Heterogeneous Diffusion, Stability Analysis, and Solution Profiles for a MHD Darcy–Forchheimer Model

In the presented analysis, a heterogeneous diffusion is introduced to a magnetohydrodynamics (MHD) Darcy–Forchheimer flow, leading to an extended Darcy–Forchheimer model. The introduction of a generalized diffusion was proposed by Cohen and Murray to study the energy gradients in spatial structures. In addition, Peletier and Troy, on one side, and Rottschäfer and Doelman, on the other side, have introduced a general diffusion (of a fourth-order spatial derivative) to study the oscillatory patterns close the critical points induced by the reaction term. In the presented study, analytical conceptions to a proposed problem with heterogeneous diffusions are introduced. First, the existence and uniqueness of solutions are provided. Afterwards, a stability study is presented aiming to characterize the asymptotic convergent condition for oscillatory patterns. Dedicated solution profiles are explored, making use of a Hamilton–Jacobi type of equation. The existence of oscillatory patterns may induce solutions to be negative, close to the null equilibrium; hence, a precise inner region of positive solutions is obtained.

Solving the Inter-Ring Distances Optimization Problem for Pentapolar and Sextopolar Concentric Ring Electrodes Based on the Negligible Dimensions Model of the Electrode

Concentric ring electrodes are noninvasive and wearable sensors for electrophysiological measurement capable of estimating the surface Laplacian (second spatial derivative of surface potential) at each electrode. Previously, progress was made toward optimization of inter-ring distances (distances between the recording surfaces of a concentric ring electrode), maximizing the accuracy of the surface Laplacian estimate based on the negligible dimensions model of the electrode. However, this progress was limited to tripolar (number of concentric rings n equal to 2) and quadripolar (n = 3) electrode configurations only. In this study, the inter-ring distances optimization problem is solved for pentapolar (n = 4) and sextopolar (n = 5) concentric ring electrode configurations using a wide range of truncation error percentiles ranging from 1st to 25th. Obtained results also suggest consistency between all the considered concentric ring electrode configurations corresponding to n ranging from 2 to 5 that may allow estimation of optimal ranges of inter-ring distances for electrode configurations with n ≥ 6. Therefore, this study may inform future concentric ring electrode design for n ≥ 4 which is important since the accuracy of surface Laplacian estimation has been shown to increase with an increase in n.

Local existence with low regularity for the 2D compressible Euler equations

We prove the local existence, uniqueness and stability of local solutions for the Cauchy problem of two-dimensional compressible Euler equations, where the initial data of velocity, density, specific vorticity [Formula: see text] and the spatial derivative of specific vorticity [Formula: see text].

Cubic B-spline quasi-interpolation and an application to numerical solution of generalized Burgers-Huxley equation

Nonlinear partial differential equations are widely studied in Applied Mathematics and Physics. The generalized Burgers-Huxley equations play important roles in different nonlinear physics mechanisms. In this paper, we develop a kind of cubic B-spline quasi-interpolation which is used to solve Burgers-Huxley equations. Firstly, the cubic B-spline quasi-interpolation is presented. Next we get the numerical scheme by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and modified Euler scheme to approximate the time derivative of the dependent variable. Moreover, the efficiency of the proposed method is illustrated by the agreement between the numerical solution and the analytical solution which indicate the numerical scheme is quite acceptable.

Adaptive High-Order Finite Difference Analysis of 2D Quenching-Type Convection-Reaction-Diffusion Equation

Quenching characteristics based on the two-dimensional (2D) nonlinear unsteady convection-reaction-diffusion equation are creatively researched. The study develops a 2D compact finite difference scheme constructed by using the first and the second central difference operator to approximate the first-order and the second-order spatial derivative, Taylor series expansion rule, and the reminder-correction method to approximate the three-order and the four-order spatial derivative, respectively, and the forward difference scheme to discretize temporal derivative, which brings the accuracy resulted meanwhile. Influences of degenerate parameter, convection parameter, and the length of the rectangle definition domain on quenching behaviors and performances of special quenching cases are discussed and evaluated by using the proposed scheme on the adaptive grid. It is feasible for the paper to offer potential support for further research on quenching problem.

Global catalog of NOx point sources derived from the divergence of the NO2 flux

<p>Satellite observations of NO<sub>2</sub> provide valuable information on the location and strength of NO<sub>x</sub> emissions, but spatial resolution is limited by horizontal transport and smearing of temporal averages due to changing wind fields. The divergence (spatial derivative) of the mean horizontal flux, however, is highly sensitive for point sources like power plant exhaust stacks.</p><p>In a previous study, point source emissions have been identified and quantified exemplarily for Riyadh, South Africa, and Germany with a detection limit of about 0.11 kg/s down to 0.03 kg/s for ideal conditions, based on TROPOMI NO<sub>2</sub> columns and ECMWF wind fields (Beirle et al., Science Advances, 2019).</p><p>Here we extend this study and derive a global catalog of NO<sub>x</sub> emissions from point sources. The specific challenges for e.g. high latitudes (longer NO<sub>x</sub> lifetime) or coastlines (potentially persistent diurnal wind patterns) are investigated.</p>