On some aspects of casual and neutral equations used in mathematical modelling

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

Sensitivity of Functionals in Problems of Variational Assimilation of Observational Data

International audienceThe problem of the variational assimilation of observational data is stated for a nonlinear evolutionmodel as a problem of optimal control in order to find the function of initial condition. The operator of themodel, and consequently the optimal solution, depend on parameters that may contain uncertainties. A functional of the solution of the problem of variational data assimilation is considered. Using the method of second-order adjoint equations, the sensitivity of the functional in respect to the model parameters is studied.The gradient of the functional is expressed through solving a “nonstandard” (nonclassical) problem thatinvolves the coupled system of direct and adjoint equations. The solvability of the nonstandard problem usingthe Hessian initial functional of observations is studied. Numerical algorithms for solving the problem andcomputing the gradient of the functional under consideration are developed with respect to the parameters.The results of the studies are applied in the problem of variational data assimilation for a 3D ocean thermodynamic model

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

This article is not available through ChesterRep.This article was submitted to the RAE2008 for the University of Chester - Applied Mathematics

Identification of the initial function for nonlinear delay differential equations

This journal article is not available through ChesterRep.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

On error analysis in variational data assimilation problem for a nonlinear convection-diffusion model

International audienceno abstrac