Boundary value problem: Weak solutions induced by fuzzy partitions

Abstract

The aim of this paper is to propose a new methodology in the construction of spaces of test functions used in a weak formulation of the Boundary Value Problem. The proposed construction is based on the so called "two dimensional" approach where at first, we select a partition of a domain and second, a dimension of an approximating functional subspace on each partition element. The main advantage consists in the independent selection of the key parameters, aiming at achieving a requested quality of approximation with a reasonable complexity. We give theoretical justification and illustration on examples that confirm our methodology.

F-Transform Inspired Weak Solution to a Boundary Value Problem

“…This method is denoted by "GFPP" (Generalized uniform Fuzzy Partition with Polynomials), and it is compared with the two well-established numerical methods, used for solving weak BVPs. The latter are: piecewise linear Finite Element Method (FEM) and Fuzzy Partition with Polynomials (FPP); see [11]. We have selected three types of BVPs with different source (right-hand side) functions.…”

“…To demonstrate its superior performance, we compare it with the well-established piece-wise linear finite element method (FEM) and the FPP (Fuzzy Partition with Polynomials) method. The latter is proposed in our recent paper [11], and the former is selected because it is in a family of conventional approaches. To evaluate the obtained results, we consider BVPs, whose exact solutions are known.…”

“…Among various spaces of test functions, we can name trigonometric functions [7], spaces of finite elements (FEM) [1], spaces of the so-called shape functions in the meshless and generalized FEMs [8][9][10], etc. Recently, we introduced a construction of test spaces motivated by the theory of higher degree fuzzy transforms [6,11].…”

“…The test space, introduced in [6] as well as in [11], is a linear space of functions represented by the inversion formula of a higher degree F m -transform. Let us note that, in these papers, only uniform fuzzy partitions with at most two basic functions, covering a point in a given interval, were considered.…”