Hyers-Ulam Stability and Existence Criteria for the Solution of Second-Order Fuzzy Differential Equations

In this paper, existence, uniqueness, and Hyers-Ulam stability for the solution of second-order fuzzy differential equations (FDEs) are studied. To deal a physical model, it is required to insure whether unique solution of the model exists. The natural transform has the speciality to converge to both Laplace and Sumudu transforms only by changing the variables. Therefore, this method plays the rule of checker on the Laplace and Sumudu transforms. We use natural transform to obtain the solution of the proposed FDEs. As applications of the established results, some nontrivial examples are provided to show the authenticity of the presented work.

[1]  J. F. Gómez‐Aguilar,et al.  Investigation of a system of nonlinear fractional order hybrid differential equations under usual boundary conditions for existence of solution , 2020, Mathematical Methods in the Applied Sciences.

[2]  J. Ahmad,et al.  Exact Solution of Linear and Non-linear Goursat Problems , 2015 .

[3]  Soon-Mo Jung,et al.  Hyers-Ulam Stability of the Delay Equation $y'(t)=\lambda y(t-\tau)$ , 2010 .

[4]  Rosana Rodríguez-López,et al.  An existence and uniqueness result for fuzzy Goursat partial differential equation , 2019, Fuzzy Sets Syst..

[5]  Weiyin Fei.  Uniqueness of Solutions to Fuzzy Differential Equations Driven by Liu's Process with Non-Lipschitz Coefficients , 2009, 2009 Sixth International Conference on Fuzzy Systems and Knowledge Discovery.

[6]  Osmo Kaleva.  Fuzzy differential equations , 1987 .

[7]  Imre J. Rudas,et al.  First order linear fuzzy differential equations under generalized differentiability , 2007, Inf. Sci..

[8]  H. Román-Flores,et al.  On new solutions of fuzzy differential equations , 2008 .

[9]  Choonkil Park,et al.  Ulam's type stability of higher order nonlinear delay differential equations via integral inequality of Grönwall-Bellman-Bihari's type , 2019, Appl. Math. Comput..

[10]  Yonghong Shen,et al.  On the Ulam stability of first order linear fuzzy differential equations under generalized differentiability , 2015, Fuzzy Sets Syst..

[11]  D. H. Hyers.  On the Stability of the Linear Functional Equation. , 1941, Proceedings of the National Academy of Sciences of the United States of America.

[12]  Barnabás Bede,et al.  Mathematics of Fuzzy Sets and Fuzzy Logic , 2012, Studies in Fuzziness and Soft Computing.

[13]  Z. Khan,et al.  Stability analysis of a dynamical model of tuberculosis with incomplete treatment , 2020, Advances in Difference Equations.

[14]  Tofigh Allahviranloo,et al.  A new method for solving fuzzy linear differential equations , 2011, Computing.

[15]  Yian-Kui Liu,et al.  Expected value of fuzzy variable and fuzzy expected value models , 2002, IEEE Trans. Fuzzy Syst..

[16]  Robert LIN,et al.  NOTE ON FUZZY SETS , 2014 .

[17]  P. Kloeden,et al.  Metric spaces of fuzzy sets , 1990 .

[18]  Abraham Kandel,et al.  Numerical solutions of fuzzy differential and integral equations , 1999, Fuzzy Sets Syst..

[19]  T. Abdeljawad,et al.  A singular ABC-fractional differential equation with p-Laplacian operator , 2019 .

[20]  Madan M. Gupta,et al.  Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems , 2003 .

[21]  D. Dubois,et al.  Towards fuzzy differential calculus part 3: Differentiation , 1982 .

[22]  Lotfi A. Zadeh,et al.  On Fuzzy Mapping and Control , 1996, IEEE Trans. Syst. Man Cybern..

[23]  Xiaowei Chen,et al.  A New Existence and Uniqueness Theorem for Fuzzy Differential Equations , 2011 .

[24]  Osmo Kaleva,et al.  A note on fuzzy differential equations , 2006 .

[25]  R. Goetschel,et al.  Elementary fuzzy calculus , 1986 .