Analysis of Water Transport inside a Plant Xylem Vessel with Pitted Thickening

In this article,computational fluid dynamics(CFD)are used to explore the dynamics of water transport inside the pitted thickening of a plant xylem vessel.A pitted thickening model combined with the Bernoulli equation is used to analyze the influence of various factors(namely,the inner diameter,thickening width,thickening height,thickening spacing,number of laps and adjacent pit axial rotation).The pressure drop and the flow resistance coefficient are the variable parameters for our analysis.The results show that these two parameters are proportional to the thickening height and thickening width,and inversely proportional to the inner diameter,thickening spacing and number of laps.Three different wall thickening structures of the vessel are compared and the pitted thickening vessel is shown to provide the largest structural flow resistance,the annular thickening vessel has the second largest resistance and the helical thickening vessel corresponds to the smallest resistance of the three structures.

On limit fractional Volterra hierarchies

For the limit fractional Volterra(LFV)hierarchy,we construct the n-fold Darboux transformation and the soliton solutions.Furthermore,we extend the LFV hierarchy to the noncommutative LFV(NCLFV)hierarchy,and construct the Darboux transformation expressed by quasi determinant of the noncommutative version.Meanwhile,we establish the relationship between new and old solutions of the NCLFV hierarchy.Finally,the quasi determinant solutions of the NCLFV hierarchy are obtained.

A Novel Low-Dimensional Method for Analytically Solving Partial Differential Equations

This paper is concerned with a low-dimensional dynamical system model for analytically solving partial differential equations(PDEs).The model proposed is based on a posterior optimal truncated weighted residue(POT-WR)method,by which an infinite dimensional PDE is optimally truncated and analytically solved in required condition of accuracy.To end that,a POT-WR condition for PDE under consideration is used as a dynamically optimal control criterion with the solving process.A set of bases needs to be constructed without any reference database in order to establish a space to describe low-dimensional dynamical system that is required.The Lagrangian multiplier is introduced to release the constraints due to the Galerkin projection,and a penalty function is also employed to remove the orthogonal constraints.According to the extreme principle,a set of ordinary differential equations is thus obtained by taking the variational operation of the generalized optimal function.A conjugate gradient algorithm by FORTRAN code is developed to solve the ordinary differential equations.The two examples of one-dimensional heat transfer equation and nonlinear Burgers’equation show that the analytical results on the method proposed are good agreement with the numerical simulations and analytical solutions in references,and the dominant characteristics of the dynamics are well captured in case of few bases used only.