We are concerned in this thesis with the problem of how to extend standard methods of approximating analytic functions on an interval, say [0, 1], to representing with exponential accuracy the more general class of functions analytic on (0,1) and continuous on [0,1]. The particular approach we take is based upon using an exponential change of variables to transform (0, 1) to (-∞, ∞) thus in the process mapping any endpoint singularities to infinity. Under mild hypotheses, the resulting function is analytic on the real line and may therefore be approximated to exponential accuracy by a spectral interpolant. These ideas form the foundations of an established class of techniques referred to as sine methods. Vie begin by providing a thorough review of sine methods which takes the form of a description of a computational software package called Sincfun. Our experiences lead us to two key conclusions: (i) it is possible to use other bases instead of sine functions; and (ii) such methods in general seem to lead to highly inefficient representations, particularly for oscillatory functions. Regarding (i), we first set out a new convergence theory for variable transformation methods based on Chebyshev polynomials. We show that, 'when a function only has singularity at only one endpoint, mapping to a semi-infinite interval rather than an infinite one can lead to a much improved exponential rate of convergence, from C-√n to c-n2/3 We also perform a thorough analysis of related numerical methods based on so-called double-exponential transforms. Regarding (ii), we first quantify the inefficiency of exponential transform methods, showing that they are suboptimal in a certain precisely defined sense. We then proceed to derive a new' class of transforms which map conformally from an infinite strip to an infinite slit strip. The resulting numerical schemes are demonstrated to have highly desirable properties with regards to the resolution of oscillatory functions . Finally, we derive new quadrature rules, methods for indefinite integration, and spectral methods for application to singular boundary value problems. All of these techniques are based around our new approach of 'variable transformation used in conjunction with Chebyshev interpolation.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:599997 |
| Date | January 2013 |
| Creators | Richardson, Mark |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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