GENERAL BOUNDARY ELEMENT METHOD FOR NON-LINEAR PROBLEMS

In this paper the well-known non-linear equation f‴+½ff″=0 with boundary conditions f(0)=0, f′(0)=0 and f (∞)=1 is used as an example to describe the basic ideas of a kind of general boundary element method for non-linear problems whose governing equations and boundary conditions may not contain any linear terms at all.     

POPULATION EXTINCTION IN DETERMINISTICAND STOCHASTIC DISCRETE‐TIME EPIDEMIC MODELS WITH PERIODIC COEFFICIENTS WITH APPLICATIONS TO AMPHIBIAN POPULATIONS

ABSTRACT. Discrete‐time deterministic and stochastic epidemic models are formulated for the spread of disease in a structured host population. The models have applications to a fungal pathogen affecting amphibian populations. The host population is structured according to two developmental stages, juveniles and adults. The juvenile stage is a post‐metamorphic, nonreproductive stage, whereas the adult stage is reproductive. Each developmental stage is further subdivided according to disease status, either susceptible or infected. There is no recovery from disease. Each year is divided into a fixed number of periods, the first period represents a time of births and the remaining time periods there are no births, only survival within a stage, transition to another stage or transmission of infection. Conditions are derived for population extinction and for local stability of the disease‐free equilibrium and the endemic equilibrium. It is shown that high transmission rates can destabilize the disease‐free equilibrium and low survival probabilities can lead to population extinction. Numerical simulations illustrate the dynamics of the deterministic and stochastic models.     

The asymptotic solution of a family of boundary value problems involving exponentially small terms

In this paper, the authors consider the family of boundary valueproblems in the limit || 0. This problem has recently appeared as a modelfor magnetic field annihilation but the equation itself, withvariously different boundary conditions, has an extensive literature.Using a combination of asymptotic and numerical analyses, thepaper gives a comprehensive treatment of the small || problem,paying particular attention to the question of duality of solutions.For |0, this is intimately connected with the occurrence ofexponentially small terms in the asymptotic solution. When =0(1) these termsz are forced by the boundary layer at y = 1,and the techniques used to deal with this case are well knownfrom previous work on the equation. However, for small ||, acase which reveals the true nature of the duality propertiesof the asymptotic solution, these well-known methods are notapplicable, and a new approach via the initial value formulationof (*) is used. The approach is based on a scaling method whichenables the problem to be reduced to a one-parameter familyof problems of initial value type. This considerably simplifiesthe search for and construction of numerical solutions thatare used to support the asymptotic analysis. For 0, it is shownthat convergence to the =0 solution only takes place for a restrictedrange of values of a and that, for sufficiently small || thereis only one solution to the given boundary value problem.     

The Relaxed Energy Density for Isotropic Elastic Membranes

Tension field theory can be incorporated into the ordinary theoryof elastic membranes merely by replacing the strain energy densityby a suitable relaxed energy density W. When W is used as theenergy density, no compressive stress can appear in the solutionof a membrane problem, and in certain regions of strain spacethe stress is a uniaxial tension. The properties of W implya number of other interesting features of the stress-strainrelations.     

Porous Medium Flow with Both a Fractional Potential Pressure and Fractional Time Derivative

The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator.The derivative in time is also fractional and is of Caputo-type, which takes into account"memory". The precise model isD_t~αu- div(u(-Δ)~(-σ)u) = f, 0 σ 1/2.This paper poses the problem over {t ∈ R~+, x ∈ R~n} with nonnegative initial data u(0, x) ≥0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x)have exponential decay at infinity is proved. The main result is H¨older continuity for such weak solutions.     

Polarizabilities and excitation energies from the multiplicative Kohn-Sham (MKS) approach

The conventional Kohn-Sham expressions for the static isotropic polarizability and vertical excitation energy are evaluated using Kohn-Sham orbitals and eigenvalues determined directly from theoretical electron densities. For a series of small molecules, polarizabilities determined from wavefunction-based BD(T) electron densities differ, on average, by about 2% from conventional BD(T) polarizabilities when the LDA exchange-correlation integrand is used in the electric Hessian matrix. Also polarizabilities determined from Kohn-Sham densities, using the B97-2 hybrid functional, are close to the conventional B97-2 values. To quantify the dependence on the choice of exchange-correlation integrand, calculations are performed also using the HCTH integrand. Rydberg excitation energies determined from BD(T) and B97-2 densities exhibit errors due to inaccuracies in the asymptotic exchange-correlation potential. This category of excitation energy is relatively insensitive to the choice of exchange-correlation integrand; valence excitations can be more sensitive.