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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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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.

Spreadsheet tools to estimate the thermal transmittance and thermal conductivities of gas spaces of an Insulated Glazing Unit

Nammi, Sathish K.; Shirvani, Hassan; Shirvani, Ayoub; Edwards, Gerard; Dunn, Jeremy (Anglia Ruskin Research Online, 2014-03)

An Insulated Glazing unit (IGU) is constructed with two or more layers of glass panes sealed together by gas spaces in-between. IGUs are prevalent in windows, doors and rooflights, primarily due to their improved thermal resistance. Today, most IGUs are either two or three layered. Adding further layers of glass improves thermal insulation but with the penalty of increased cost and weight. Low emissivity (Low-e) film coatings, when deposited on the glass panes, reduce long-wavelength radiative heat losses. Furthermore, filling the gas spaces with the inert gases (e.g. Argon, Krypton, Xenon and SF6), further reduce conduction and natural convection across the gap. In summary, higher thermal insulation performance of an IGU can be achieved with gas fillings and Low-e coatings on glass. This report discusses spreadsheets that have been developed, capable of estimating the thermal transmittance values of IGU, as per BS EN 673. The spreadsheet tools also have the ability to estimate the thermal conductivity of the gas spaces between the panes of IGU.

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.

Mixed-type functional differential equations: A numerical approach (extended version)

Ford, Neville J.; Lumb, Patricia M. (University of Chester, 2007)

An inverse problem for delay differential equations - analysis via integral equations

Baker, Christopher T. H.; Parmuzin, Evgeny I. (University of Chester, 2006)

The numerical solution of fractional differential equations: Speed versus accuracy

Ford, Neville J.; Simpson, A. Charles (Manchester Centre for Computational Mathematics, 2003-05-23)

This paper discusses the development of efficient algorithms for a certain fractional differential equation.

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.

Existence theory for a class of evolutionary equations with time-lag, studied via integral equation formulations

Baker, Christopher T. H.; Lumb, Patricia M. (University of Chester, 2006)

In discussions of certain neutral delay differential equations in Hale’s form, the relationship of the original problem with an integrated form (an integral equation) proves to be helpful in considering existence and uniqueness of a solution and sensitivity to initial data. Although the theory is generally based on the assumption that a solution is continuous, natural solutions of neutral delay differential equations of the type considered may be discontinuous. This difficulty is resolved by relating the discontinuous solution to its restrictions on appropriate (half-open) subintervals where they are continuous and can be regarded as solutions of related integral equations. Existence and unicity theories then follow. Furthermore, it is seen that the discontinuous solutions can be regarded as solutions in the sense of Caratheodory (where this concept is adapted from the theory of ordinary differential equations, recast as integral equations).

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