Soft Computing https://doi.org/10.1007/s00500-018-3529-7 METHODOLOGIES AND APPLICATION Fuzzy linear systems via boundary value problem Hewayda M. S. Lotfy 1 · Azza A. Taha 1 · I. K. Youssef 1 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 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 2n × 2n 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. Keywords Compact storage · Iterative methods · Fuzzy system of linear equations 1 Introduction Systems of linear equations play important roles in sev- eral domains such as in the commercial, scientifical, and engineering domains. Usually, problems in these domains get pruned and reduced into linear systems of equations. When the problem structure becomes imprecise, the linear system of equations becomes no more crisp. Vague values of the system parameters can be demonstrated using prob- ability distributions, intervals, or fuzzy values. Fuzziness deals with imprecise, ambiguous, and unclear inputs that is why it is extremely important to create mathematical models and numerical methods for fuzzy linear system of equations (FLS). Communicated by V. Loia. B Azza A. Taha Azzataha@sci.asu.edu.eg Hewayda M. S. Lotfy hewayda_lotfy@sci.asu.edu.eg I. K. Youssef kaoud22@sci.asu.edu.eg 1 Department of Mathematics, Faculty of Science, Ain Shams University, Abbassia, Cairo 11566, Egypt A general model for solving an arbitrary n × n FLS whose coefficients matrix are crisp and its right-hand side column is an arbitrary fuzzy number vector is given (Friedman et al. 1998). They proposed an embedding approach where the original n × n FLS is replaced by 2n × 2n crisp linear system. Moreover, another embedding approach (Allahvi- ranloo and Hashemi 2014) replaces the original n × n FLS by two n × n crisp linear systems. In both of the mentioned embedding techniques, the size of the computational work is doubled. Solving such crisp large linear systems is a prob- lem in itself, and therefore, the use of iterative techniques becomes extremely useful. The iterative numerical solutions for fuzzy linear system are studied (Allahviranloo 2005; Feng 2008; Yin and Wang 2009). Compact storage is a technique for storing sparse matrices. The main idea is to store only the nonzero elements and to be able to perform the neces- sary matrix operations. Compact storage schemes allocate contiguous storage in memory for the nonzero elements of the matrix. This requires also a scheme for knowing the places of the elements in the original matrix. There are many alternatives for the compact storage schemes, mentioned for instance (Saad 2003; Day 1977; Barrett et al. 1994). In this paper, we propose a general model for solving an n × n FLS whose coefficients are crisp with a fuzzy right- 123