Uncertain Static Problems

Abstract

Under static conditions, the governing differential equations of various science and engineering problems lead to systems of simultaneous equations (linear and nonlinear). In this chapter, we focus on the solution for a system of linear equations. In mathematics, the theory of linear systems is the basis, and a fundamental part, of linear algebra. Also, the computational algorithms for finding the solutions for the system of linear equations are an important part of numerical linear algebra and play a prominent role in various fields viz. engineering, physics, chemistry, computer science and economics. For simplicity and easy computation, all the involved parameters and variables of the linear system are usually considered as deterministic or exact. But as a practical matter, due to the uncertain environment, one may have imprecise, incomplete, insufficient, or vague information about the parameters because of several errors. Traditionally, such uncertainty or vagueness may be modeled through a probabilistic approach. But a large amount of data is required for the traditional probabilistic approach. Without a sufficient amount of experimental data, the probabilistic methods may not deliver reliable results at the required precision. Therefore, intervals and/or fuzzy numbers may become efficient tools to handle uncertain and vague parameters when there is an insufficient amount of data available. In this regard, uncertain static problems may be modeled through an interval system of linear equations (ISLE) ([A][x] = [b]) and/or fuzzy system of linear equations (FSLE) \(\left( {\tilde A\tilde x = \tilde b} \right)\).