The identification of energy scales in the region of Isoscalar Giant Quadrupole Resonance (ISGQR) is motivated by their potential use in understanding how an ordered collective motion transforms into a disordered motion of intrinsic single-particle degrees-of-freedom in many-body quantum systems. High energy-resolution measurements of the ISGQR were obtained by proton inelastic scattering at Ep= 200 MeV using the K600 magnetic Spectrometer at iThemba LABS. The nuclei 58Ni, 90Zr, 120Sn and 208Pb, associated with closed shells, were investigated. Both the Fourier transform and Wavelet analysis were used to extract characteristic energy scales and were later compared with the results from the theoretical microscopic Quasi-particle Phonon Model (QPM), including contributions from collective and non-collective states. The scales found in the experimental data were in good agreement with the QPM. This provides a strong argument that the observed energy scales result from the decay of the collective modes into 2p-2h states. The different scale regions were tested directly by reconstruction of measured energy spectra using the Inverse Fourier Transform and the Continuous Wavelet Transform (CWT), together with a comparison to a previously available reconstruction using the Discrete Wavelet Transform (DWT).

This dissertation considers an approximation strategy using a wavelet reconstruction scheme for solving elliptic problems. The foci of the work are on (1) the approximate solution of differential equations using multiresolution analysis based on wavelet transforms and (2) the homogenization process for solving one and two-dimensional problems, to understand the solutions of second order elliptic problems. We employed homogenization to compute the average formula for permeability in a porous medium. The structure of the associated multiresolution analysis allows for the reconstruction of the approximate solution of the primary variable in the elliptic equation. Using a one-dimensional wavelet reconstruction algorithm proposed in this work, we are able to numerically compute the approximations of the pressure variables. This algorithm can directly be applied to elliptic problems with discontinuous coefficients.We also implemented Java codes to solve the two dimensional elliptic problems using our methods of solutions. Furthermore, we propose homogenization wavelet reconstruction algorithm, fast transform and the inverse transform algorithms that use the results from the solutions of the local problems and the partial derivatives of the pressure variables to reconstruct the solutions.