On solving first-kind integral equations using wavelets on a bounded interval

The conventional method of moments (MoM), when applied directly to integral equations, leads to a dense matrix which often becomes computationally intractable. To overcome the difficulties, wavelet-bases have been used previously which lead to a sparse matrix. The authors refer to "MoM with wavelet bases" as "wavelet MoM". There have been three different ways of applying the wavelet techniques to boundary integral equations: 1) wavelets on the entire real line which requires the boundary conditions to be enforced explicitly, 2) wavelet bases for the bounded interval obtained by periodizing the wavelets on the real line, and 3) "wavelet-like" basis functions. Furthermore, only orthonormal (ON) bases have been considered. The present authors propose the use of compactly supported semi-orthogonal (SO) spline wavelets specially constructed for the bounded interval in solving first-kind integral equations. They apply this technique to analyze a problem involving 2D EM scattering from metallic cylinders. It is shown that the number of unknowns in the case of wavelet MoM increases by m-1 as compared to conventional MoM, where m is the order of the spline function. Results for linear (m=2) and cubic (m=4) splines are presented along with their comparisons to conventional MoM results. It is observed that the use of cubic spline wavelets almost "diagonalizes" the matrix while maintaining less than 1.5% of relative normed error. The authors also present the explicit closed-form polynomial representation of the scaling functions and wavelets.<
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