Optical combustion measurement and analysis systems using multiple sensors have received considerable attention.

We present the image segmentation model using the modified Allen？Cahn equation with a fractional Laplacian.

We present the image segmentation model using the modified Allen？Cahn equation with a fractional Laplacian. The motion of the interface for the classical Allen？Cahn equation is known as the mean curvature flows, whereas its dynamics is changed to the macroscopic limit of Levy process by replacing the Laplacian operator with the fractional one.

We analyze the effective time step size of a nonlinear convex splitting scheme for the Cahn？Hilliard (CH) equation. The convex splitting scheme is unconditionally stable, which implies we can use arbitrary large time-steps and get stable numerical solutions.

We present a compact scheme to solve the Cahn？Hilliard equation with a periodic boundary condition, which is fourth-order accurate in space. We introduce schemes for two and three dimensions, which are derived from the one-dimensional compact stencil.

The recovery of 3D left ventricle(LV) shape using 2D echocardiography is very attractable topic in the field of ultrasound imaging. In this paper, we propose a mathematical model to determine the 3D position of LV contours extracted from multiple 2D echocardiography images.

In this paper, a locally conservative, lowest order staggered discontinuous Galerkin method is developed for the Stokes equations. The proposed method allows rough grids and is based on the partition of the domain into arbitrary shapes of quadrilaterals or polygons, which makes the method highly desirable for practical applications.

A discontinuous Galerkin method with skeletal multipliers (DGSM) is developed for diffusion problem. Skeletal multiplier is introduced on the edge/face of each element through the definition of a weak divergence and a weak derivative in the method.

During the winter of 2016-2017, an epidemic of highly pathogenic avian influenza (HPAI) ledto high mortality in poultry and put a serious burden on the poultry industry of the Republic of Korea. Effective control measures considering spatial heterogeneity to mitigate the HPAI epidemic is still a challenging issue. Here we develop a spatial-temporal compartmental model that incorporates the culling rate as a function of the reported farms and farm density in each town. The epidemiological and geographical data of two species, chickens and ducks, from the farms in the sixteen towns in Eumseong-gun and Jincheon-gun are used to find the best-fitted parameters of the metapopulation model. The best culling radius to maximize the final size of the susceptible farms and minimize the total number of culled farms is calculated from the model.

For a nondegenerate finite subscheme Γ in Pc？, let reg(Γ) and ℓ(Γ) be, respectively, the regularity of Γ and the largest integer ℓ such that there exists an ℓ-secant line to Γ？.