• Analysis via integral equations of an identification problem for delay differential equations

      Baker, Christopher T. H.; Parmuzin, Evgeny I.; University College Chester ; Institute of Numerical Mathematics, Russian Academy of Sciences (Rocky Mountain Mathematics Consortium, 2004)
    • Identification of the initial function for discretized delay differential equations

      Baker, Christopher T. H.; Parmuzin, Evgeny I.; University College Chester ; Institute of Numerical Mathematics, Russian Academy of Sciences (Elsevier, 2005-09-15)
      In the present work, we analyze a discrete analogue for the problem of the identification of the initial function for a delay differential equation (DDE) discussed by Baker and Parmuzin in 2004. The basic problem consists of finding an initial function that gives rise to a solution of a discretized DDE, which is a close fit to observed data.
    • Identification of the initial function for nonlinear delay differential equations

      Baker, Christopher T. H.; Parmuzin, Evgeny I.; University College Chester ; Institute of Numerical Mathematics, Russian Academy of Sciences (de Gruyter, 2005)
      We consider a 'data assimilation problem' for nonlinear delay differential equations. Our problem is to find an initial function that gives rise to a solution of a given nonlinear delay differential equation, which is a close fit to observed data. A role for adjoint equations and fundamental solutions in the nonlinear case is established. A 'pseudo-Newton' method is presented. Our results extend those given by the authors in [(C. T. H. Baker and E. I. Parmuzin, Identification of the initial function for delay differential equation: Part I: The continuous problem & an integral equation analysis. NA Report No. 431, MCCM, Manchester, England, 2004.), (C. T. H. Baker and E. I. Parmuzin, Analysis via integral equations of an identification problem for delay differential equations. J. Int. Equations Appl. (2004) 16, 111–135.)] for the case of linear delay differential equations.
    • An inverse problem for delay differential equations - analysis via integral equations

      Baker, Christopher T. H.; Parmuzin, Evgeny I.; University of Chester (University of Chester, 2006)
    • On some aspects of casual and neutral equations used in mathematical modelling

      Baker, Christopher T. H.; Bocharov, Gennady; Parmuzin, Evgeny I.; Rihan, F. A. R.; University of Chester (University of Chester, 2007)
      The problem that motivates the considerations here is the construction of mathematical models of natural phenomena that depend upon past states. The paper divides naturally into two parts: in the first, we expound the inter-connection between ordinary differential equations, delay differential equations, neutral delay-differential equations and integral equations (with emphasis on certain linear cases). As we show, this leads to a natural hierarchy of model complexity when such equations are used in mathematical and computational modelling, and to the possibility of reformulating problems either to facilitate their numerical solution or to provide mathematical insight, or both. Volterra integral equations include as special cases the others we consider. In the second part, we develop some practical and theoretical consequences of results given in the first part. In particular, we consider various approaches to the definition of an adjoint, we establish (notably, in the context of sensitivity analysis for neutral delay-differential equations) roles for well-defined ad-joints and ‘quasi-adjoints’, and we explore relationships between sensitivity analysis, the variation of parameters formulae, the fundamental solution and adjoints.