On weighted generalized functions associated with quadratic forms

In this article we consider certain types of weighted generalized functions associated with nondegenerate quadratic forms. Such functions and their derivatives are used for constructing fundamental solutions of iterated ultra-hyperbolic equations with Bessel operator and for constructing negative real powers of ultra-hyperbolic operators with Bessel operator.Comment: 16 page

Toroidal and poloidal energy in rotating Rayleigh-B\'enard convection

We consider rotating Rayleigh-B\'enard convection of a fluid with a Prandtl number of $Pr = 0.8$ in a cylindrical cell with an aspect ratio $\Gamma = 1/2$. Direct numerical simulations were performed for the Rayleigh number range $10^5 \leq Ra \leq 10^9$ and the inverse Rossby number range $0 \leq 1/Ro \leq 20$. We propose a method to capture regime transitions based on the decomposition of the velocity field into toroidal and poloidal parts. We identify four different regimes. First, a buoyancy dominated regime occurring as long as the toroidal energy $e_{tor}$ is not affected by rotation and remains equal to that in the non-rotating case, $e^0_{tor}$. Second, a rotation influenced regime, starting at rotation rates where $e_{tor} > e^0_{tor}$ and ending at a critical inverse Rossby number $1/Ro_{cr}$ that is determined by the balance of the toroidal and poloidal energy, $e_{tor} = e_{pol}$. Third, a rotation dominated regime, where the toroidal energy $e_{tor}$ is larger than both, $e_{pol}$ and $e^0_{tor}$. Fourth, a geostrophic turbulence regime for high rotation rates where the toroidal energy drops below the value of non-rotating convection

The general form of the Euler--Poisson--Darboux equation and application of transmutation method

In the paper we find solution representations in the compact integral form to the Cauchy problem for a general form of the Euler--Poisson--Darboux equation with Bessel operators via generalized translation and spherical mean operators for all values of the parameter $k$, including also not studying before exceptional odd negative values. We use a Hankel transform method to prove results in a unified way. Under additional conditions we prove that a distributional solution is a classical one too. A transmutation property for connected generalized spherical mean is proved and importance of applying transmutation methods for differential equations with Bessel operators is emphasized. The paper also contains a short historical introduction on differential equations with Bessel operators and a rather detailed reference list of monographs and papers on mathematical theory and applications of this class of differential equations.Comment: Electronic Journal of Differential Equations, 201