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A collocation and Cartesian grid methods using new radial basis function to solve class of partial differential equations
Faculty
Science
Year:
2010
Type of Publication:
Article
Pages:
1349-1362
Authors:
Ahmed, S. G, Mekey, M. L
DOI:
10.1080/00207160802322316
Journal:
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS TAYLOR \& FRANCIS LTD
Volume:
87
Research Area:
Mathematics
ISSN
ISI:000277399400014
Keywords :
meshless methods, thin plate method, multiquadraic, radial basis functions, Cartesian grid methods
Abstract:
In the present paper, a class of partial differential equation represented by Poisson's type problems are solved using a proposed Cartesian grid method and a collocation technique using a newradial basis function. The advantage of using this new radial basis function represented by overcoming singularity from the diagonal elements when thin plate radial basis function is used. The new function is a combination of both multiquadric and thin plate radial basis functions. The new radial basis function contains a control parameter epsilon, that takes one when evaluating the singular elements and equals zero elsewhere. Collocation of the approximate solution of the potential over the governing and boundary condition equations leads to a double linear system of equations. A proposed algebraic procedure is then developed to solve the double system. Examples of Poisson and Helmholtz equations are solved and the present results are compared with the their analytical solutions. A good agreement with analytical results is achieved.
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