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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.

An integral of motion is a function of the states of a dynamical system that is constant along the system’s trajectories. Integrals are known for their utility as a means of reducing the dimension of a system, effectively leaving only one differential – or in some cases algebraic – equation to be solved. Invariants of dynamical systems have also proven useful in other contexts, such as in estimation, numerical integration and optimal control. Regardless of the manner in which an integral is employed, finding an analytic form for the integrals of a system generally requires solution of a system of non-linear partial differential equations, with the exception of cases in which certain symmetries of the system are apparent. The objective of this work is to investigate a generalized method for determining motion integrals for non-linear dynamical systems. This method will not work for all nonlinear systems. Indeed, it is expected that the results will test the limitations of this method. In this we consider a method for determining integrals of motion for a small class of dynamical systems akin to the traditional series expansion method for solving partial differential equations. This method involves posing a candidate integral of motion as a series expansion in terms of some set of polynomials. The coefficients of the candidate polynomial are treated as the unknowns in a system of equations. The system of equations is constructed by sampling simulated trajectories of the dynamical system in question. Then the coefficients are solved for using singular value decomposition. There are a number of parameters that can potentially affect this method’s ability to generate an integral of motion that effectively approximates the phase space of the full nonlinear dynamical system. Part of this thesis proposes what some of these parameters might be and investigates how they affect the outcome. A couple of well-known systems are used to conduct these tests: the simple pendulum and the rotating rigid body. The simple pendulum is one of the simplest examples of a non-linear system, and examples of the rotating rigid body in aerospace engineering are ubiquitous.

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Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 136-142). Available also in a digital version from Dissertation Abstracts.

Thèse--Université de Paris.

Thesis (doctoral)--University of Tartu, 2004. / Includes bibliography (p. 123-128).

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1932. / Vita. Sonderabdruck aus den "Schriften des Mathematischen Seminars und des Instituts für angewandte Mathematik der Universität Berlin", Bd. I.

Thesis (Ph. D.)--University of Hong Kong, 1994. / Includes bibliographical references (leaves 257-259).