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The Generalized Fourier Series Solutions of the Euler-Bernoulli Beam Equation
Published in Wen L. Li, Weiming Sun, Fourier Methods in Science and Engineering, 2023
In Sec. 4.3, the beam displacement is assumed to vanish identically at each end. As a consequence, its odd-periodic extension is ensured to have C1 continuity, and its first two derivatives can be directly obtained from term-by-term differentiations. According to Theorem 2.3, the Fourier series (4.35) can be ensured to converge at the speed of λm2Am→0 as m→∞. However, due to the presence of the λm4-like terms in the coefficient matrix, it may not be guaranteed that the characteristic equation (4.54) will lead to the converged solution.
New approaches to solving the Saint-Venant equations
Published in Wim Uijttewaal, Mário J. Franca, Daniel Valero, Victor Chavarrias, Clàudia Ylla Arbós, Ralph Schielen, Alessandra Crosato, River Flow 2020, 2020
which is identical to eq. ((21)) except for the coefficient of the pressure-like terms for the faces on the LHS and the construction of the pressure-like spatial gradient that is the first term on the RHS. These terms are the analogs of splitting the I1 and I2 pressure terms in the Cunge-Liggett derivation, but are significantly simpler to implement. Equations (21) and (22) are both conservative finite-volume equations that are consistent with the differential form of eq. (12).
Modelling a Conversion of a Confined to an Unconfined Aquifer Flow with Classical and Fractional Derivatives
Published in Abdon Atangana, Mathematical Analysis of Groundwater Flow Models, 2022
Awodwa Magingi, Abdon Atangana
Addition of integration intervals and subtraction of like terms into Equation (22.10): utn+1−utn=∫tntn+1Fuτdτ
Structural differences between unannealed and expanded high-density amorphous ice based on isotope substitution neutron diffraction
Published in Molecular Physics, 2019
Katrin Amann-Winkel, Daniel T. Bowron, Thomas Loerting
The neutron scattering experiments with H/D isotopic substitution were performed at the ISIS spallation neutron source (Chilton, Didcot, UK) and follow the protocol established in earlier neutron scattering studies on amorphous ices [8]. The scattering data were collected on the SANDALS Diffractometer [33] and reduced to the interference differential scattering cross section F(Q) using the GudrunN routines [34]. These routines perform the essential background, container scattering, multiple scattering, attenuation and inelastic scattering corrections, and finally normalise the data to the scattering from a known vanadium calibration standard. For measurement, the amorphous ice samples were, under liquid nitrogen, pestled to a powder and loaded into parallel sided TiZr cells for data collection at 80 K and ambient pressure. As the samples were formed from powdered material, the data were also corrected for powder packing fraction as described in Ref. [8]. Data were analysed in the Q-range of 0.5–30 Å−1. The subsequent structural modelling of the data was performed using Empirical Potential Structure Refinement (EPSR) [35]. The whole procedure is described in detail by Bowron et al. [8]. In brief, F(Q) can be written as with Sαβ(Q) being the site-site partial structure factors (Faber-Ziman) between atoms of type α and β, where the coefficients cα/cβ and bα/bβ represent concentrations and scattering lengths for the atom types, respectively. To avoid double counting of the like terms within the summation, δαβ is the Kronecker delta function. For the case of pure water, the combination of data from three isotopically substituted samples makes it possible for us to extract the three partial structure factors that fully define the atomic pair correlations in the system, SHH(Q), SOH(Q) and SOO(Q) [36]. In all samples studied, the measured neutron diffraction patterns show no sign of Bragg peaks that would have indicated the presence of crystalline material, in accordance with the X-ray data shown in Figure 2.