... the fictitious FD points x₋₁, x_N+2. One approach to doing this is to follow on from discretizing the Sturm–Liouville ODEs by discretizing the differential operators themselves, i.e., approximate the first- and second-order differential operators present...

... approximation for the FD case and knot sequence, order of B-splines). At the point at which the spatial derivatives in the partial-wave grid method have been approximated explicitly, the time-propagation algorithm is common for both the spectral and grid representations; it is the matrix structure of H...

... G⁰ is the Hartree Green function. As usual, for real systems, these formally exact equations are not solvable in practice but they provide a convenient starting point for approximations. The GWA corresponds to neglecting the vertex δΣ/δG in Eq. ( 4.1 ), keeping only the linear term...

...-loop control of proppant bank height while minimizing uncertainty in the measurements ( Siddhamshetty and Kwon, 2019 ). Apart from being used in MPC design, the MOESP technique has also been used in the design of Approximate dynamic programming (ADP) based controller for the hydraulic fracturing...

...., 2022 ). To handle plant-model mismatch and further improve the approximation of nonlinear dynamics in KLMPC, offset-free MPC is combined with KLMPC ( Son et al., 2020 , 2021 , 2022 ). Among regression-based modeling approaches, sparse regression is receiving growing attention in modeling...

... , P. D. , “ Economic model predictive control of nonlinear process systems using lyapunov techniques ,” AIChE. J.   58 , 855 – 870 ( 2012 ). 10.1002/aic.v58.3 Huang , Z. , Liu , Q. , Liu , J. , and Huang , B. , “ A comparative study of model approximation methods applied...

... the diameter of dots. The calculated energy bandgap values of indirect (Γ→L) and (Γ→X) and direct (Γ→Γ) directions using Engel–Vosko generalized gradient approximation (EVGGA) for the investigated Si at varied diameters of dots are displayed in Table 6.1 , corresponding to other experimental results ( Zhang...

... curve ( Baihly et al., 2010 ); and (4) the production peak is independent of the timing of the well-restimulation operation ( Tavassoli et al., 2013 ; and Cafaro et al., 2016 ). Sample average approximation We are given a set of sampled wells and each well has known initial...

... (or a working group). It might also be important to stress that even if there is in a certain sense an objective mathematical description of the world, this can mostly only be an approximation of the world because of unavoidable idealizations ( Quale, 2011 ). Communication Mathematics has its own signs...

... or He⁺) with one electron. In case of many-electron systems, the presence of electron correlations forbids an exact solution. Perturbative treatments and other approximate methods such as the Hartree–Fock approach that reduce the many-body problem into an effective one-electron problem...

.... According to the Teaching and Learning International Survey (TALIS) ( OECD, 2020 ), which included 260 000 teachers in 15 000 schools across 48 countries and economies worldwide, approximately 90% of teachers (mainly lower-second level) experienced all three components in their initial teacher education...

.... , and Ernzerhof , M. , “ Generalized gradient approximation made simple ,” Phys. Rev. Lett.   77 , 3865 – 3868 ( 1996 ). 10.1103/PhysRevLett.77.3865 Rached , D. , Rabah , M. , Benkhettou , N. , Driz , M. , and Soudini , B. , “ Calculated band structures and optical properties...

... signaling pathway through correction term estimation and neural network modeling ,” PLoS Comput. Biol.   16 ( 12 ), e1008472 ( 2020 ). 10.1371/journal.pcbi.1008472 Leo , T. Y. , Kravaris , C. , and Seinfeld , J. H. , “ History matching by spline approximation and regularization...

... source with high potential compared to fossil fuels such as coal, oil, and conventional reserves of natural gas. The global quantity of technically recoverable natural gas from hydrate sediments is estimated to be approximately 3 × 10¹³ m³ ( Koh et al., 2015 ). Although...

...: a “ball like” movement of “sound molecules,” whereby molecules fly in approximately straight lines from the source to the receiver, and a “jerky” two steps forward one step back movement of air molecules. They also identified a depiction of the same idea on the macroscopic level, whereby sound is like...

... system grid side and the PV system side need to be satisfied to ensure the safety of the PV installer and the reliability of the utility grid ( Eltawil and Zhao, 2010 ). The design optimization of PV systems can offer the most approximate production scheme, system configuration/location, size...

... of thermodynamics.” Other students had difficulties with the concept of efficiency and the constraints it places on devices. Approximately 15% of students answered that the thermodynamic efficiency simply prohibits engines that are 100% efficient. Others placed an arbitrary limit on the efficiency without referring...

... simple equation, such as Colebrook's one, can be precious for students to increase their computational skills. Colebrook's equation is implicit in the flow friction factor, so it is necessary to solve it by using iteratively or explicit approximations; these, in turn, need to be developed using different...

... ( 7.3 ) is unique. Once the optimal steady-state is obtained, it is passed to the lower-layer MPC. MPC is an online optimization-based control algorithm. It approximates the solution of a constrained infinite-time optimal control problem by solving a finite-time optimization problem at every sampling...