Fuzzy Linear Systems via Boundary Value Problem

Abstract

Reduction in storage and number of operations are considered through avoiding the representation of zeros in storage as well as in the calculations. The importance of this approach has its effect in large problems that appear in numerical treatments of boundary value problems in general and becomes more effective when fuzzy concepts are considered. We introduce an extended embedding solution model named fuzzy compact storage Gauss–Seidel (FCGS) for solving linear systems of equations with a fuzzy-based right-hand side. The model starts by applying the embedding approach to the n× n fuzzy linear system, a compact storage technique is then applied to the resultant 2 n× 2 n de-fuzzification matrix, and finally, a Gauss–Seidel method is applied to the system. The FCGS experimental results and algorithm are clarified on some numerical examples including a fuzzy boundary value problem (FBVP). The error improvements through Gauss–Seidel iterations of fuzzy solution computations are reported. The fuzzy solutions at α-cuts are shown and compared to the exact solutions. FCGS achieved a reduction of at least 50% of storage by using the compact storage concepts and consequently obtain a reduction in the mathematical operations and accordingly the running time especially in FBVP applications.