Some aspects of causal & neutral equations used in modelling

Christopher T H Baker, Gennady Bocharov, Eugene Parmuzin, Fathalla Rihan

    Research output: Contribution to journalArticlepeer-review


    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) rôles for well-defined adjoints and 'quasi-adjoints', and we explore relationships between sensitivity analysis, the variation of parameters formulae, the fundamental solution and adjoints. © 2008 Elsevier B.V. All rights reserved.
    Original languageEnglish
    Pages (from-to)335-349
    Number of pages14
    JournalJournal of Computational and Applied Mathematics
    Issue number2
    Publication statusPublished - 15 Jul 2009


    • Adjoints
    • Analysis of models
    • Computational modelling
    • Delay & neutral delay-differential equations
    • Fundamental solutions
    • Model selection
    • Resolvents
    • Sensitivity
    • Variation of parameters
    • Volterra integral equations


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