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Direct solution of ill‐posed boundary value problems by radial basis function collocation method
Author(s) -
Cheng A. H.D.,
Cabral J. J. S. P.
Publication year - 2005
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1362
Subject(s) - collocation (remote sensing) , well posed problem , boundary value problem , mathematics , convergence (economics) , radial basis function , singular boundary method , collocation method , iterative method , mathematical optimization , function (biology) , boundary (topology) , basis (linear algebra) , stability (learning theory) , computer science , mathematical analysis , finite element method , boundary element method , differential equation , ordinary differential equation , geometry , physics , machine learning , evolutionary biology , artificial neural network , biology , economics , thermodynamics , economic growth
Numerical solution of ill‐posed boundary value problems normally requires iterative procedures. In a typical solution, the ill‐posed problem is first converted to a well‐posed one by assuming the missing boundary values. The new problem is solved by a conventional numerical technique and the solution is checked against the unused data. The problem is solved iteratively using optimization schemes until convergence is achieved. The present paper offers a different procedure. Using the radial basis function collocation method, we demonstrate that the solution of certain ill‐posed problems can be accomplished without iteration. This method not only is efficient and accurate, but also circumvents the stability problem that can exist in the iterative method. Copyright © 2005 John Wiley & Sons, Ltd.