Article Preview
TopIntroduction
Many problems in applied sciences and engineering require the solution of fuzzy differential equations where uncertainty or vagueness prevails. In some cases they are subject to fuzzy initial or boundary conditions, see (O’Regana and Lakshmikantham, 2003). This gives rise to fuzzy initial value problems and fuzzy two-point boundary value problems. In general, it is too complicated to obtain a closed form solution for such problems. This leads to numerical approximations.
In solving fuzzy two point boundary value problems, different approaches were adopted by many researchers. One approach transforms the fuzzy boundary value problem to a crisp one by interpreting it as a family of differential inclusions, see (Hullermeier, 1997; Li et al. 2011). Another approach is based on Zadeh’s extension principle. In this approach, the associated crisp problem is solved and the boundary fuzzy values are substituted in the solution rather than the real constants, see (Guo et al., 2013; Zadeh, 1975). The third approach assumes the solution is a fuzzy function and the derivatives involved must be considered fuzzy derivatives even if only the boundary values are fuzzy, see (O’Regana and Lakshmikantham, 2003; Lakshmikantham et al., 2001).
Numerically, many authors dealt with fuzzy differential equations(FDE). To list some, first order FDEs under strongly generalized derivatives were considered by (Bede et al., 2007). Euler method was applied for solving initial value problems for FDEs in (Palligkinis, et al., 2009). The four-stage order Runge-Kutta methods for FDEs were developed in for example (Allahviranloo, et al., 2007), while Adams and Nystrorm methods and predictor-corrector methods for solving FDEs can be found in (Khastan and Ivaz, 2009; Khastan and Nieto, 2010). Existence and uniqueness of numerical solutions was considered by (Fard et al., 2017), solving FDE using Bernstein neural network was done by (Jafari et al., 2017) and numerical simulation of fuzzy initial value problems using general linear method and B-series was done in (Rabiei et al., 2017). Numerical methods for a linear second order boundary value problem was considered by (Chalco-Cano and Roman, 2008), where the authors used the concept of generalized Hakuhara differentiability. While in (Glufatullayev et al., 2013), the authors considered initial value solvers and Gaussian iteration to solve the systems involved. Runge-Kutta methods were used by (Rabiei et al., 2012). The use of reducing kernal Hilbert space method for solving fuzzy differential equations and fuzzy two point boundary value problems was considered by (Abu-Arqub et al. 2016; Abu-Arq et al. 2017).
This article considers the fuzzy two-point boundary value problem of the form, see (Li et al. 2011; Guo et al., 2013):
subject to the fuzzy boundary conditions:
where
is continuous fuzzy valued function,
are fuzzy numbers and
represents the set of fuzzy numbers.