Generalized Fuzzy Partition in Galerkin Method for the Boundary Valued Problem

This paper introduces a novel construction of test spaces in the Ritz-Galerkin method. These spaces are established on the basis of linear spaces of functions that are linear combinations of polynomials and fuzzy sets of a generalized uniform fuzzy partition.

[1]  G. Fairweather,et al.  Numerical Methods for Two – Point Boundary Value Problems , 2008 .

[2]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[3]  C. Fletcher.  Computational Galerkin Methods , 1983 .

[4]  Martina Danková,et al.  Towards a higher degree F-transform , 2011, Fuzzy Sets Syst..

[5]  Michal Holcapek,et al.  Suppression of High Frequencies in Time Series Using Fuzzy Transform of Higher Degree , 2016, IPMU.

[6]  Martins Kokainis,et al.  Approximation properties of higher degree F-transforms based on B-splines , 2015, 2015 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE).

[7]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[8]  Ulrich Reif,et al.  Weighted Extended B-Spline Approximation of Dirichlet Problems , 2001, SIAM J. Numer. Anal..

[9]  Vladik Kreinovich,et al.  Necessary and sufficient conditions for generalized uniform fuzzy partitions , 2015, Fuzzy Sets Syst..

[10]  J. Melenk.  On Approximation in Meshless Methods , 2005 .

[11]  B. Reddy,et al.  Introductory Functional Analysis: With Applications to Boundary Value Problems and Finite Elements , 1997 .

[12]  Guirong Liu,et al.  A point interpolation meshless method based on radial basis functions , 2002 .

[13]  J. McWhirter.  Variational Methods in Mathematics, Science and Engineering , 1978 .

[14]  I. Babuska,et al.  Acta Numerica 2003: Survey of meshless and generalized finite element methods: A unified approach , 2003 .

[15]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[16]  Kevin W. Cassel,et al.  Variational Methods with Applications in Science and Engineering: Dedication , 2013 .

[17]  Irina Perfilieva,et al.  Fuzzy transforms: Theory and applications , 2006, Fuzzy Sets Syst..

[18]  H. Brezis.  Functional Analysis, Sobolev Spaces and Partial Differential Equations , 2010 .