Time-Fractional KdV Equation: Formulation and Solution using Variational Methods

In this work, the semi-inverse method has been used to derive the Lagrangian of the Korteweg-de Vries (KdV) equation. Then, the time operator of the Lagrangian of the KdV equation has been transformed into fractional domain in terms of the left-Riemann-Liouville fractional differential operator. The variational of the functional of this Lagrangian leads neatly to Euler-Lagrange equation. Via Agrawal's method, one can easily derive the time-fractional KdV equation from this Euler-Lagrange equation. Remarkably, the time-fractional term in the resulting KdV equation is obtained in Riesz fractional derivative in a direct manner. As a second step, the derived time-fractional KdV equation is solved using He's variational-iteration method. The calculations are carried out using initial condition depends on the nonlinear and dispersion coefficients of the KdV equation. We remark that more pronounced effects and deeper insight into the formation and properties of the resulting solitary wave by additionally considering the fractional order derivative beside the nonlinearity and dispersion terms.Comment: The paper has been rewritten, 12 pages, 3 figure

BIOGEOCHEMICAL SPECIALIZATION OF MACROPHY TESAND THEIR ROLE AS A BIOFILTER IN THE SELENGA DELTA

This study aims to evaluate the biofiltration ability of higher aquatic vegetation of the Selenga delta as a barrier for heavy metals and metalloids (HMM) flows into the Lake Baikal. Main aquatic vegetation species have been collected from deltaic channels and inner lakes: Nuphar pumila, Potamogeton perfoliatus, P. pectinatus, P. natans, P. friesii, Butomus umbellatus, Myriophyllum spicatum, Ceratophyllum demersum, Phragmites australis . Analysis of the obtained data showed that regardless of the place of growth hydatophytes spiked water-milfoil ( M. spicatum ) and the fennel-leaved pondweed ( P. pectinatus ) most actively accumulate metals. Opposite tendencies were found for helophytes reed (Ph. australis) and flowering rush ( B. umbellatus ), which concentrate the least amount of elements. This supports previous findings that the ability to concentrate HMM increases in the series of sur face - floating - submerged plants. Regarding river water, the studied macrophyte species are enriched with Mn and Co, regarding suspended matter - Mo, Mn and B, regarding bottom sediments - Mn, Mo and As. We identified two associations of chemical elements: S-association with the predominant suspended form of migration (Be, V, Co, Ni, W, Pb, Bi, Mn, Fe and Al) and D-association with the predominant dissolved form of migration (B, U, Mo, Cr, Cu, Zn, As, Cd, Sn and Sb). Due to these associations three groups of macrophytes were distinguished - flowering rush and reed with a low HMM content; small yellow pond-lily and common floating pondweed with a moderate accumulation of S-association and weak accumulation of D-association elements; and clasping-leaved pondweed, fennel-leaved pondweed, and pondweed Friesii accumulating elements of both S and D groups. The results suggest that macrophytes retain more than 60% of the total Mn flux that came into the delta, more than 10% - W, As, and from 3 to 10% B, Fe, Co, Mo, Cd, V, Ni, Bi, Be, Cu, Zn, Cr, U, Al. The largest contribution is made by the group of hydatophytes (spiked water-milfoil and pondweed), which account for 74 to 96% of the total mass of substances accumulated by aquatic plants

Biogeochemical specialization of macrophytes and their role as a biofilter in the selenga delta

This study aims to evaluate the biofiltration ability of higher aquatic vegetation of the Selenga delta as a barrier for heavy metals and metalloids (HMM) flows into the Lake Baikal. Main aquatic vegetation species have been collected from deltaic channels and inner lakes: Nuphar pumila, Potamogeton perfoliatus, P. pectinatus, P. natans, P. friesii, Butomus umbellatus, Myriophyllum spicatum, Ceratophyllum demersum, Phragmites australis. Analysis of the obtained data showed that regardless of the place of growth hydatophytes spiked water-milfoil (M. spicatum) and the fennel-leaved pondweed (P. pectinatus) most actively accumulate metals. Opposite tendencies were found for helophytes reed (Ph. australis) and flowering rush (B. umbellatus), which concentrate the least amount of elements. This supports previous findings that the ability to concentrate HMM increases in the series of surface – floating – submerged plants. Regarding river water, the studied macrophyte species are enriched with Mn and Co, regarding suspended matter – Mo, Mn and B, regarding bottom sediments – Mn, Mo and As. We identified two associations of chemical elements: S-association with the predominant suspended form of migration (Be, V, Co, Ni, W, Pb, Bi, Mn, Fe and Al) and D-association with the predominant dissolved form of migration (B, U, Mo, Cr, Cu, Zn, As, Cd, Sn and Sb). Due to these associations three groups of macrophytes were distinguished – flowering rush and reed with a low HMM content; small yellow pond-lily and common floating pondweed with a moderate accumulation of S-association and weak accumulation of D-association elements; and clasping-leaved pondweed, fennel-leaved pondweed, and pondweed Friesii accumulating elements of both S and D groups. The results suggest that macrophytes retain more than 60% of the total Mn flux that came into the delta, more than 10% – W, As, and from 3 to 10% B, Fe, Co, Mo, Cd, V, Ni, Bi, Be, Cu, Zn, Cr, U, Al. The largest contribution is made by the group of hydatophytes (spiked water-milfoil and pondweed), which account for 74 to 96% of the total mass of substances accumulated by aquatic plants. © 2019, Lomonosov Moscow State University. All rights reserved