Interval linear systems as a necessary step in fuzzy linear systems

International audienceThis article clarifies what it means to solve a system of fuzzy linear equations, relying on the fact that they are a direct extension of interval linear systems of equations, already studied in a specific interval mathematics literature. We highlight four distinct definitions of a systems of linear equations where coefficients are replaced by intervals, each of which based on a generalization of scalar equality to intervals. Each of the four extensions of interval linear systems has a corresponding solution set whose calculation can be carried out by a general unified method based on a relatively new concept of constraint intervals. We also consider the smallest multidimensional intervals containing the solution sets. We propose several extensions of the interval setting to systems of linear equations where coefficients are fuzzy intervals. This unified setting clarifies many of the anomalous or inconsistent published results in various fuzzy interval linear systems studies

Mathematical Fuzzy Logic in the Emerging Fields of Engineering, Finance, and Computer Sciences

Mathematical fuzzy logic (MFL) specifically targets many-valued logic and has significantly contributed to the logical foundations of fuzzy set theory (FST). It explores the computational and philosophical rationale behind the uncertainty due to imprecision in the backdrop of traditional mathematical logic. Since uncertainty is present in almost every real-world application, it is essential to develop novel approaches and tools for efficient processing. This book is the collection of the publications in the Special Issue “Mathematical Fuzzy Logic in the Emerging Fields of Engineering, Finance, and Computer Sciences”, which aims to cover theoretical and practical aspects of MFL and FST. Specifically, this book addresses several problems, such as:- Industrial optimization problems- Multi-criteria decision-making- Financial forecasting problems- Image processing- Educational data mining- Explainable artificial intelligence, etc

Numerical Solution of Some Uncertain Diffusion Problems

Diffusion is an important phenomenon in various fields of science and engineering. These problems depend on various parameters viz. diffusion coefficients, geometry, material properties, initial and boundary conditions etc. Governing differential equations with deterministic parameters have been well studied. But, in real practice these parameters may not be crisp (exact) rather it involves vague, imprecise and incomplete information about the system variables and parameters. Uncertainties occur due to error in measurements, observations, experiments, applying different operating conditions or it may be due to maintenance induced errors, etc. As such, it is an important concern to model these type of uncertainties. Traditionally uncertain problems are modelled through probabilistic approach. But probabilistic methods may not able to deliver reliable results at the required precision without sufficient data. In this context, interval and fuzzy theory may be used to manage such uncertainties. Accordingly, the system parameters and variables are represented here as interval and fuzzy numbers. Generally, we get interval or fuzzy system of equations for uncertain steady state problems with interval or fuzzy parameters whereas interval or fuzzy eigenvalue problems may be obtained for unsteady state. This thesis redefined interval or fuzzy arithmetic in order to handle the uncertain problems. The proposed arithmetic has been used to solve fuzzy and interval system of equations and eigenvalue problems. Various numerical methods viz. Finite Element Method (FEM), Wavelet Method (WM), Euler Maruyama and Milstein Methods are studied by introducing interval or fuzzy theory. The proposed arithmetic has been combined with FEM and WM to develop Interval or Fuzzy Finite Element Method (I/FFEM) and Interval or Fuzzy Wavelet Method (I/FWM). Further, it may be pointed out that sometimes systems may possess uncertainties due to randomness and fuzziness of the parameters. As such, here we have hybridized the concept of fuzziness as well as stochasticity to develop numerical fuzzy stochastic methods viz. interval or Fuzzy Euler Maruyama and Interval/Fuzzy Milstein. These methods are also been used to solve various diffusion problems. Numerical examples and different application problems are solved to demonstrate the efficiency and capabilities of the developed methods. In this respect, imprecisely defined diffusion problems such as heat conduction and conjugate heat transfer in rod, homogeneous and non-homogeneous fin and plate, along with one group, multi group and point kinetic neutron diffusion with interval or fuzzy uncertainties have been investigated. The convergence of the field variables have been investigated with respect to the number of element discretization of the domain in case of I/FEM. Accordingly, convergence of the proposed interval or fuzzy FEM has been studied for unsteady heat conduction in a cylindrical rod. For conjugate heat transfer problems, the convergence of uncertain temperature distributions with respect to the number of element discretizations has also been studied. Further, various combinations of uncertain parameters are considered and the sensitivity of these parameters has been reported. Next, one group and two group problems have been solved and the sensitivity of the uncertain parameters in the context of fast and thermal neutrons are presented. The hybrid fuzzy stochastic methods have also been used to investigate uncertain stochastic point kinetic neutron diffusion problem. Uncertain variation of neutron populations are analysed by considering two random samples. Developed interval or fuzzy WM has also been used to solve uncertain differential equation. Finally obtained results for the said problems are compared in special cases for the validation of proposed methods