Pseudo-Differential Operators in a Wave Diffraction Problem with Impedance Conditions

Mathematics Subject Classification: 35J05, 35J25, 35C15, 47H50, 47G30We consider an impedance boundary-value problem for the Helmholtz equation which models a wave diffraction problem with imperfect conductivity on a strip. Pseudo-differential operators are used to deal with this wave diffraction problem. Therefore, single and double layer potentials allow a reformulation of the problem into a system of integral equations. By using operator theoretical methods, the well-posedness of the problem is obtained for a set of impedance parameters, and in a framework of Bessel potential spaces.∗ This work is partially supported by the Portuguese Science Foundation (FCT–Fundação para a Ciência ea Tecnologia) through Unidade de Investigação Matemática e Aplicações of the University of Aveiro, Portugal. * The second author is supported by the Portuguese Science Foundation through grant number SFRH/BPD/20524/2004

Diffraction by a half-plane with different face impedances on an obstacle perpendicular to the boundary

The paper is devoted to study classes of plane wave diffraction problems by a region which involves a crack with impedance boundary conditions. Conditions on the wave number and impedance parameters are found to ensure the well-posedness of the problems in a scale of Bessel potential spaces. Under such conditions, representations of the solutions are also obtained upon the consideration of some associated operators which, in a sense, combine operators of Wiener-Hopf and Hankel type

A recent strong X-ray flaring activity of 1ES 1959+650 with possibly less efficient stochastic acceleration

We present an X-ray flaring activity of 1ES 1959+650 in 2015 August-2016 January, which was the most powerful and prolonged during the 10.75 yr period since the start of its monitoring with X-ray Telescope onboard Swift. A new highest historical 0.3-10 keV count rate was recorded three times that makes this object the third BL Lacertae source exceeding the level of 20 counts s-1. Along with the overall variability by a factor of 5.7, this epoch was characterized by fast X-ray flares by a factor of 2.0-3.1, accompanied with an extreme spectral variability. The source also shows a simultaneous flaring activity in the optical - UV and 0.3-100 GeV bands, although a fast γ-ray flare without significant optical - X-ray counterparts is also found. In contrast to the X-ray flares in the previous years, the stochastic acceleration seems be less important for the electrons responsible for producing X-ray emission during this flare that challenges the earlier suggestion that the electrons in the jets of TeV-detected BL Lacertae objects should undergo an efficient stochastic acceleration resulting in a lower X-ray spectral curvature

X-RAY FLARING ACTIVITY OF MRK 421 IN THE FIRST HALF OF 2013

We present the results of the Swift and NuSTAR observations of the nearby BL Lac object Mrk 421 during 2013 January-June. The source exhibited a strong long-term variability in the 0.3-10 keV and 3-79 keV bands with the maximum-to-minimum daily-binned flux ratios of 22 and 95, respectively, in about 3 months, mainly due to unprecedented strong X-ray outbursts by more than an order of magnitude in both bands within 2 weeks in 2013 April when the 0.3-10 keV count rate exceeded the level of 200 cts s-1 for the first time, and Mrk 421 became one of the brightest sources in the X-ray sky. The source was also very active on intra-day timescales, and it showed flux doubling and halving timescales of 1.16-7.20 hr and 1.04-3.54 hr, respectively. On some occasions, the flux varied by 4%-23% within 300-840 s. During this period, the source also exhibited some of the most extreme X-ray spectral variability ever reported for BL Lacs—the location of the synchrotron spectral energy distribution peak shifted from a few eV to ∼10 keV, and the photon index at 1 keV and curvature parameter varied on timescales from a few weeks down to intervals shorter than 1 ks. MAGIC and First G-APD Cherenkov Telescope observations also revealed a very strong very high energy (VHE) flare during April 11-17. The UV and HE γ-ray flares were much weaker compared to their X-ray counterparts, and they generally showed significantly stronger correlation with each other than with the X-ray fluxes

Swift Observations of Mrk 421 in Selected Epochs. I. The Spectral and Flux Variability in 2005–2008

We present detailed results of Swift observations of the nearby TeV-detected blazar Mrk 421, based on the rich archival data obtained during 2005 March-2008 June. The best fits of the 0.3-10 keV spectra were mainly obtained using the log-parabolic model, yielding low spectral curvatures expected in the case of the efficient stochastic acceleration of particles. During strong X-ray flares, the position of the synchrotron spectral energy distribution peak {E}{{p}} was beyond 8 keV for 41 spectra, while it sometimes was situated at the UV frequencies in quiescent states. The photon index at 1 keV exhibited a broad range, and the values a< 1.70 were observed during the strong flares, hinting at the possible presence of a jet hadronic component. The spectral parameters were correlated in some periods, expected in the framework of the first- and second-order Fermi accelerations of X-ray emitting particles, as well as in the case of turbulence spectrum. The 0.3-10 keV flux and spectral parameters sometimes showed very fast variability down to the fluctuations by 6-20% in 180-960 s, possibly related to the small-scale turbulent areas containing strongest magnetic fields. X-ray and very high-energy fluxes often showed correlated variability, although several occurrences of more complicated variability patterns are also revealed, indicating that the multifrequency emission of Mrk 421 could not be generated in a single zone

A heat conduction problem of 2D unbounded composites with imperfect contact conditions

We consider a steady-state heat conduction problem in 2D unbounded doubly periodic composite materials with temperature independent conductivities of their components. Imperfect contact conditions are assumed on the boundaries between the matrix and inclusions. By introducing complex potentials, the corresponding boundary value problem for the Laplace equation is transformed into a special R-linear boundary value problem for doubly periodic analytic functions. The method of functional equations is used for obtaining a solution. Thus, the R-linear boundary value problem is transformed into a system of functional equations which is analysed afterwards. A new improved algorithm for solving this system is proposed. It allows to compute the average property and reconstruct the temperature and the flux at an arbitrary point of the composite. Computational examples are presented

Effective conductivity of a composite material with stiff imperfect contact conditions

We present an analytic and numerical analysis of several properties of a composite material with stiff imperfect interface conditions. Spaces of functions are identified where we can guarantee existence and uniqueness of solutions. In particular, formulas for the temperature distribution and flux are exhibited. Numerical calculations of the material characteristics such as temperature, flux and the effective conductivity are also performed and interpreted