Radial basis approximation and its application to biharmonic equation
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Radial basis approximation and its application to biharmonic equation Xin Li
Received: 4 March 2008 / Accepted: 10 October 2008 / Published online: 15 November 2008 © Springer Science + Business Media, LLC 2008
Abstract Order of approximating functions and their derivatives by radial bases on arbitrarily scattered data is derived. And then radial bases are used to construct solutions of biharmonic equations that approximate potential integrals for the exact solutions with the order of approximation derived. Keywords Radial basis approximation · Biharmonic equation · Arbitrarily scattered data Mathematics Subject Classifications (2000) 41A25 · 31B30 · 35C15
1 Introduction Radial basis functions (RBF), with simple structures in mathematics, have been successfully applied in several scientific fields such as neural networks, computer modeling, interpolation on scattered data, etc. (cf. [2, 3, 9] and many others). And more recently they have proved to be powerful tools in solving differential equations numerically (cf. [1, 4, 7, 8, 11] for instance). In this paper we will generalize the approximation scheme in [5] to arbitrarily scattered data and derive more accurate order of approximating functions and their derivatives, which is especially suitable for radial basis approximation. And then, as in [6] for Poisson’s equations, we will use radial
Communicated by Juan Manuel Peña. X. Li (B) Department of Mathematical Sciences, University of Nevada, Las Vegas, Las Vegas, NV 89154-4020, USA e-mail: [email protected]
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X. Li
bases to construct solutions of biharmonic equations that approximate the exact solutions in potential form with the order of approximation derived. In Section 2 approximation result on arbitrarily scattered data will be established. And solutions of biharmonic equations will be constructed by radial bases in Section 3. The order of approximation of the constructed solutions to the exact solutions in potential form will be derived in Section 4.
2 Approximation on scattered data Let I = [−1, 1]s be the unit cube of Rs . For a bounded domain D in Rs and δ > 0, let Dδ = D + δ I := {x + y; x ∈ D, y ∈ δ I}. For any integer n, set s j j+1 s ∩ Dδ = ∅ , In (Dδ ) = j ∈ Z ; , n n where 1 = (1, · · · , 1) ∈ Z s . Suppose that X = {xi ; 1 ≤ i ≤ N} is an arbitrary set of scattered points in Dδ . For each j ∈ In (Dδ ), we let Xj be the subset of points s in X that lie in nj , j+1 , or n s j j+1 , Xj = X ∩ . n n Let Nj be the number of points in Xj (or the cardinality of Xj ), which from now on we assume Nj ≥ 1
(1)
for any j ∈ In (Dδ ). Clearly X = ∪j∈In (Dδ ) Xj and N = j∈In (Dδ ) Nj . Let C(D) be the set of all continuous and bounded functions on D with f C(D) = sup | f (x)|. x∈D
For k = (k1 , · · · , ks ) ∈ Z , where ki ≥ 0 for 1 ≤ i ≤ s, let |k| = k1 + · · · + ks |k| and write Dk f = k∂1 f ks as usual. For an integer m ≥ 0, denote by Cm (D) the s
∂ x1 ···∂ xs
space of all functions f whose partial derivatives Dk f for |k| ≤ m are bounded and continuous in D with
Dk f
, f C
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