On solutions of initial-boundary value problem for fuzzy partial differential equations

Abstract

In this paper, we investigate linear partial differential equations with fuzzy source function, and with fuzzy initial and boundary conditions. Usually, researchers consider solutions of fuzzy differential equations in the form of fuzzy-valued functions. On the contrary, in this study, we are looking for a solution in the form of fuzzy set (bunch) of real functions. To demonstrate the proposed approach we use Dirichlet problem for the heat equation. We assume the source function, and the initial and boundary conditions to be in a special form, which we name as triangular fuzzy function. We show that the uncertainties of the solution due to these parameters are triangular fuzzy functions too. The solution for the example, which we discuss in the paper, is expressed by an analytical formula. If we use numerical methods, we can find the solution in the suggested sense for each problem from the examined class.