This thesis is concerned with the qualitative and quantitative properties of solutions of certain classes of ordinary di erential equations (ODEs); in particular linear boundary value problems of second order ODE's and non-linear ODEs of order at most four. The Lyapunov's second method of special functions called Lyapunov functions are employed extensively in this thesis. We construct suitable complete Lyapunov functions to discuss the qualitative properties of solutions to certain classes of non-linear ordinary di erential equations considered. Though there is no unique way of constructing Lyapunov functions, We adopt Cartwright's method to construct complete Lyapunov functions that are required in this thesis. Su cient conditions were established to discuss the qualitative properties such as boundedness, convergence, periodicity and stability of the classes of equations of our focus. Another aspect of this thesis is on the quantitative properties of solutions. New scheme based on interpolation and collocation is derived for solving initial value problem of ODEs. This scheme is derived from the general method of deriving the spline functions. Also by exploiting the Trigonometric identity property of the Chebyshev polynomials, We develop a new scheme for approximating the solutions of two-point boundary value problems. These schemes are user-friendly, easy to develop algorithm (computer program) and execute. They compare favorably with known standard methods used in solving the classes of problems they were derived for
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:ufh/vital:11588 |
| Date | January 2009 |
| Creators | Ogundare, Babatunde Sunday |
| Publisher | University of Fort Hare, Faculty of Science & Agriculture |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis, Doctoral, PhD (Applied Mathematics) |
| Format | 120 leaves; 30 cm, pdf |
| Rights | University of Fort Hare |
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