ON THE SOLUTION OF TWO DIMENSIONAL LAPLACE EQUATION WITH FUZZY BOUNDARIES

In the present work, we investigate two dimensional Laplace equation with Neu-mann boundary conditions which are considered to be uncertain. Laplace equationarises in the study of a plethora of physical phenomena, including electrostatic orgravitational potential, the displacement eld of a two or three dimensional elasticmembrane and the velocity potential of an incompressible uid ow. Since di erentialequation arises from physics rules, it is deterministic and the fuzziness is consideredonly on the boundary functions which come from experience or measurement. In theliterature there are two di erent de nition for fuzzy function: the rst is a functionfrom classical space to a fuzzy space, called fuzzy set valued function and the secondis a fuzzy subset of a space of functions, called fuzzy bunch of functions. In this study,we will consider the boundary conditions are bunches of fuzzy functions. We de nesolution for fuzzy boundary value problem under study, as a fuzzy bunch of functions,i.e as a set of crisp functions that every function has a certain membership degree tothe solution. This membership degree is de ned by using the membership degrees ofthe boundary values. Then, we prove that this solution is well de ned. Furthermore,since Laplace equation is a linear equation, the superposition principle is satis ed andwe can divide the problem into crisp and uncertain boundary value problem. Finally,a numerical example is proposed in order to show e ciency of the method.

ON THE SOLUTION OF TWO DIMENSIONAL LAPLACE EQUATION WITH FUZZY BOUNDARIES Keywords:

Laplace equation , Bunch of functions , Boundary value problem