Existence and Uniqueness of Solutions for Fuzzy Boundary Value Problems Under Granular Differentiability

Abstract

This paper considers fuzzy boundary value problems associated with second-order fuzzy differential equations under granular differentiability. Using a horizontal membership function, we present the notion of second-order granular differentiability for fuzzy functions. Using the granular differentiability concept, we interpret two kinds of two-point boundary value problems for second-order fuzzy differential equations. Sufficient conditions are established for the existence and uniqueness of solutions to these fuzzy boundary value problems. An algorithm is presented for solving non-linear fuzzy boundary value problems under granular differentiability. We provide one example and two engineering applications to demonstrate the algorithm’s effectiveness and results.

Keywords

• granular differentiability
• horizontal membership function
• contraction mapping
• fuzzy boundary value problem