Modification of Interval Arithmetic for Modelling and Solving Uncertainly Defined Problems by Interval Parametric Integral Equations System

Abstract

In this paper we present the concept of modeling and solving uncertainly defined boundary value problems described by 2D Laplace’s equation. We define uncertainty of input data (shape of boundary and boundary conditions) using interval numbers. Uncertainty can be considered separately for selected or simultaneously for all input data. We propose interval parametric integral equations system (IPIES) to solve so-define problems. We obtain IPIES in result of PIES modification, which was previously proposed for precisely (exactly) defined problems. For this purpose we have to include uncertainly defined input data into mathematical formalism of PIES. We use pseudo-spectral method for numerical solving of IPIES and propose modification of directed interval arithmetic to obtain interval solutions. We present the strategy on examples of potential problems. To verify correctness of the method, we compare obtained interval solutions with analytical ones. For this purpose, we obtain interval analytical solutions using classical and directed interval arithmetic.