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

peer-reviewedA 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.ACCEPTEDpeer-reviewe

Longitudinal Outcomes of Gender Identity in Children (LOGIC): study protocol for a retrospective analysis of the characteristics and outcomes of children referred to specialist gender services in the UK and the Netherlands

INTRODUCTION: Specialist gender services for children and young people (CYP) worldwide have experienced a significant increase in referrals in recent years. As rates of referrals increase, it is important to understand the characteristics and profile of CYP attending these services in order to inform treatment pathways and to ensure optimal outcomes. METHODS AND ANALYSIS: A retrospective observational study of clinical health records from specialist gender services for CYP in the UK and the Netherlands. The retrospective analysis will examine routinely collected clinical and outcome measures data including demographic, clinical, gender identity-related and healthcare resource use information. Data will be reported for each service and also compared between services. This study forms part of a wider programme of research investigating outcomes of gender identity in children (the Longitudinal Outcomes of Gender Identity in Children study). ETHICS AND DISSEMINATION: The proposed study has been approved by the Health Research Authority and London-Hampstead Research Ethics Committee as application 19/LO/0181. The study findings will be published in peer-reviewed journals and presented at both conferences and stakeholder events

Clinical course, characteristics and prognostic indicators in patients presenting with back and leg pain in primary care. The ATLAS study protocol

Low-back related leg pain with or without nerve root involvement is associated with a poor prognosis compared to low back pain (LBP) alone. Compared to the literature investigating prognostic indicators of outcome for LBP, there is limited evidence on prognostic factors for low back-related leg pain including the group with nerve root pain. This 1 year prospective consultation-based observational cohort study will describe the clinical, imaging, demographic characteristics and health economic outcomes for the whole cohort, will investigate differences and identify prognostic indicators of outcome (i.e. change in disability at 12 months), for the whole cohort and, separately, for those classified with and without nerve root pain. In addition, nested qualitative studies will provide insights on the clinical consultation and the impact of diagnosis and treatment on patients' symptom management and illness trajectory

Acute otitis externa: Consensus definition, diagnostic criteria and core outcome set development.

OBJECTIVE: Evidence for the management of acute otitis externa (AOE) is limited, with unclear diagnostic criteria and variably reported outcome measures that may not reflect key stakeholder priorities. We aimed to develop 1) a definition, 2) diagnostic criteria and 3) a core outcome set (COS) for AOE. STUDY DESIGN: COS development according to Core Outcome Measures in Effectiveness Trials (COMET) methodology and parallel consensus selection of diagnostic criteria/definition. SETTING: Stakeholders from the United Kingdom. SUBJECTS AND METHODS: Comprehensive literature review identified candidate items for the COS, definition and diagnostic criteria. Nine individuals with past AOE generated further patient-centred candidate items. Candidate items were rated for importance by patient and professional (ENT doctors, general practitioners, microbiologists, nurses, audiologists) stakeholders in a three-round online Delphi exercise. Consensus items were grouped to form the COS, diagnostic criteria, and definition. RESULTS: Candidate COS items from patients (n = 28) and literature (n = 25) were deduplicated and amalgamated to a final candidate list (n = 46). Patients emphasised quality-of-life and the impact on daily activities/work. Via the Delphi process, stakeholders agreed on 31 candidate items. The final COS covered six outcomes: pain; disease severity; impact on quality-of-life and daily activities; patient satisfaction; treatment-related outcome; and microbiology. 14 candidate diagnostic criteria were identified, 8 reaching inclusion consensus. The final definition for AOE was 'diffuse inflammation of the ear canal skin of less than 6 weeks duration'. CONCLUSION: The development and adoption of a consensus definition, diagnostic criteria and a COS will help to standardise future research in AOE, facilitating meta-analysis. Consulting former patients throughout development highlighted deficiencies in the outcomes adopted previously, in particular concerning the impact of AOE on daily life

Impact of Cultural Tourism Upon Urban Economies: An Econometric Exercise

In recent years, interest in tourism has spread rapidly throughout many small and medium European cities, which previously have not necessarily considered themselves as tourist destinations. Tourism is increasingly seen as a potential lever towards high economic growth, measured both in terms of income and employment. In the present Working Paper we report the analysis on the economic impact undertaken in the framework of the PICTURE Project, showing the results of a novel econometric exercise to statistically assess the impacts of cultural tourism upon European municipalities. More precisely the analysis aims at estimating the effects of tourism specialisation on local income and prices. The Working Paper is built as follows. Section 1 presents and discusses secondary data about tourism facts and figures, including the economic impact of tourism upon European economies, with a focus on cultural tourism. An extensive review of literature, which identifies the main categories of impacts and the currently available methodologies to assess them, is undertaken. Section 2 focuses on the state of the art. Section 3 describes the database built for the analysis, sources and variables. In order to visually represent the spatial variability of the main parameters, a series of thematic maps at NUTS 3 level(Maps of European tourism), using GIS (Geographical Information System) are also included in the Working Paper. Section 4 shows the results of the econometric analysis of European panel data for the estimation of the effects of tourism specialisation on both local incomes and prices. Section 5 concludes

A Fitted Scheme for a Caputo Initial-Boundary Value Problem

In this paper we consider an initial-boundary value problem with a Caputo time derivative of order (Formula presented.). The solution typically exhibits a weak singularity near the initial time and this causes a reduction in the orders of convergence of standard schemes. To deal with this singularity, the solution is computed with a fitted difference scheme on a graded mesh. The convergence of this scheme is analysed using a discrete maximum principle and carefully chosen barrier functions. Sharp error estimates are proved, which show an enhancement in the convergence rate compared with the standard L1 approximation on uniform meshes, and also indicate an optimal choice for the mesh grading. This optimal mesh grading is less severe than the optimal grading for the standard L1 scheme. Furthermore, the dependence of the error on the final time forms part of our error estimate. Numerical experiments are presented which corroborate our theoretical results