbasbandy, A fuzzy generalized power series method under generalized Hukuhara differentiability for solving fuzzy Legendre differential equation, Soft Computing, Vol. 24, pp. 8763-8779 (2020).
• E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations I, 2nd edn., Oxford University Press, London, (1962).
• V.F. Wasques, E. Esmi, L.C. Barros and P. Sussner, Numerical Solutions for Bidimensional Initial Value Problem with Interactive Fuzzy Numbers, Fuzzy Information Processing, Springer, Cham., (2018).
• L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-III, Information Sciences, Vol. 8, N. 3, pp. 199-249 (1975).
• H. Gültekin, N. Altınışık, On Solution of Two point Fuzzy Boundary Value Peoblem, The Bulletin of Society for Mathematical Services and Standards, Vol. 11, pp 31-39, (2014).
• A. Armand, Z. Gouyandeh, Solving two-point fuzzy boundary value problem using variational iteration method, Communications on Advanced Com Science with Applications, Vol. 2013, pp. 1-10 (2013).
• M. H. Suhhiem, R. I. Khwayyit, Approximate Polynomial Solution for Two-Point Fuzzy Boundary Value Problems, Vol. 7 No. 1, pp. 64-79 (2024)