A Variational Formulation for Anisotropic Wave Traveling in a Porous Medium

Abstract

An anisotropic wave in a porous medium is a hot topic in the coastal protection. A fractal derivative model is established, and a variational principle is established for the anisotropic wave traveling. The variational principle reveals an energy conservation law during the traveling process.

Eigenvalues of non‐Hermitian matrices: A dynamical and an iterative approach—Application to a truncated Swanson model

“…This is the case, for instance, when the real parts (respectively, the norms) of the eigenvalues coincide; see Section 2.1 (respectively, Section 3). We are planning to consider a possible extension of the dynamical approach adopting (or extending) the fractal variational principle already used in literature for a quite wide class of applications (previous studies [23][24][25] ). Also, we are interested in extending our ideas to infinite-dimensional matrices and to compare our results with those in Erickson et al 26 This is particularly interesting for us, in view of our interest for solving Schrödinger equations for concrete systems living in infinite-dimensional Hilbert spaces.…”

New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle

“…Before we establish a variational formulation for the previous problem in a fractal space, we give the following theorem [96][97][98][99].…”

Fractal diffusion of silver ions in hollow cylinders with unsmooth inner surface

“…This paradox can be solved using the twoscale thermodynamics. 20,21 Actually the inner surface is not smooth enough (see Figure 1), so a continuum model with smooth boundary on a large scale leads to a wrong result; however, if we observe the problem on a smaller scale so that the unsmooth inner surface can be measured, the paradox can be completely solved by the fractal calculus, 22,23 which is to study various phenomena in discontinuous space, and has widely applied in electrochemistry, 24 biomechanics, 25,26 Tsunami model, 27 wool fiber, 28 thermal insulation, 29 fractal solitary wave, 30 and fractal convection-diffusion model. 31 As shown in Figure 1, the inner surface of the hollow fiber is not smooth, and the inner boundary of the cross section is a coastline-like curve with fractal dimensions larger than 1, so the fractal calculus has to be adopted in our study.…”