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Limited By:	Author = Stynes, Martin;

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A parameter-robust numerical method for a system of reaction–diffusion equations in two dimensions (2018)

Kellogg, R.B.; Madden, Niall; Stynes, Martin

A parameter-robust numerical method for a system of reaction–diffusion equations in two dimensions (2018)

Kellogg, R.B.; Madden, Niall; Stynes, Martin

http://hdl.handle.net/10379/12194

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An analysis of the Grünwald–Letnikov scheme for initial-value problems with weakly singular solutions (2019)

Chen, Hu; Holland, Finbarr; Stynes, Martin

An analysis of the Grünwald–Letnikov scheme for initial-value problems with weakly singular solutions (2019)

Chen, Hu; Holland, Finbarr; Stynes, Martin

Abstract:

A convergence analysis is given for the Grünwald–Letnikov discretisation of a Riemann–Liouville fractional initial-value problem on a uniform mesh tm=mτ with m=0,1,…,M. For given smooth data, the unknown solution of the problem will usually have a weak singularity at the initial time t=0. Our analysis is the first to prove a convergence result for this method while assuming such non-smooth behaviour in the unknown solution. In part our study imitates previous analyses of the L1 discretisation of such problems, but the introduction of some additional ideas enables exact formulas for the stability multipliers in the Grünwald–Letnikov analysis to be obtained (the earlier L1 analyses yielded only estimates of their stability multipliers). Armed with this information, it is shown that the solution computed by the Grünwald–Letnikov scheme is O(τtmα−1) at each mesh point tm; hence the scheme is globally only O(τα) accurate, but it is O(τ) accurate for mesh points tm that are bounded away f...

http://hdl.handle.net/10468/7479

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An efficient collocation method for a Caputo two-point boundary value problem (2015)

Kopteva, Natalia; Stynes, Martin

An efficient collocation method for a Caputo two-point boundary value problem (2015)

Kopteva, Natalia; Stynes, Martin

Abstract:

A two-point boundary value problem is considered on the interval , where the leading term in the differential operator is a Caputo fractional-order derivative of order with . The problem is reformulated as a Volterra integral equation of the second kind in terms of the quantity , where is the solution of the original problem. A collocation method that uses piecewise polynomials of arbitrary order is developed and analysed for this Volterra problem; then by postprocessing an approximate solution of is computed. Error bounds in the maximum norm are proved for and . Numerical results are presented to demonstrate the sharpness of these bounds.
ACCEPTED
peer-reviewed

http://hdl.handle.net/10344/5605

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Post-primary students' images of mathematics: a focus on the qualitative data (2013)

Lane, Ciara; Stynes, Martin; O'Donoghue, John

Post-primary students' images of mathematics: a focus on the qualitative data (2013)

Lane, Ciara; Stynes, Martin; O'Donoghue, John

Abstract:

A questionnaire survey was conducted as part of a PhD study investigating post-primary students' images of mathematics in Ireland. A definition of image of mathematics was adopted from Lim (1999) and Wilson (2011). Students' images of mathematics were hypothesized to include attitudes, beliefs, motivation, self-concept, emotions and past experiences regarding mathematics. A questionnaire was created with both quantitative and qualitative aspects. This paper focuses on the qualitative facet by reviewing students' responses to the open-ended questions according to the five categories of image of mathematics found by Lim (1999). The qualitative data provides a more in-depth insight into Irish students' images of mathematics. Findings from the qualitative data afford an innovative insight into post-primary mathematics education from the student perspective, thus offering a means for mathematics educators to respond to students' needs and encourage Irish post...

http://hdl.handle.net/10344/7246

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Reflection-on-action in qualitative research processes: deconstructing research and developing an honest critical self-appraisal rubric. (2018)

Stynes, Martin; Murphy, Timothy; McNamara, Gerry; O'Hara, Joe

Reflection-on-action in qualitative research processes: deconstructing research and developing an honest critical self-appraisal rubric. (2018)

