Convergence analysis of two finite element methods for the modified Maxwell's Steklov eigenvalue problem

The modified Maxwell's Steklov eigenvalue problem is a new problem arising from the study of inverse electromagnetic scattering problems. In this paper, we investigate two finite element methods for this problem and perform the convergence analysis. Moreover, the monotonic convergence of the discrete eigenvalues computed by one of the methods is analyzed.

On Modified Second Paine–de Hoog–Anderssen Boundary Value Problem

This article deals with a special case of the Sturm–Liouville boundary value problem (BVP), an eigenvalue problem characterized by the Sturm–Liouville differential operator with unknown spectra and the associated eigenfunctions. By examining the BVP in the Schrödinger form, we are interested in the problem where the corresponding invariant function takes the form of a reciprocal quadratic form. We call this BVP the modified second Paine–de Hoog–Anderssen (PdHA) problem. We estimate the lowest-order eigenvalue without solving the eigenvalue problem but by utilizing the localized landscape and effective potential functions instead. While for particular combinations of parameter values that the spectrum estimates exhibit a poor quality, the outcomes are generally acceptable although they overestimate the numerical computations. Qualitatively, the eigenvalue estimate is strikingly excellent, and the proposal can be adopted to other BVPs.

Threshold Dynamics of a Diffusive Herpes Model Incorporating Fixed Relapse Period in a Spatial Heterogeneous Environment

In this paper, we aim to establish the threshold-type dynamics of a diffusive herpes model that assumes a fixed relapse period and nonlinear recovery rate. It turns out that when considering diseases with a fixed relapse period, the diffusion of recovered individuals will lead to nonlocal recovery term. We characterize the basic reproduction number, ℜ 0 , for the model through the next generation operator approach. Moreover, in a homogeneous case, we calculate the ℜ 0 explicitly. By utilizing the principal eigenvalue of the associated eigenvalue problem or equivalently by ℜ 0 , we establish the threshold-type dynamics of the model in the sense that the herpes is either extinct or close to the epidemic value. Numerical simulations are performed to verify the theoretical results and the effects of the spatial heterogeneity on disease transmission.