A Relativistic Orbit Model for Temporal Properties of AGN

We present a unified model for X-ray quasi-periodic oscillations (QPOs) seen in Narrow-line Seyfert 1 (NLSy1) galaxies, γ-ray and optical band QPOs that are seen in Blazars. The origin of these QPOs is attributed to the plasma motion in corona or jets of these AGN. In the case of X-ray QPOs, we applied the general relativistic precession model for the two simultaneous QPOs seen in NLSy1 1H 0707-945 and deduce orbital parameters, such the radius of the emission region, and spin parameter a for a circular orbit; we obtained the Carter’s constant Q, a, and the radius in the case of a spherical orbit solution. In other cases where only one X-ray QPO is seen, we localized the orbital parameters for NLSy1 galaxies REJ 1034+396, 2XMM J123103.2+110648, MS 2254.9-3712, Mrk 766, and MCG-06-30-15. By applying the lighthouse model, we found that a kinematic origin of the jet based γ-ray and optical QPOs, in a relativistic MHD framework, is possible. Based on the inbuilt Hamiltonian formulation with a power-law distribution in the orbital energy of the plasma consisting of only circular or spherical trajectories, we show that the resulting Fourier power spectral density (PSD) has a break corresponding to the energy at ISCO. Further, we derive connection formulae between the slopes in the PSD and that of the energy distribution. Overall, given the preliminary but promising results of these relativistic orbit models to match the observed QPO frequencies and PSD at diverse scales in the inner corona and the jet, it motivates us to build detailed models, including a transfer function for the energy spectrum in the corona and relativistic MHD jet models for plasma flow and its polarization properties.

Sixth-Kind Chebyshev Spectral Approach for Solving Fractional Differential Equations

The basic aim of this paper is to develop new numerical algorithms for solving some linear and nonlinear fractional-order differential equations. We have developed a new type of Chebyshev polynomials, namely, Chebyshev polynomials of sixth kind. This type of polynomials is a special class of symmetric orthogonal polynomials, involving four parameters that were constructed with the aid of the extended Sturm–Liouville theorem for symmetric functions. The proposed algorithms are basically built on reducing the fractional-order differential equations with their initial/boundary conditions to systems of algebraic equations which can be efficiently solved. The new proposed algorithms are supported by a detailed study of the convergence and error analysis of the sixth-kind Chebyshev expansion. New connection formulae between Chebyshev polynomials of the second and sixth kinds were established for this study. Some examples were displayed to illustrate the efficiency of the proposed algorithms compared to other methods in literature. The proposed algorithms have provided accurate results, even using few terms of the proposed expansion.