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The Korteweg–de Vries (KdV) equation is an essential model to characterize shallow water waves in fluid mechanics. Here, we investigated the generalized time and time-space fractional KdV equation with fractional derivative of Riemann–Liouville. At the beginning of, we applied the fractional Lie symmetry scheme to derive their symmetry, respectively. We found that the vector fields of these considered equations decrease as the independent variables fractionalize. Subsequently, the one-parameter Lie transformation groups of these concerned models were yielded. At the same time, they can be reduced into fractional order ordinary differential equations with the Erdélyi–Kober fractional operators. Finally, by obtaining the nonlinear self-adjointness, conservation laws of the generalized time-space fractional KdV equation were also found. These good results provide a basis for us to further understand the phenomenon of shallow water waves. [ABSTRACT FROM AUTHOR]

Woody biomaterials are widely available, renewable, and environmentally friendly. The wood adsorbents are classified into five categories: woody biomass, woody activated carbon, wood ceramics, wood aerogel, and woody activated carbon fiber, according to their preparation process methods and physical properties, and according to economic costs, physical properties, and other process characteristics. Investigate the modification of wood adsorbents and their coupling with other treatment methods for removing pollutants from water and enhancing the adsorption of specific pollutants by imparting new properties to the wood adsorbents for specific pollutants. The adsorption isotherm and kinetic models are presented, focusing on the theory of model assumptions and model characteristics. Thereby, the microscopic adsorption mechanism is macroscopically represented through the fitting of adsorption data and points out the problems of some wood adsorbents in removing pollutants from water. [ABSTRACT FROM AUTHOR]

Based on the Hirota bilinear form of the generalized (2+1)-dimensional Boussinesq equation, which can be expressed as the shallow water wave mechanism appearing in fluid mechanics, we applied the new polynomial functions to construct the rational solutions and rogue wave-type solutions. Next, the system parameters control on the rational solutions and rogue wave-type solutions were also shown. As a result, we found the following basic facts: (i) these parameters may affect the wave shapes, amplitude, and bright/dark for this considered equation, (ii) the solitary wave interaction rogue waves and triplet rogue wave-type solutions can be viewed on t − y , t − x , and x − y planes, respectively. Their nonlinear dynamic behaviors were presented by numerical simulation of the 2D- and 3D-plots. [ABSTRACT FROM AUTHOR]