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Subjectsfractional differential equations (20)Energy harvesting (18)XPS (18)numerical methods (16)Ageing (10)atmospheric chemistry (8)Error estimates (8)stability (8)Virtual Reality (8)delay differential equations (7)View MoreJournalJournal of Computational and Applied Mathematics (11)Journal of Physics: Conference Series (8)Applied Numerical Mathematics (6)Atmospheric Chemistry and Physics (5)Journal of computational and applied mathematics (5)View MoreAuthorsFord, Neville J. (72)Smith, Graham C. (38)Yan, Yubin (30)Lawrence, Jonathan (23)Waugh, David G. (23)Baker, Christopher T. H. (21)Jia, Yu (20)Mc Auley, Mark T. (19)Du, Sijun (17)Phillips, Gavin J. (17)View MoreTypesArticle (328)Meetings and Proceedings (30)Book chapter (20)Technical Report (20)Doctoral (14)View More

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The numerical solution of fractional and distributed order differential equations

Connolly, Joseph A. (University of Liverpool (University College Chester), 2004-12)

Fractional Calculus can be thought of as a generalisation of conventional calculus in the sense that it extends the concept of a derivative (integral) to include non-integer orders. Effective mathematical modelling using Fractional Differential Equations (FDEs) requires the development of reliable flexible numerical methods. The thesis begins by reviewing a selection of numerical methods for the solution of Single-term and Multi-term FDEs. We then present: 1. a graphical technique for comparing the efficiency of numerical methods. We use this to compare Single-term and Multi-term methods and give recommendations for which method is best for any given FDE. 2. a new method for the solution of a non-linear Multi-term Fractional Dif¬ferential Equation. 3. a sequence of methods for the numerical solution of a Distributed Order Differential Equation. 4. a discussion of the problems associated with producing a computer program for obtaining the optimum numerical method for any given FDE.

A posteriori error estimates for fully discrete schemes for the time dependent Stokes problem

Baensch, Eberhard; Karakatsani, Fotini; Makridakis, Charalambos (Springer, 2018-05-02)

This work is devoted to a posteriori error analysis of fully discrete finite element approximations to the time dependent Stokes system. The space discretization is based on popular stable spaces, including Crouzeix–Raviart and Taylor–Hood finite element methods. Implicit Euler is applied for the time discretization. The finite element spaces are allowed to change with time steps and the projection steps include alternatives that is hoped to cope with possible numerical artifices and the loss of the discrete incompressibility of the schemes. The final estimates are of optimal order in L∞(L2) for the velocity error.

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.

Identifying Criegee intermediates as potential oxidants in the troposphere

Novelli, Anna; Hens, Korbinian; Tatum-Ernest, Cheryl; Martinez, Monica; Noelscher, Anke C.; Sinha, Vinayak; Paasonen, Paasonen; Petaja, Tuukka; Sipila, Mikko; Elste, Thomas; Plass-Dulmer, Christian; Phillips, Gavin J.; Kubistin, Dagmar; Williams, Jonathan; Vereecken, Luc; Lelieveld, Jos; Harder, Hartwig (Copernicus Publications, 2017-06-29)

We analysed the extensive dataset from the HUMPPA-COPEC 2010 and the HOPE 2012 field campaigns in the boreal forest and rural environments of Finland and Germany, respectively, and estimated the abundance of stabilised Criegee intermediates (SCI) in the lower troposphere. Based on laboratory tests, we propose that the background OH signal observed in our IPI-LIF-FAGE instrument during the afore-mentioned campaigns is caused at least partially by SCI. This hypothesis is based on observed correlations with temperature and with concentrations of unsaturated volatile organic compounds and ozone. The background OH concentration also complements the previously underestimated production rate of sulfuric acid and is consistent with its scavenging through the addition of sulphur dioxide. A central estimate of the SCI concentration of ~ 5 × 104 molecules cm−3 (with an order of magnitude uncertainty) is calculated for the two environments. This implies a very low ambient concentration of SCI, though, over the boreal forest, significant for the conversion of SO2 into H2SO4. The large uncertainties in these calculations, owing to the many unknowns in the chemistry of Criegee intermediates, emphasise the need to better understand these processes and their potential effect on the self-cleaning capacity of the atmosphere.

Bifurcations in approximate solutions of stochastic delay differential equations

Baker, Christopher T. H.; Ford, Judith M.; Ford, Neville J. (World Scientific Publishing Company, 2004)

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

Baker, Christopher T. H.; Parmuzin, Evgeny I. (Rocky Mountain Mathematics Consortium, 2004)

CAB - Collaboration across borders: Peer evaluation for collaborative learning

Whatley, Janice; Bell, Frances; Shaylor, Jan P.; Zaitseva, Elena; Zakrzewska, Danuta (The Informing Science Institute, 2005)

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.

On Halanay-type analysis of exponential stability for the theta-Maruyama method for stochastic delay differential equations

Baker, Christopher T. H.; Buckwar, Evelyn (World Scientific Publishing, 2009-05-08)

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