In this paper, we introduce a fuzzy Mellin transform method for solving Hermite fuzzy differential equations. The fuzzy Mellin transform reduce the problem of solving a Hermite fuzzy differential equation to a problem of solving a difference equation, whose inverse transform gives the solution of the fuzzy differential equation at hands. Under some conditions, we also give some Hyers-Ulam stability result of Hermite fuzzy differential equations.
S.M.Ulam, A Collection of Mathematical Problems, New York, 1968.
2.
D.H.Hyers, On the stability of the linear functional equation, Proceedings of the National Academy of Sciences of the United States of America27 (1941), 222–224.
3.
T.Aoki, On the stability of the linear transformation in Banach spaces, Journal of the Mathematical Society of Japan2 (1950), 64–66.
4.
T.M.Rassias, On the stability of the linear mapping in Banach spaces, Proceedings ofthe American Mathematical Society72 (1978), 297–300.
5.
J.M.Rassias, On approximation of approximately linear mappings by linear mappings, Journal of Functional Analysis46 (1982), 126–130.
6.
T.Z.Xu, J.M.Rassias and W.X.Xu, Stability of a general mixed additive-cubic functional equation in non-Archimedean fuzzy normed spaces, Journal of Mathematical Physics51 (2010), 1–19.
7.
Z.P.Yang, T.Z.Xu and M.Qi, Ulam-hyers stability for fractional differential equations in quaternionic analysis, Advances in Applied Clifford Algebras26 (2016), 1–10.
8.
T.Z.Xu, Z.P.Yang and J.M.Rassias, Direct and fixed point approaches to the stability of an AQ-functional equation in non-Archimedean normed spaces, Journal of Computional Analysis and Applications17 (2014), 697–706.
9.
S.M.Jung and J.Roh, Hyers-Ulam stability of the time independent Schrodinger equations, Applied Mathematics Letters74 (2017), 147–153.
10.
S.M.Jung, D.Popa and T.M.Rassias, On the stability of the linear functional equation in a single variable on complete metric groups, Journal of Global Optimization59 (2014), 165–171.
11.
S.M.Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, Journal ofMath-ematical Analysis and Applications222 (1998), 126–137.
12.
J.R.Wang, L.L.Lii and Y.Zhou, New concepts and results in stability of fractional differential equations, Communications in Nonlinear Science and Numerical Simulation17 (2012), 2530–2538.
13.
Y.H.Shen, On the Ulam stability of first order linear fuzzy differential equations under generalized differentiability, Fuzzy Sets and Systems280 (2015), 27–57.
14.
S.Abbasbandy and A.Shirzadi, Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems, Numerical Algorithms54 (2010), 521–532.
15.
T.Allahviranloo, S.Abbasbandy, S.Salahshour and A.Hakimzadeh, A new method for solving fuzzy linear differential equations, Soft Computing92 (2011), 181–197.
16.
T.Allahviranloo and S.Salahshour, Euler method for solving hybrid fuzzy differential equation, Soft Computing15 (2011), 1247–1253.
17.
R.P.Agarwal, V.Lakshmikantham and J.J.Nieto, On the concept of solution for fractional differential equations with uncertainty.Nonlinear Analysis: Theory, Methods and Applications72 (2010), 2859–2862.
18.
B.Bede and S.G.Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems151 (2005), 581–599.
19.
H.C.Wu, The improper fuzzy Riemann integral and its numerical integration, Information Sciences111 (1999), 109–137.
20.
M.Friedman, M.Ma and A.Kandel, Numerical solution of fuzzy differential and integral equations, Fuzzy Sets and Systems106 (1999), 35–48.
21.
A.G.Muhammad, S.Ahmed, K.Majid and I.B.Syeda, A novel analytical solution of a fractional diffusion problem by homotopy analysis transform method, Neural Computing and Applications23 (2013), 1643–1647.
22.
L.Stefanini and B.Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations.Nonlinear Analysis: Theory, Methods and Applications71 (2009), 1311–1328.
23.
J.J.Buckley and T.Feuring, Introduction to fuzzy partial differential equations, Fuzzy Sets and Systems105 (1999), 241–248.
24.
M.Qi, Z.P.Yang and T.Z.Xu, A reproducing kernel method for solving nonlocal fractional boundary value problems with uncertainty, Soft Computing21 (2017), 4019–4028.
25.
T.Allahviranloo, A.Armand and Z.Gouyandeh, Fuzzy fractional differential equations under generalized fuzzy Caputo derivative, Journal ofIntelligent & Fuzzy Systems26 (2014), 1481–1490.
26.
S.Abbasbandy and T.Allahviranloo, The Adomian decomposition method applied to the fuzzy system of Fredholm integral equation of the second kind, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems14 (2006), 101–110.
27.
T.Allahviranloo, S.Abbasbandy, N.Ahmady and E.Ahmady, Improved predictoraAScorrector method for solving fuzzy initial value problems, Information Sciences179 (2009), 945–955.
28.
S.Salahshour and T.Allahviranloo, Application of fuzzy differential transform method for solving fuzzy Volterra integral equations, Applied Mathematical Modelling37 (2013), 1016–1027.
29.
S.Salahshour and E.Haghi, Solving fuzzy heat equation by fuzzy Laplace transforms, Communications in Computer and Information Science81 (2010), 512–521.
30.
S.Salahshour, T.Allahviranloo and S.Abbasbandy, Solving fuzzy fractional differential equation by fuzzy Laplace transforms, Communications in Nonlinear Science and Numerical Simularion17 (2012), 1372–1381.
31.
S.Butera and M.D.Paola, Fractional differential equations solved by using Mellin transform, Communications in Nonlinear Science and Numerical Simularion19 (2014), 2220–2227.
32.
P.L.Butzer, A.A.Kilbas and J.J.Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, Journal of Mathematical Analysis and Applcations269 (2002), 1–27.
33.
P.L.Butzer and S.Jansche, A direct approach to the Mellin transform, Jounral of Fourier Analysis and Applications3 (1997), 325–375.
34.
M.B.Ahmadi, N.A.Kiani and N.Mikaeilvand, Laplace transform formula on fuzzy nth-order derivative and its application in fuzzy ordinary differential equations, Soft Computing18 (2014), 2461–2469.
35.
S.Salahshour and T.Allahviranloo, Applications of fuzzy Laplace transforms, Soft Computing17 (2013), 145–158.
36.
Z.Gouyandeh, T.Allahviranloo, S.Abbasbandy and A.Armand, A fuzzy solution of heat equation under generalized Hukuhara differentiability by fuzzy Fourier transform, Fuzzy Sets and Systems (2016). 10.1016/ j.fss.2016.04.010
37.
X.Sun and Z.P.Yang, A Mellin transform method for solving fuzzy differential equations, Advances in Difference Equations2016 (2016), 296.
38.
O.Taloa and F.Basarb, Determination of the duals of classical sets of sequences of fuzzy numbers and related matrix transformations, Computers and Mathematics with Applications58 (2009), 717–733.
