Numerical solutions of fuzzy partial differential equations and its applications in computational mechanics

Abstract

Calculation of the solution of fuzzy partial differential equations is in general very difficult. We can find the exact solution only in some special cases. Fortunately, in most of engineering applications relations between the solutions and uncertain parameters are monotone (we can assume that, when the uncertainty of the parameters is sufficiently small). In this case, the exact solution can be calculated using only endpoints of given intervals. In order to improve the efficiency of calculation we can apply sensitivity analysis. In this paper, a very efficient algorithm of solution was presented. This algorithm is based on finite element method (or any other numerical method of solution PDE like for example FEM or BEM) and sensitivity analysis. Using this method we can solve engineering problems with thousands degree of freedom. Fuzzy partial differential equations can be applied for modeling of mechanical system (structures) with uncertain parameters. To construct the fuzzy membership function random sets can be applied. This theory contains fuzzy sets and probability theory as special cases. Using algorithms, which are described in this paper we can solve partial differential equations with random and fuzzy parameters.