Sentence examples for most analytical solutions from inspiring English sources
Most analytical solutions to engineering or geological problems are limited to simple geometries.
Most analytical solutions obtained for fractional differential problems are given in terms of special functions, which make numerical evaluation difficult and almost impossible.
Most analytical solutions to image forces could be found in literatures for two-dimensional (2D) solids with or without the consideration of surface stress.
Most analytical solutions are based on the assumption that the ground is isotropic, which may not be realistic, as e.g. depositional processes may result in engineering properties that depend on the direction of deposition.
However, due to the problems for which the solutions are sought which are complex in most of the cases, most available analytical solutions in multi-dimensional media are not given in their closed forms.
Indeed, most of analytical solutions contain infinite series and special functions, and thus are complicated and inconvenient for computational and numerical situations.
Most previous analytical solutions are based on integral transform techniques, meaning that the initial condition is restricted in the sense that the choice of initial condition has an important impact on whether or not the inverse transform can be calculated exactly.
We have added this discussion at the end of section A of the Results, on the most favourable analytical solution: "The discrepancy between the configurations observed and the symmetric solutions additional layers would just propagate further this situation without leading to the appearance of FCC and HCP mixtures".
4) Results section, subsection headed "Which is the most favorable analytical solution?": The authors focus on HCP and FCC, but it is unclear here to which degree do they see also other solutions, besides HCP and FCC, e.g. random order of layers: ABCABABACBCBA… which is an arrangement that pack 3D space just as well as FCC or HCP but does not have any large-scale structure along the z-axis.
The proposed analytical solution is a generalization of most of existing analytical solutions.
However, it is difficult to derive the analytical solutions to most of the fractional equations.
