Numerical analysis of delay differential and integro-differential equations

Zhang, Wenkui (1998) Numerical analysis of delay differential and integro-differential equations. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

In this thesis, we analyze the collocation-based continuous Runge-Kutta methods for delay differential equations and delay Volterra integro-differential equations. We will look at the global convergence and local superconvergence properties of collocation solutions. We also consider the possible extensions of these results to neutral type delay equations and higher order equations. -- In Chapter 2, we give the resolvent representations for solutions to Volterra integral and integro-differential equations with constant delay, and discuss their relevance for the superconvergence order problem. We prove that the resolvent representation does not exist for the proportional delay case. We then analyze the impact of discontinuities in solutions on our numerical methods. We show that discontinuities occur in higher order derivatives for delay integro-differential equations than for delay differential equations. We also prove that discontinuities arising in solutions to neutral delay integro-differential equations are different from those for neutral delay differential equations. Similar results hold for delay Volterra integral equation and delay Volterra integro-differential equation. We also give the discontinuity properties for solutions to state-dependent delay equations. -- In Chapter 3, we discuss collocation solutions to various equations with constant delay, and survey global and local convergence results. Some extensions to neutral type constant delay problems are also described. -- In Chapter 4, we introduce collocation methods for differential and integro-differential equations with variable delay, especially proportional delay. We prove that the global convergence order equals the number of collocation parameters used for first order differential equations with proportional delay. We give concrete representations for collocation solutions after the first step, and conduct some numerical experiments which suggest that superconvergence does exist in the proportional delay case. An extension to second order DDE is also given. -- In Chapter 5 we suggest a new approach, standard embedding, to the superconvergence order problem of collocation solutions to differential equations with proportional delay, and are able to prove that superconvergence results again do exist under certain conditions.

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