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Recent surveys have revealed that the majority of numerical methods for the solution of integral equations use one of two main techniques for generating a set of simultaneous equations for their solution. Either the unknown function is expanded as a combination of basis set functions and the resulting coefficients found, or the integral is discretized using quadrature formulae. The latter results in simultaneous equations for the solution at the quadrature abscissae. The thesis proposes techniques based on various direct iterative methods, including refinements of residual correction which hold no restrictions for nonlinear integral equations. New implementations of successive approximations and Newton's method appear. The latter compares particularly well with other versions as the evaluation of the Jacobian can be made equivalent to the solution of matrix equations of relatively small dimensions. The method can be adapted to the solution of first-kind equations and has been applied to systems of integral equations. The schemes are designed to be adaptive with the aid of the progressive quadrature rules of Patterson or Clenshaw and Curtis and interpolation formulae. The Clenshaw-Curtis rule is particularly favoured as it delivers error estimates. A very powerful routine for the solution of a wide range of integral equations has resulted with the inclusion of a new efficient method for calculating singular integrals. Some work is devoted to the conversion of differential to integral or integro-differential equations and comparing the merits of solving a problem in its original and converted forms. Many equations are solved as test examples throughout the thesis of which several are of physical significance. They include integral equations for the slowing down of neutrons, the Lane-Emden equation, an equation arising from a chemical reactor problem, Chandrasekhar's isotropic scatter ing of radiation equation and the Blasius equation in boundary layer theory.

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Bibliography: leaves 220-230. / iv, 230, [25] leaves : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / This thesis performs a modeling study on a number of sterically constrained aryl-alkyne type systems with the results being used to guide subsequent synthetic efforts. Disulfide, silylene, stannyl, oxalate and 1,2-dioxaethyl and 1,3-dioxapropyl linkages are investigated as candidates to constrain the rotation of the tolane substructure. A preliminary Z-scan experiment is developed and is demonstrated as being able to detect optical nonliniarities. / Thesis (Ph.D.)--University of Adelaide, Dept. of Chemistry, 1997?

Copies of author's previously published works inserted. / Bibliography: p. 119-127. / 12, 127, [8], 35 p., [3] leaves : ill. (chiefly col.) ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / This thesis demonstrates the combined use of stimulated Brillouin scattering (SBS) for aberration correction, auto alignment and pulse compression as an ideal technology for studies of nonlinear optical processes, in particular nonlinear optical scattering (NLS). / Thesis (Ph.D.)--University of Adelaide, Dept. of Physics and Mathematical Physics, 1998

Doctor of Philosophy (PhD). / The term `Lax pair' refers to linear systems (of various types) that are related to nonlinear equations through a compatibility condition. If a nonlinear equation possesses a Lax pair, then the Lax pair may be used to gather information about the behaviour of the solutions to the nonlinear equation. Conserved quantities, asymptotics and even explicit solutions to the nonlinear equation, amongst other information, can be calculated using a Lax pair. Importantly, the existence of a Lax pair is a signature of integrability of the associated nonlinear equation. While Lax pairs were originally devised in the context of continuous equations, Lax pairs for discrete integrable systems have risen to prominence over the last three decades or so and this thesis focuses entirely on discrete equations. Famous continuous systems such as the Korteweg de Vries equation and the Painleve equations all have integrable discrete analogues, which retrieve the original systems in the continuous limit. Links between the different types of integrable systems are well known, such as reductions from partial difference equations to ordinary difference equations. Infinite hierarchies of integrable equations can be constructed where each equation is related to adjacent members of the hierarchy and the order of the equations can be increased arbitrarily. After a literature review, the original material in this thesis is instigated by a completeness study that finds all possible Lax pairs of a certain type, including one for the lattice modified Korteweg de Vries equation. The lattice modified Korteweg de Vries equation is subsequently reduced to several q-discrete Painleve equations, and the reductions are used to form Lax pairs for those equations. The series of reductions suggests the presence of a hierarchy of equations, where each equation is obtained by applying a recursion relation to an earlier member of the hierarchy, this is confirmed using expansions within the Lax pairs for the q-Painleve equations. Lastly, some explorations are included into fake Lax pairs, as well as sets of equivalent nonlinear equations with similar Lax pairs.

Thesis (Ph.D.)--Hong Kong University of Science and Technology, 2007. / Includes bibliographical references (leaves 153-177). Also available in electronic version.

Thesis (Ph.D.) -- University of Adelaide, Dept. of Physics and Mathematical Physics, 1998. / Copies of author's previously published works inserted. Bibliography: p. 119-127.