Numerical treatment of oscillatory delay and mixed functional differential equations arising in modelling
AuthorsMalique, Md A.
AdvisorsFord, Neville J.
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AbstractThe pervading theme of this thesis is the development of insights that contribute to the understanding of whether certain classes of functional differential equation have solutions that are all oscillatory. The starting point for the work is the analysis of simple (linear autonomous) ordinary differential equations where existing results allow a full explanation of the phenomena. The Laplace transform features as a key tool in developing a theoretical background. The thesis goes on to explore the corresponding theory for delay equations, advanced equations and functional di erential equations of mixed type. The focus is on understanding the links between the characteristic roots of the underlying equation, and the presence or otherwise of oscillatory solutions. The linear methods are used as a class of numerical schemes which lead to discrete problems analogous to each of the classes of functional differential equation under consideration. The thesis goes on to discuss the insights that can be obtained for discrete problems in their own right, and then considers those new insights that can be obtained about the underlying continuous problem from analysis of the oscillatory behaviour of the analogous discrete problem. The main conclusions of the work are some semi-automated computational approaches (based upon the Principle of the Argument) which allow the prediction of oscillatory solutions to be made. Examples of the effectiveness of the approach are provided, and there is some discussion of its theoretical basis. The thesis concludes with some observations about further work and some of the limitations of existing analytical insights which restrict the reliability with which the approach developed can be applied to wider classes of problem.
CitationFord, N. J., Yan, Y., & Malique, M. A. (2010). Numerical treatment of oscillary functional differential equations. Journal of Computational and Applied Mathematics, 234(9), 2010, 2757-2767
Ferreira, J. M., Ford, N. J., Malique, M. A., Pinelas, S., & Yan, Y. (2011). Analytical and numerical treatment of oscillatory mixed differential equations with differentiable delays and advances. Journal of Computational and Applied Mathematics, 235(17), 2011, 5112-5130
Malique, M. A. (2007). Numerical methods for oscillatory delay differential equations. Paper given at the HERCMA international conference in Althens, 2007
Ford, N. J., Yan, Y., & Malique, M. A. (2010, May 27). Numerical methods of oscillatory mixed-type functional di erential equations. Paper given at First International Workshop on Differential and Integral Equations with Application in Biology and Medicine in Karlovassi, Greece.
TypeThesis or dissertation
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