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The work in this paper concerns the study of different approximations for one-dimensional one-phase Stefan-like problems with a space-dependent latent heat. It is considered two different problems, which differ from each other in their boundary condition imposed at the fixed face: Dirichlet and Robin conditions. The approximate solutions are obtained by applying the heat balance integral method (HBIM), the modified HBIM and the refined integral method (RIM). Taking advantage of the exact analytical solutions, we compare and test the accuracy of the approximate solutions. The analysis is carried out using the dimensionless generalised Stefan number (Ste) and Biot number (Bi). It is also studied the case when Bi goes to infinity in the problem with a convective condition, recovering the approximate solutions when a temperature condition is imposed at the fixed face. Some numerical simulations are provided in order to assert which of the approximate integral methods turns out to be optimal. Moreover, we pose an approximate technique based on minimising the least-squares error, obtaining also approximate solutions for the classical Stefan problem. [ABSTRACT FROM AUTHOR]

The objectives of this retrospective observational study were to investigate the association between BCS at 21 d before calving with prepartum and postpartum DMI, energy balance (EB), and milk yield. Data from 427 multigravid cows from 11 different experiments conducted at the University of Florida (Gainesville, FL) were used. Cows were classified according to their BCS at 21 d before calving as fat (BCS ≥ 4.00; n = 83), moderate (BCS = 3.25-3.75; n = 287), and thin (BCS ≤ 3.00; n = 57). Daily DMI from -21 to -1 and from +1 to +28 DIM was individually recorded. Energy balance was calculated as the difference between net energy for lactation consumed and required. Dry matter intake in fat cows was lower than that in moderate and thin cows both prepartum (fat = 9.97 ± 0.21, moderate = 11.15 ± 0.14, and thin = 11.92 ± 0.22 kg/d) and postpartum (fat = 14.35 ± 0.49, moderate = 15.47 ± 0.38, and thin = 16.09 ± 0.47 kg/d). Dry matter intake was also lower for moderate cows compared with thin cows prepartum, but not postpartum. Energy balance in fat cows was lower than in moderate and thin cows both prepartum (fat = -4.16 ± 0.61, moderate = -1.20 ± 0.56, and thin = 0.88 ± 0.62 Mcal/d) and postpartum (fat = -12.77 ± 0.50, moderate = -10.13 ± 0.29, and thin = -6.14 ± 0.51 Mcal/d). Energy balance was also lower for moderate cows compared with thin cows both prepartum and postpartum. There was a quadratic association between BCS at 21 d before calving and milk yield. Increasing BCS from 2.5 to 3.5 was associated with an increase in daily milk yield of 6.0 kg and 28 d cumulative milk of 147 kg. Increasing BCS from 3.5 to 4.5 was associated with a decrease in daily milk yield of 4.4 kg and 28 d cumulative milk of 116 kg. In summary, a moderate BCS at 21 d before calving was associated with intermediate DMI and EB pre- and postpartum but greater milk yield compared with thinner and fatter cows. Our findings indicate that a moderate BCS is ideal for ensuring a successful lactation., (The Authors. Published by Elsevier Inc. on behalf of the American Dairy Science Association®. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).)