On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution

In this paper, the solvability of some inverse problems for a nonlocal analogue of a fourth-order parabolic equation was studied. For this purpose, a nonlocal analogue of the biharmonic operator was introduced. When defining this operator, transformations of the involution type were used. In a parallelepiped, the eigenfunctions and eigenvalues of the Dirichlet type problem for a nonlocal biharmonic operator were studied. The eigenfunctions and eigenvalues for this problem were constructed explicitly and the completeness of the system of eigenfunctions was proved. Two types of inverse problems on finding a solution to the equation and its righthand side were studied. In the two problems, both of the righthand terms depending on the spatial variable and the temporal variable were obtained by using the Fourier variable separation method or reducing it to an integral equation. The theorems for the existence and uniqueness of the solution were proved

Dunkl-Poisson Equation and Related Equations in Superspace

In this paper, we investigate the Almansi expansion for solutions of Dunkl-polyharmonic equations by the 0-normalized system for the Dunkl-Laplace operator in superspace. Moreover, applying the 0-normalized system, we construct solutions to the Dunkl-Helmholtz equation, the Dunkl-Poisson equation, and the inhomogeneous Dunkl-polyharmonic equation in superspace

Differential rotation measurement of soft X-Ray corona

The aim of this paper is to study the latitudinal variation in the solar rotation in soft X-ray corona. The time series bins are formed on different latitude regions of the solar full disk (SFD) images that extend from 80 degree South to 80 degree North. These SFD images are obtained with the soft X-ray telescope (SXT) on board the Yohkoh solar observatory. The autocorrelation analyses are performed with the time series that track the SXR flux modulations in the solar corona. Then for each year, extending from 1992 to 2001, we obtain the coronal sidereal rotation rate as a function of the latitude. The present analysis from SXR radiation reveals that; (i) the equatorial rotation rate of the corona is comparable to the rotation rate of the photosphere and the chromosphere, (ii) the differential profile with respect to the latitude varies throughout the period of the study; it is more in the year 1999 and least in 1994 and (iii) the equatorial rotation period varies systematically with sunspot numbers and indicates its dependence on the phases of the solar activity cycle.Comment: 9 Pages, 4 Figures, Accepted for Publication in MNRA

Comparison of the sidereal angular velocity of subphotospheric layers and small bright coronal structures during the declining phase of solar cycle 23

Context. We compare solar differential rotation of subphotospheric layers derived from local helioseismology analysis of GONG++ dopplergrams and the one derived from tracing small bright coronal structures (SBCS) using EIT/SOHO images for the period August 2001 - December 2006, which correspond to the declining phase of solar cycle 23. Aims. The study aims to find a relationship between the rotation of the SBCS and the subphotospheric angular velocity. The northsouth asymmetries of both rotation velocity measurements are also investigated. Methods. Subphotospheric differential rotation was derived using ring-diagram analysis of GONG++ full-disk dopplergrams of 1 min cadence. The coronal rotation was derived by using an automatic method to identify and track the small bright coronal structures in EIT full-disk images of 6 hours cadence. Results. We find that the SBCS rotate faster than the considered upper subphotospheric layer (3Mm) by about 0.5 deg/day at the equator. This result joins the results of several other magnetic features (sunspots, plages, faculae, etc.) with a higher rotation than the solar plasma. The rotation rate latitudinal gradients of the SBCS and the subphotospheric layers are very similar. The SBCS motion shows an acceleration of about 0.005 deg/day/month during the declining phase of solar cycle 23, whereas the angular velocity of subsurface layers does not display any evident variation with time, except for the well known torsional oscillation pattern. Finally, both subphotospheric and coronal rotations of the southern hemisphere are predominantly larger than those of the northern hemisphere. At latitudes where the north-south asymmetry of the angular velocity increases (decreases) with activity for the SBCS, it decreases (increases) for subphotospheric layers.Comment: 6pages, 8 figures, Accepted for publication in Astronomy and Astrophysic

Solar coronal differential rotation from XBPs in Hinode/XRT and Yohkoh/SXT images

Our aim is to identify and trace the X-ray Bright Points (XBPs) over the disk and use them as tracers to determine the coronal rotation. This investigation will help to clarify and understand several issues: whether (i) the corona rotates differentially; (ii) the rotation depends on the sizes of the XBPs; and (iii) dependence on phases of the solar magnetic cycle. We analysed the daily full-disk soft X-ray images observed with (i) X-Ray Telescope (XRT) on-board the Hinode mission during January, March and April, 2007 and (ii) Soft X-ray Telescope (SXT) on-board the Yohkoh from 1992 to 2001 using SSW in IDL. We have used the tracer method to trace the passage of XBPs over the solar disk with the help of overlaying grids and derived the sidereal angular rotation velocity and the coordinates (latitude and longitude) of the XBPs. We have determined the position of a large number of XBPs both in Hinode/XRT and Yohkoh/SXT images and followed them over the solar disk as a function of time. We derived the coronal sidereal angular rotation velocity and compared it with heliocentric latitude and as a function of solar activity cycle. In addition, we measured the sizes of all the XBPs and related them with the coronal rotation. The important results derived from these investigations are: (i) the solar corona rotates differentially like the photosphere and chromosphere; (ii) the sidereal angular rotation velocity is independent of the sizes of the XBPs; (iii) the sidereal angular rotation velocity does not depend on phases of the solar magnetic cycle; and (iv) the differential rotation of the corona is present throughout the solar magnetic cycle.Comment: 9 pages, 4 figure

Periodicities in the coronal rotation and sunspot numbers

The present study is an attempt to investigate the long term variations in coronal rotation by analyzing the time series of the solar radio emission data at 2.8 GHz frequency for the period 1947 - 2009. Here, daily adjusted radio flux (known as Penticton flux) data are used. The autocorrelation analysis shows that the rotation period varies between 19.0 to 29.5 sidereal days (mean sidereal rotation period is 24.3 days). This variation in the coronal rotation period shows evidence of two components in the variation; (1) 22-years component which may be related to the solar magnetic field reversal cycle or Hale's cycle, and (3) a component which is irregular in nature, but dominates over the other components. The crosscorrelation analysis between the annual average sunspots number and the coronal rotation period also shows evidence of its correlation with the 22-years Hale's cycle. The 22-years component is found to be almost in phase with the corresponding periodicities in the variation of the sunspots number.Comment: 9 pages, 5 figures, Accepted for publication in MNRA

Differential coronal rotation using radio images at 17 GHz

In the present work, we perform time-series analysis on the latitude bins of the solar full disk (SFD) images of Nobeyama Radioheliograph (NoRH) at 17 GHz. The flux modulation method traces the passage of radio features over the solar disc and the autocorrelation analysis of the time-series data of SFD images (one per day) for the period 1999-2001 gives the rotation period as a function of latitude extending from 60 degree S to 60 degree N. The results show that the solar corona rotates less differentially than the photosphere and chromosphere, i.e., it has smaller gradient in the rotation rate.Comment: 5 pages, 5 figures, Accepted for publication in MNRAS letter