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Proceedings Paper

Exact generation of continuous-wavelet transforms and reconstruction for Sturm-Liouville ODEs and PDEs
Author(s): Carlos R Handy; Romain Murenzi
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Paper Abstract

We have developed a highly effective continuous wavelet transform formalism for Sturm-Liouville eigenvalue problems, -(partial)x2(Psi)(x) + V(x)(Psi)(x) equals E(Psi)(x), involving an arbitrary rational polynomial potential function V(x). Our method enables the exact generation of wavelet transforms of the type W(Psi)(a,b) equals (integral) dx(1/(square root)a)W((x-b)/a)(Psi)(x), where W(x) equals -N(partial)xie-Q(x) and Q(x) equals (Sigma)n equals 02N (Xi)nxn, provided i (is greater than or equal to) 1, and (Xi)2N (is greater than) 0. This first principles approach emphasizes the use of dilated and translated power moments, (mu)a,b(p) equals (integral) dxxpe-Q(x/a)(Psi)(x + b). For the broad class of problems defined above, a finite number of the moments satisfy a closed, coupled, set of first order differential equations with respect to the inverse scale variable: (partial)(1/a)(mu) a,b(k) equals (Sigma)l equals 0(m(subscript)s)ME,a,b(k,l)(mu)a,b(l), where ms is problem dependent. Using moment eigenvalue methods, one can solve the infinite scale problem, a equals (infinity) , and proceed to numerically integrate the coupled first order equations. For the class of wavelet functions being considered, the wavelet transform, W(Psi) , is a linear superposition of the moments and therefore trivially obtainable. Reconstruction of the corresponding solution, (Psi) , ensues by either using well known dyadic wavelet reconstruction methods, or evaluating the a yields 0 limit of certain moments. The second approach is equivalent to a wavelet reconstruction that is based upon integrating over all values of the scale and translation variables. We also discus a generalization of this formalism permitting its application to 2D Sturm-Liouville PDEs.

Paper Details

Date Published: 3 April 1997
PDF: 10 pages
Proc. SPIE 3078, Wavelet Applications IV, (3 April 1997); doi: 10.1117/12.271760
Show Author Affiliations
Carlos R Handy, Clark Atlanta Univ. (United States)
Romain Murenzi, Clark Atlanta Univ. (United States)

Published in SPIE Proceedings Vol. 3078:
Wavelet Applications IV
Harold H. Szu, Editor(s)

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