TY - JOUR

T1 - Some aspects of causal & neutral equations used in modelling

AU - Baker, Christopher T.H.

AU - Bocharov, Gennady

AU - Parmuzin, Eugene

AU - Rihan, Fathalla

N1 - Funding Information:
The first author is emeritus professor in the University of Manchester, the second author is supported in part by a Leverhulme visiting professorship at the University of Chester, UK and in part by the Russian Foundation for Basic Research; he is now honorary visiting professor at Chester. The third author is supported by INTAS Fellowship YS04-83-2818 held at the University of Chester, UK and in part by the Russian Foundation for Basic Research.

PY - 2009/7/15

Y1 - 2009/7/15

N2 - 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.

AB - 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.

KW - Adjoints

KW - Analysis of models

KW - Computational modelling

KW - Delay & neutral delay-differential equations

KW - Fundamental solutions

KW - Model selection

KW - Resolvents

KW - Sensitivity

KW - Variation of parameters

KW - Volterra integral equations

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U2 - 10.1016/j.cam.2008.04.001

DO - 10.1016/j.cam.2008.04.001

M3 - Article

AN - SCOPUS:67349195508

SN - 0377-0427

VL - 229

SP - 335

EP - 349

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

IS - 2

ER -