Resolvent Estimates for Boundary Value Problems on Large Intervals Via the Theory of Discrete Approximations

In many applications such as the stability analysis of traveling waves, it is important to know the spectral properties of a linear differential operator on the whole real line. We investigate the approximation of this operator and its spectrum by finite interval boundary value problems from an abstract point of view. Under suitable assumptions on the boundary operators, we prove that the approximations converge regularly (in the sense of discrete approximations) to the all line problem, which has strong implications for the behavior of resolvents and spectra. As an application, we obtain resolvent estimates for abstract coupled hyperbolic–parabolic equations. Furthermore, we show that our results apply to the FitzHugh–Nagumo system.     

A Delay Second-Order Set-Valued Differential Equation with Hukuhara Derivatives

In this article, we study a second-order differential equation with three-point boundary conditions with the notion of Hukuhara derivatives. The existence and uniqueness of a solution is given under a Lipschitz condition on the right-hand side in the second and third variables.     

Boundary Integrals and Approximations of Harmonic Functions

Steklov expansions for a harmonic function on a rectangle are derived and studied with a view to determining an analog of the mean value theorem for harmonic functions. It is found that the value of a harmonic function at the center of a rectangle is well approximated by the mean value of the function on the boundary plus a very small number (often 3 or fewer) of specific further boundary integrals. These integrals are coefficients in the Steklov representation of the function. Similar approximations are found for the central values of solutions of Robin and Neumann boundary value problems. The results follow from analyses of the explicit expressions for the Steklov eigenvalues and eigenfunctions.     

Superconvergence of Solution Derivatives for the Shortley–Weller Difference Approximation of Poisson's Equation. II. Singularity Problems

Abstract

This is a continued analysis on superconvergence of solution derivatives for the Shortley–Weller approximation in Li (Li, Z. C., Yamamoto, T., Fang, Q. ([2003] Li, Z. C., Yamamoto, T. and Fang, Q. 2003. Superconvergence of solution derivatives for the Shortley–Weller difference approximation of Poisson's equation, Part I. Smoothness problems. J. Comp. and Appl. Math., 152(2): 307–333.  [Google Scholar]): Superconvergence of solution derivatives for the Shortley–Weller difference approximation of Poisson's equation, Part I. Smoothness problems. J. Comp. and Appl. Math. 152(2):307–333), which is to explore superconvergence for unbounded derivatives near the boundary. By using the stretching function proposed in Yamamoto (Yamamoto, T. ([2002] Yamamoto, T. 2002. Convergence of consistant and inconsistent finite difference schemes and an acceleration technique. J. Comp. Appl. Math., 140: 849–866. [Crossref], [Web of Science ®] , [Google Scholar]): Convergence of consistant and inconsistent finite difference schemes and an acceleration technique. J. Comp. Appl. Math. 140:849–866), the second order superconvergence for the solution derivatives can be established. Moreover, numerical experiments are provided to support the error analysis made. The analytical approaches in this article are non-trivial, intriguing, and different from Li, Z. C., Yamamoto, T., Fang, Q. ([2003] Li, Z. C., Yamamoto, T. and Fang, Q. 2003. Superconvergence of solution derivatives for the Shortley–Weller difference approximation of Poisson's equation, Part I. Smoothness problems. J. Comp. and Appl. Math., 152(2): 307–333.  [Google Scholar]). This article also provides the superconvergence analysis for the bilinear finite element method and the finite difference method with nine nodes.     

On the stochastic versions of Neumann and oblique derivative problems

In some areas, for instance geodesy, one finds the need of suitably defining the solution of certain boundary value problems (BVPs), for instance, the Laplace equation, where boundary data are very irregular and can be described as fields of random variables, with suitable regularity constraints (Rummel and Sansò, Lecture Notes on Earth Sciences, Vol. 65, 1997). This item has been attacked in the literature, although mainly for the case of the Dirichlet problem while much less material is available, for instance, for the Neumann and the Oblique Derivative Problem. In studying these stochastic problems in detail, the authors have found fairly general criteria which provide an automatic translation of a deterministic result into the corresponding stochastic one.     

Further calculations for Israeli options

Recently Kifer introduced the concept of an Israeli (or Game) option. That is a general American-type option with the added possibility that the writer may terminate the contract early inducing a payment not less than the holder's claim had they exercised at that moment. Kifer shows that pricing and hedging of these options reduces to evaluating a stochastic saddle point problem associated with Dynkin games. Kyprianou, A.E. (2004) "Some calculations for Israeli options", Fin. Stoch. 8, 73-86 gives two examples of perpetual Israeli options where the value function and optimal strategies may be calculated explicity. In this article, we give a third example of a perpetual Israeli option where the contingent claim is based on the integral of the price process. This time the value function is shown to be the unique solution to a (two sided) free boundary value problem on (0, ∞) which is solved by taking an appropriately rescaled linear combination of Kummer functions. The probabilistic methods we appeal to in this paper centre around the interaction between the analytic boundary conditions in the free boundary problem, Itô's formula with local time and the martingale, supermartingle and submartingale properties associated with the solution to the stochastic saddle point problem.     

MnO2/SiO2–SO3H nanocomposite as hydrogen peroxide scavenger for durability improvement in proton exchange membranes

Membrane durability was a key problem to the development of proton exchange membrane fuel cells (PEMFCs). A novel nanocomposite MnO₂/SiO₂–SO₃H was prepared to mitigate the hydrogen peroxide attack to the membranes at fuel cell condition. The nanocomposites were synthesized by the wet chemical method and three-step functionalization. The crystal structure was characterized by X-ray powder diffraction (XRD), the crystallite size and the distribution of the nanocomposites were investigated by TEM. SEM-EDX was used to analyze the elemental distribution on the surface of the nanocomposite. And the surface functional groups (–SO₃H) were evaluated by FT-IR. The amount of sulfonic acid groups introduced onto the silica surface was determined by titration method. The radical scavenging ability was estimated by UV–VIS spectroscopy using dimethyl sulfoxide (DMSO) as the trapping agent. The membrane durability was investigated via ex situ Fenton test and in situ open circuit voltage (OCV) accelerated test. In these tests, the fluoride emission rate (FER) reduced by nearly one order of magnitude with the dispersion of MnO₂/SiO₂–SO₃H nanocomposites into Nafion membrane, suggesting that MnO₂/SiO₂–SO₃H nanocomposites had a promising application to mitigate the degradation of the proton exchange membrane.     

Neumann Casimir effect: A singular boundary-interaction approach

Dirichlet boundary conditions on a surface can be imposed on a scalar field, by coupling it quadratically to a δ-like potential, the strength of which tends to infinity. Neumann conditions, on the other hand, require the introduction of an even more singular term, which renders the reflection and transmission coefficients ill-defined because of UV divergences. We present a possible procedure to tame those divergences, by introducing a minimum length scale, related to the nonzero ‘width’ of a nonlocal term. We then use this setup to reach (either exact or imperfect) Neumann conditions, by taking the appropriate limits. After defining meaningful reflection coefficients, we calculate the Casimir energies for flat parallel mirrors, presenting also the extension of the procedure to the case of arbitrary surfaces. Finally, we discuss briefly how to generalize the worldline approach to the nonlocal case, what is potentially useful in order to compute Casimir energies in theories containing nonlocal potentials; in particular, those which we use to reproduce Neumann boundary conditions.     

The three-dimensional origin of the classifying algebra

It is known that reflection coefficients for bulk fields of a rational conformal field theory in the presence of an elementary boundary condition can be obtained as representation matrices of irreducible representations of the classifying algebra, a semisimple commutative associative complex algebra.