Incorporating optimisation technique into Euler’s method for solving differential equations with fuzzy initial values

Abstract

Fuzzy set provides a powerful technique to introduce uncertainty into numerical methods. However, the computations of fuzzy sets often face difficult problems. This is due to non-applicability of common existing methods, severe overestimation in computation, or very high computational complexity. This paper proposes a new strategy to introduce uncertainty into Euler’s method. It consists of two parts. First, we develop a new fuzzy version of Euler’s method, which takes into account the dependency problem that arises in the classical Euler’s method. Second, we perform optimisation technique to approximate the solution of differential equations with fuzzy initial values. This combination gives accurate results and requires only few function evaluations. Several examples are provided to show the capability of our proposed methods compared to the conventional fuzzy version of Euler’s method proposed in the literature.