Exact generation of continuous-wavelet transforms and reconstruction for Sturm-Liouville ODEs and PDEs

Carlos R Handy; Romain Murenzi

doi:10.1117/12.271760

3 April 1997 Exact generation of continuous-wavelet transforms and reconstruction for Sturm-Liouville ODEs and PDEs

Carlos R Handy, Romain Murenzi

Proceedings Volume 3078, Wavelet Applications IV; (1997) https://doi.org/10.1117/12.271760
Event: AeroSense '97, 1997, Orlando, FL, United States

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 0}^2N (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) dxx^pe^-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)}M_E,a,b(k,l)(mu)_a,b(l), where m_s 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.

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Carlos R Handy and Romain Murenzi "Exact generation of continuous-wavelet transforms and reconstruction for Sturm-Liouville ODEs and PDEs", Proc. SPIE 3078, Wavelet Applications IV, (3 April 1997); https://doi.org/10.1117/12.271760

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KEYWORDS

Wavelets

Continuous wavelet transforms

Palladium

Differential equations

Superposition

Transform theory

Wavelet transforms

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