SOLUTION OF DIRICHLET BOUNDARY VALUE PROBLEM FOR THE POISSON EQUATION BASED ON A FUZZY SYSTEM

摘要

A fuzzy system solution of the Dirichlet boundary value problem for the Poisson equation ▽~2 f(x,y) = φ(x,y) is described. The solution we obtained is a cubic spline function ∑_(i,j) C_(ij)B_i(x)B_j(y) which approximates f(x,y), where B_i(x) and B_j (y) are cubic B-splines. The Dirichlet boundary conditions along with the solution at grid points are expressed as fuzzy rules and the Laplacian at the grid points are expressed as a linear combination of the solutions C_(ij) 's corresponding to the grid point and its 8 immediate neighbor grid points. We show that the approximation error is O(h~2) which is better than that of finite difference approximation solutions. The calculation results for a few examples are included to verify out method.