Some aspects of causal & neutral equations used in modelling

  • Authors:

  • Christopher T. H. Baker;Gennady Bocharov;Eugene Parmuzin;Fathalla Rihan

  • Affiliations:

  • Department of Mathematics, University of Chester, Chester CH1 4BJ, UK and The University of Manchester, M13 9PL, UK;Institute of Numerical Mathematics, Russian Academy of Science, Moscow, Russian Federation;Institute of Numerical Mathematics, Russian Academy of Science, Moscow, Russian Federation;Faculty of Science, Helwan University, Egypt

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2009

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Abstract

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 adjoints and 'quasi-adjoints', and we explore relationships between sensitivity analysis, the variation of parameters formulae, the fundamental solution and adjoints.