Premium
A basis function for approximation and the solutions of partial differential equations
Author(s) -
Tian H. Y.,
Reutskiy S.,
Chen C. S.
Publication year - 2008
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20304
Subject(s) - mathematics , eigenfunction , basis function , basis (linear algebra) , mathematical analysis , helmholtz equation , partial differential equation , trigonometric series , series (stratigraphy) , type (biology) , function (biology) , differential equation , boundary value problem , eigenvalues and eigenvectors , geometry , paleontology , ecology , physics , quantum mechanics , evolutionary biology , biology
In this article, we introduce a type of basis functions to approximate a set of scattered data. Each of the basis functions is in the form of a truncated series over some orthogonal system of eigenfunctions. In particular, the trigonometric eigenfunctions are used. We test our basis functions on recovering the well‐known Franke's and Peaks functions given by scattered data, and on the extension of a singular function from an irregular domain onto a square. These basis functions are further used in Kansa's method for solving Helmholtz‐type equations on arbitrary domains. Proper one level and two level approximation techniques are discussed. A comparison of numerical with analytic solutions is given. The numerical results show that our approach is accurate and efficient. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008