Numerical resolution of the hyperbolic heat equation using smoothed mathematical functions instead of Heaviside and Dirac delta distributions

The hyperbolic bioheat equation (HBE) has been used to model heating applications involving very short power pulses. This equation includes two mathematical distributions (Heaviside and Delta) which have to be necessarily substituted for smoothed mathematical functions when the HBE is solved by numerical methods. This study focuses on which type of smoothed functions would be suitable for this purpose, i.e. those which would provide solutions similar to those obtained analytically from the original Heaviside and Delta distributions. The logistic function was considered as a substitute for the Heaviside function, while its derivative and the probabilistic Gaussian function were considered as substitutes for the Delta distribution. We also considered polynomial interpolation functions, in particular, the families of smoothed functions with continuous second derivative without overshoot used by COMSOL Multiphysics. All the smoothed functions were used to solve the HBE by the Finite Element Method (COMSOL Multiphysics), and the solutions were compared to those obtained analytically from the original Heaviside and Delta distributions. The results showed that only the COMSOL smoothed functions provide a numerical solution almost identical to the analytical one. Finally, we demonstrated mathematically that in order to find a suitable smoothed function (f) that must adequately substitute any mathematical distribution (D) in the HBE, the difference D - f must have compact support. (c) 2013 Elsevier Ltd. All rights reserved.This work received financial support from the Spanish "Plan Nacional de I + D + I del Ministerio de Ciencia e Innovacion" Grant No. TEC2011-27133-C02-01 and from Universitat Politenica de Valencia (PAID-06-11 Ref. 1988). V. Romero Garcia is grateful for the support of "Programa de Contratos Post-Doctorales con Movilidad UPV del Campus de Excelencia (CEI-01-11)" and FEDER Project MAT2009-09438.Rivera Ortun, MJ.; Trujillo Guillen, M.; Romero García, V.; López Molina, JA.; Berjano Zanón, E. (2013). Numerical resolution of the hyperbolic heat equation using smoothed mathematical functions instead of Heaviside and Dirac delta distributions. International Communications in Heat and Mass Transfer. 46:7-12. https://doi.org/10.1016/j.icheatmasstransfer.2013.05.017S7124

Physical activity differences between children from migrant and native origin

BACKGROUND: Children from migrant origin are at higher risk for overweight and obesity. As limited physical activity is a key factor in this overweight and obesity risk, in general, the aim of this study is to assess to what degree children from migrant and native Dutch origin differ with regard to levels of physical activity and to determine which home environment aspects contribute to these differences. METHODS: A cross-sectional survey among primary caregivers of primary school children at the age of 8–9 years old (n = 1943) from 101 primary schools in two urban areas in The Netherlands. We used bivariate correlation and multivariate regression techniques to examine the relationship between physical and social environment aspects and the child’s level of physical activity. All outcomes were reported by primary caregivers. Outcome measure was the physical activity level of the child. Main independent variables were migrant background, based on country of birth of the parents, and variables in the physical and social home environment which may enhance or restrict physical activity: the availability and the accessibility of toys and equipment, as well as sport club membership (physical environment), and both parental role modeling, and supportive parental policies (social environment). We controlled for age and sex of the child, and for socio-economic status, as indicated by educational level of the parents. RESULTS: In this sample, physical activity levels were significantly lower in migrant children, as compared to children in the native population. Less physical activity was most often seen in Turkish, Moroccan, and other non-western children (p < .05). CONCLUSIONS: Although traditional home characteristics in both the physical, and the social environment are often associated with child’s physical activity, these characteristics provided only modest explanation of the differences in physical activity between migrant and non-migrant children in this study. The question arises whether interventions aimed at overweight and obesity should have to focus on home environmental characteristics with regard to physical activity