Stynes, Martin; Murphy, Timothy; McNamara, Gerry; O'Hara, Joe

Abstract:

In this paper, four critical friends meet to discuss qualitative research practices. Together they put one of their own case studies under the knife and deconstruct it to investigate the possibilities that knowledge work is complicated not only by the dynamics of socially constructed enterprises and the actors involved therein, but by the positioning of the researcher. The case describes an evaluative study of a university programme where students engaged in directed experiential learning in group integrated learning settings. The researcher was also the course lead-tutor and this gave rise to some concern, on later reflection and in discussions among critical friends, when issues of researcher positioning were considered. Together, through questioning the topic, the literature, the research experience and the role of the researcher, we developed a reflection-on-action rubric. In a research arena where subjective, interpretative and messy examples abound, as they should, this paper ...

http://doras.dcu.ie/23100/

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Reflection-on-action in qualitative research: a critical self-appraisal rubric for deconstructing research (2018)

Stynes, Martin; Murphy, Timothy; McNamara, Gerry; O'Hara, Joe

Reflection-on-action in qualitative research: a critical self-appraisal rubric for deconstructing research (2018)

Stynes, Martin; Murphy, Timothy; McNamara, Gerry; O'Hara, Joe

Abstract:

In this paper, four critical friends meet to discuss qualitative research practices. Together they put one of their own case studies under the knife and deconstruct it to investigate the possibilities that knowledge work is complicated not only by the dynamics of socially constructed enterprises and the actors involved therein, but by the positioning of the researcher. The case describes an evaluative study of a university program where students engaged in directed experiential learning in group-integrated learning settings. The researcher was also the course lead-tutor and this gave rise to some concern, on later reflection and in discussions among critical friends, when issues of researcher positioning were considered. Together, through questioning the topic, the literature, the research experience and the role of the researcher, we developed a reflection-on-action rubric. In a research arena where subjective, interpretative and messy examples abound, as they should, this paper of...

http://hdl.handle.net/10344/6731

Marked Mark

Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction-diffusion problem (2011)

Kopteva, Natalia; Stynes, Martin

Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction-diffusion problem (2011)

Kopteva, Natalia; Stynes, Martin

Abstract:

A semilinear reaction-diffusion two-point boundary value problem, whose second-order derivative is multiplied by a small positive parameter epsilon(2), is considered. It can have multiple solutions. The numerical computation of solutions having interior transition layers is analysed. It is demonstrated that the accurate computation of such solutions is exceptionally difficult. To address this difficulty, we propose an artificial-diffusion stabilization. For both standard and stabilised finite difference methods on suitable Shishkin meshes, we prove existence and investigate the accuracy of computed solutions by constructing discrete sub- and super-solutions. Convergence results are deduced that depend on the relative sizes of epsilon and N, where N is the number of mesh intervals. Numerical experiments are given in support of these theoretical results. Practical issues in using Newton's method to compute a discrete solution are discussed.
ACCEPTED
peer-reviewed

http://hdl.handle.net/10344/5772

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Walk a mile in my shoes: A case study of the everyday lives and work experiences of a group of Irish primary school Principals (2014)

Stynes, Martin

Walk a mile in my shoes: A case study of the everyday lives and work experiences of a group of Irish primary school Principals (2014)

Stynes, Martin

Abstract:

While much has been written about theories and practices of management and leadership in education in recent years, what school Principals actually do on a daily basis is relatively unresearched in Ireland. Moreover, how Principals experience the job personally goes largely unnoticed. To investigate such questions, the researcher adopts a case study approach to gathering data from a group of 31 Irish primary school Principals. Researcher-driven diaries offer opportunities for self-observation by recording Principals’ personal reflections on management and leadership activities as they arise during the day. On completion, all of the diaries are collated into a single bound volume and a copy of the booklet is returned to each participant. Principals report that it is both interesting and worthwhile to read the entries of others and to gain insights into the daily work practices and lived experiences of colleagues. 21 of the 31 Principals are available in the following weeks for a sec...

http://doras.dcu.ie/20154/