Mixed-type functional differential equations: A numerical approach

This is a PDF version of a preprint submitted to Elsevier. The definitive version was published in Journal of computational and applied mathematics and is available at www.elsevier.comThis preprint discusses mixed-type functional equations

An algorithm to detect small solutions in linear delay differential equations

This is a PDF version of a preprint submitted to Elsevier. The definitive version was published in the Journal of computational and applied mathematics and is available at www.elsevier.comThis preprint discusses an algorithm that provides a simple reliable mechanism for the detection of small solutions in linear delay differential equations.This article was submitted to the RAE2008 for the University of Chester - Applied Mathematics

Analytical and numerical investigation of mixed-type functional differential equations

NOTICE: this is the author’s version of a work that was accepted for publication in Journal of computational and applied mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of computational and applied mathematics, 234 (2010), doi: 10.1016/j.cam.2010.01.028This journal article is concerned with the approximate solution of a linear non-autonomous functional differential equation, with both advanced and delayed arguments

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

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

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

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)

Delay differential equations: Detection of small solutions

This thesis concerns the development of a method for the detection of small solutions to delay differential equations. The detection of small solutions is important because their presence has significant influence on the analytical prop¬erties of an equation. However, to date, analytical methods are of only limited practical use. Therefore this thesis focuses on the development of a reliable new method, based on finite order approximations of the underlying infinite dimen¬sional problem, which can detect small solutions. Decisions (concerning the existence, or otherwise, of small solutions) based on our visualisation technique require an understanding of the underlying methodol¬ogy behind our approach. Removing this need would be attractive. The method we have developed can be automated, and at the end of the thesis we present a prototype Matlab code for the automatic detection of small solutions to delay differential equations

A review of the methods for the solution of DEAs

Differential-Algebraic equation systems (DAEs) occur in a variety of applications in science and engineering and interest in them has grown considerably during the latter part of the twentieth century. The aims of this thesis are to provide the interested reader with a comprehensible and informative introduction to DAEs, whilst assuming no prior knowledge of the subject area; to give an indication to the reader with a DAE to solve whether or not they can hope to be successful; to introduce the reader to possible methods of solution (either analytical, numerical or both

Delay differential equations : detection of small solutions

This thesis concerns the development of a method for the detection of small solutions to delay differential equations. The detection of small solutions is important because their presence has significant influence on the analytical prop¬erties of an equation. However, to date, analytical methods are of only limited practical use. Therefore this thesis focuses on the development of a reliable new method, based on finite order approximations of the underlying infinite dimen¬sional problem, which can detect small solutions. Decisions (concerning the existence, or otherwise, of small solutions) based on our visualisation technique require an understanding of the underlying methodol¬ogy behind our approach. Removing this need would be attractive. The method we have developed can be automated, and at the end of the thesis we present a prototype Matlab code for the automatic detection of small solutions to delay differential equations.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Computational methods for a mathematical model of propagation of nerve impulses in myelinated axons

NOTICE: this is the author’s version of a work that was accepted for publication in Applied Numerical Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Applied Numerical Mathematics, 85, November 2014, pp. 38-53. DOI: 1016/j.apnum.2014.06.0046.004This paper is concerned with the approximate solution of a nonlinear mixed type functional differential equation (MTFDE) arising from nerve conduction theory. The equation considered describes conduction in a myelinated nerve axon. We search for a monotone solution of the equation defined in the whole real axis, which tends to given values at ±∞. We introduce new numerical methods for the solution of the equation, analyse their performance, and present and discuss the results of the numerical simulations