In this thesis we consider two types of ill-posed problem: (13 the numerical solution of Fredholm integral equations of the first kind of convolution type, with non-negativity constraints on the solution, and (2) the numerical inversion of some truncated integral transforms of nonnegative functions. In (2) the transforms are with respect to simple kernel functions of the form cos ax and sin(ax)/(ax), a constant. In solving both types of problem we extend ideas of Biraud for solving convolution type integral equations. This method is based on extrapolation and in Chapter 1a brief resume of extrapolation methods is given by way of introduction. In Chapter 2 we discuss Biraud's method and the choice of approximating function spaces. In this thesis we consider two distinct types of approximations; band-limited approximations, which-are not analytic in Fourier space, and also approximations which are analytic in Fourier space. In Chapter 3, which deals with numerical deconvolution, Wahba's idea of weighted cross-validation is used to determine an optimal rectangular filter, and the resulting cut-off point. is compared on several' test examples, with the best cut-off point from which to perform Biraud extrapolation. In many cases the two cut-off point coincide, or very nearly. In Chapter 4, trigonometric Biraud extrapolation is used to numerically invert several autocorrelation functions arising in laser technology. In Chapter 5, two of these problems are repeated using Hermite wave-functions (HWFs) as bases. 8oth. linear and Biraud extrapolation 'methods are compared. A new formula for the convolutions of HWFs is developed and used.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:549162 |
| Date | January 1981 |
| Creators | Al-Faour, Omar Mohamad |
| Publisher | Aberystwyth University |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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