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All of ChesterRepCommunitiesTitleAuthorsPublication DateSubmit DateSubjectsPublisherJournalThis CommunityTitleAuthorsPublication DateSubmit DateSubjectsPublisherJournal

Subjectsnumerical methods (4)delay differential equations (3)computational modelling (2)fractional differential equations (2)integral equations (2)numerical solutions (2)small solutions (2)adjoint equations (1)analysis of models (1)Applications (1)View MoreJournalInternational Journal of Energy and Power Engineering (1)AuthorsFord, Neville J. (10)Baker, Christopher T. H. (7)Lumb, Patricia M. (3)Bocharov, Gennady (2)Edwards, Gerard (2)Nammi, Sathish K. (2)Parmuzin, Evgeny I. (2)Rihan, F. A. R. (2)Shirvani, Ayoub (2)Shirvani, Hassan (2)View MoreTypes

Technical Report (20)

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Characterising small solutions in delay differential equations through numerical approximations

Ford, Neville J.; Lunel, Sjoerd M. V. (Manchester Centre for Computational Mathematics, 2003-05-23)

This paper discusses how the existence of small solutions for delay differential equations can be predicted from the behaviour of the spectrum of the finite dimensional approximations.

Theory and numerics for multi-term periodic delay differential equations, small solutions and their detection

Ford, Neville J.; Lumb, Patricia M. (University of Chester, 2006)

We summarise a theoretical treatment that analyses whether the equation has small solutions. We consider discrete equations that arise when a numerical method with fixed step size is applied to approximate the solution to (†) and we develop a corresponding theory. Our results show that small solutions can be detected reliably by the numerical scheme. We conclude with some numerical examples.

Fixed point theroms and their application - discrete Volterra applications

Baker, Christopher T. H.; Song, Yihong (University of Chester, 2006)

The existence of solutions of nonlinear discrete Volterra equations is established. We define discrete Volterra operators on normed spaces of infinite sequences of finite-dimensional vectors, and present some of their basic properties (continuity, boundedness, and representation). The treatment relies upon the use of coordinate functions, and the existence results are obtained using fixed point theorems for discrete Volterra operators on infinite-dimensional spaces based on fixed point theorems of Schauder, Rothe, and Altman, and Banach’s contraction mapping theorem, for finite-dimensional spaces.

Numerical modelling of qualitative behaviour of solutions to convolution integral equations

Diogo, Teresa; Ford, Judith M.; Ford, Neville J.; Lima, Pedro M. (University of Chester, 2006)

We consider the qualitative behaviour of solutions to linear integral equations of the form where the kernel k is assumed to be either integrable or of exponential type. After a brief review of the well-known Paley-Wiener theory we give conditions that guarantee that exact and approximate solutions of (1) are of a specific exponential type. As an example, we provide an analysis of the qualitative behaviour of both exact and approximate solutions of a singular Volterra equation with infinitely many solutions. We show that the approximations of neighbouring solutions exhibit the correct qualitative behaviour.

Noise-induced changes to the bifurcation behaviour of semi-implicit Euler methods for stochastic delay differential equations

Ford, Neville J.; Norton, Stewart J. (University of Chester, 2007)

We are concerned with estimating parameter values at which bifurcations occur in stochastic delay differential equations. After a brief review of bifurcation, we employ a numerical approach and consider how bifurcation values are influenced by the choice of numerical scheme and the step length and by the level of white noise present in the equation. In this paper we provide a formulaic relationship between the estimated bifurcation value, the level of noise, the choice of numerical scheme and the step length. We are able to show that in the presence of noise there maybe some loss of order in the accuracy of the approximation to the true bifurcation value compared to the use of the same approach in the absence of noise.

Halanay-type theory in the context of evolutionary equations with time-lag

Baker, Christopher T. H. (University of Chester, 2009)

We consider extensions and modifications of a theory due to Halanay, and the context in which such results may be applied. Our emphasis is on a mathematical framework for Halanay-type analysis of problems with time lag and simulations using discrete versions or numerical formulae. We present selected (linear and nonlinear, discrete and continuous) results of Halanay type that can be used in the study of systems of evolutionary equations with various types of delayed argument, and the relevance and application of our results is illustrated, by reference to delay-differential equations, difference equations, and methods.

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, 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.

Concerning periodic solutions to non-linear discrete Volterra equations with finite memory

Baker, Christopher T. H.; Song, Yihong (University of Chester, 2007)

In this paper we discuss the existence of periodic solutions of discrete (and discretized) non-linear Volterra equations with finite memory. The literature contains a number of results on periodic solutions of non linear Volterra integral equations with finite memory, of a type that arises in biomathematics. The “summation” equations studied here can arise as discrete models in their own right but are (as we demonstrate) of a type that arise from the discretization of such integral equations. Our main results are in two parts: (i) results for discrete equations and (ii) consequences for quadrature methods applied to integral equations. The first set of results are obtained using a variety of fixed point theorems. The second set of results address the preservation of properties of integral equations on discretizing them. An expository style is adopted and examples are given to illustrate the discussion.

Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations

Ford, Neville J.; Connolly, Joseph A. (University of Chester, 2007)

We give a comparison of the efficiency of three alternative decomposition schemes for the approximate solution of multi-term fractional differential equations using the Caputo form of the fractional derivative. The schemes we compare are based on conversion of the original problem into a system of equations. We review alternative approaches and consider how the most appropriate numerical scheme may be chosen to solve a particular equation.

Neutral delay differential equations in the modelling of cell growth

Baker, Christopher T. H.; Bocharov, Gennady; Rihan, F. A. R. (University of Chester, 2008)

In this contribution, we indicate (and illustrate by example) roles that may be played by neutral delay differential equations in modelling of certain cell growth phenomena that display a time lag in reacting to events. We explore, in this connection, questions involving the sensitivity analysis of models and related mathematical theory; we provide some associated numerical results.

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