ON SOLVING FIRST-KIND INTEGRAL-EQUATIONS USING WAVELETS ON A BOUNDED INTERVAL

Conventional method of moments (MoM), when applied directly to integral equations, leads to a dense matrix which often becomes computationally intractable due to the large memory requirement and high computation time necessary to invert a dense matrix. To overcome these difficulties, wavelet-bases have been used recently which, primarily because of their local supports and vanishing moment properties, lead to a sparse matrix. We will 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. In this paper, we propose the use of compactly supported semi-orthogonal (SO) spline wavelets specially constructed for the bounded interval in solving first-kind integral equations. We apply this technique to analyze a problem involving two-dimensional electromagnetic 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, We also present the explicit closed-form polynomial representation of the scaling functions and wavelets.