On a solution of the fuzzy Dirichlet problem for the heat equation

Abstract

In real-world applications, the behavior of the system is determined by physics laws and described by differential equations. In particular, the heat transfer is determined by Fourier's law of heat conduction and described by partial differential equation of parabolic type. When we construct a mathematical model, we need several parameters, a lot of them are obtained from measurements, or observations. In general, no measurement is perfect and these parameters become uncertain. Fuzzy sets are a useful tool to model such uncertainties. Thus, mathematical models arise, where due to physical laws the dynamics is crisp (certain) but some parameters (such as the source term, initial and boundary values) are fuzzy. In this paper, we consider such a model. Namely, we investigate fuzzy Dirichlet problem for the heat equation with fuzzy source function and fuzzy initial-boundary conditions.Most of the researchers assume a solution of a fuzzy differential equation as a fuzzy-valued function. But this approach is accompanied with some known difficulties. We are motivated by the fact that the fuzzy-valued function is not the only tool to model the uncertainties, changing with time. We are looking for a solution in the form of a fuzzy set (bunch) of real functions. We assume that the source term and initial-boundary conditions are modeled by triangular fuzzy functions, which are a special kind of fuzzy bunches. To determine the solution, we split the given fuzzy Dirichlet problem to three subproblems. The first subproblem provides the crisp solution. The other two subproblems give the uncertainties due to initial-boundary conditions and due to source function. We show that these uncertainties are triangular fuzzy functions. On the basis of the obtained results, we establish the existence and uniqueness theorem for the solution, under commonly accepted conditions.We propose a solution method and explain it on numerical examples.