On the properties and applications of fuzzy analytic equations

Abstract

This paper presents the fuzzy power series method to solve the second-order differential equation under generalized Hukuhara differentiability with fuzzy and real coefficients. For this purpose, different types of fuzzy analytic functions by attention to generalized Hukuhara differentiability, ordinary and singular points are introduced. Since in the discussion of fuzzy power series, the concept of the fuzzy convergence radius is one of the most essential and fundamental concepts, the fuzzy convergence radius under the generalized division is defined. Fundamental theorems, such as fuzzy ratio tests, the convergence of the fuzzy geometric series, are expressed and proven. In addition, it has been shown that the fuzzy convergence radius of fuzzy power series does not change concerning derivatives under operators such as derivatives and integrals. It has been shown that the fuzzy analytic functions are still fuzzy analytic functions concerning derivatives under fuzzy operators, summation, multiplication, and generalized division. Then, the uniqueness of the solution of the second-order fuzzy differential equations with fuzzy and real coefficients in the form of a fuzzy power series by attention to the type of generalized differentiability is shown. Finally, there are examples to demonstrate the effectiveness of the method.