Stability of Euler Methods for Fuzzy Differential Equation

Abstract

The Liu process is a fuzzy process whose membership function is a symmetric function on an expected value. The object of this paper was a fuzzy differential equation driven by Liu process. Since the existing fuzzy Euler solving methods (explicit Euler scheme, semi-implicit Euler scheme, and implicit Euler scheme) have the same convergence, to compare them, we presented four stabilities, i.e., asymptotical stability, mean square stability, exponential stability, and A stability. By choosing special fuzzy differential equation as a test equation, we deduced that mean square stability is equivalent to exponential stability. Furthermore, an explicit fuzzy Euler scheme and semi-implicit fuzzy Euler scheme showed asymptotical stability and mean square stability, while an explicit fuzzy Euler scheme failed to meet A stability but that an implicit fuzzy Euler scheme is A stable, and whether semi-implicit fuzzy Euler scheme is A stable depends on the values of α and λ.

Incorporating Fuzziness in the Traditional Runge–Kutta Cash–Karp Method and Its Applications to Solve Autonomous and Non-Autonomous Fuzzy Differential Equations

“…Moreover, Georgieva [21] presented the double fuzzy Sumudu transform to solve the partial Volterra fuzzy integro-differential equation with the convolution kernel under H-differentiability. In [22], You, Cheng, and Ma studied the stability of the fuzzy Euler method related to the FDE. The stabilities involved are the asymptotical stability, the mean square (MS) stability, the exponential stability and the A stability.…”