Related papers: Weighted spherical means generated by generalized …
The generalized Euler-Poisson-Darboux (EPD) equation with complex parameter $\alpha$ is given by $$ \Delta_x u=\frac{\partial^2 u}{\partial t^2}+\frac{n-1+2\alpha}{t}\,\frac{\partial u}{\partial t}, $$ where $u(x,t)\in \mathscr E(\mathbb…
Asgeirsson's theorem establishes a mean value property for solutions of the ultrahyperbolic equation. In the case of four variables, it states that the integrals of a solution over certain pairs of conjugate circles are the same. In this…
We show how to define and go from the spin-s spherical harmonics to the tensorial spin-s harmonics. These quantities, which are functions on the sphere taking values as Euclidean tensors, turn out to be extremely useful for many…
Various superpositions of Bessel-Gaussian beams and modified Bessel Gaussian beams are considered. Two selected parameters characterizing these beams, with respect to which the superpositions are constructed, are the topological index $n$…
A kind of generalized Gelfand pair is introduced via a Banach algebra consisting of bi-invariant functions in a weighted Lebesgue space. The related spherical functions and the Fourier transformation are constructed. The multipliers of the…
A gravitational potential has the spherical property when the field outside any uniform spherical shell is indistinguishable from that of a point mass at the center. We present the general potentials that possess this property on constant…
Generalized integral formulas involving the generalized Bessel-Maitland function are considered and it expressed in terms of generalized Wright hypergeometric functions. By assuming appropriate values of the parameters in the main results,…
In Einstein's theory of relativity, the interaction of two collinearly polarized plane gravitational waves can be described by a Goursat problem for the Euler--Darboux equation in a triangular domain. In this paper, using a representation…
A surface that is the pointwise sum of circles in Euclidean space is either coplanar or contains no more than 2 circles through a general point. A surface that is the pointwise product of circles in the unit-quaternions contains either 2,…
In this paper a nonlinear Euler-Poisson-Darboux system is considered. In a first part, we proved the genericity of the hypergeometric functions in the development of exact solutions for such a system in some special cases leading to Bessel…
We provide hyperbolic analogues of some classical theorems in spherical geometry due to Menelaus, Euler, Lexell, Ceva and Lambert. Some of the spherical results are also made more precise.
The solution in hyperspherical coordinates for $N$ dimensions is given for a general class of partial differential equations of mathematical physics including the Laplace, wave, heat and Helmholtz, Schr\"{o}dinger, Klein-Gordon and…
The Bargmann-Wigner formalism is adapted to spherical surfaces embedded in three to eleven dimensions. This is demonstrated to generate wave equations in spherical space for a variety of antisymmetric tensor fields. Some of these equations…
The general solution of M\o ller's field equations in case of spherical symmetry is derived. The previously obtained solutions are verified as special cases of the general solution.
In this note, we prove that the solutions obtained to the spherically symmetric Euler equations in the recent works [2, 3] are weak solutions of the multi-dimensional compressible Euler equations. This follows from new uniform estimates…
Here, Darboux's classical results about transformations with differential substitutions for hyperbolic equations are extended to the case of parabolic equations of the form $L u = \big(D^2_{x} + a(x,y) D_x + b(x,y) D_y + c(x,y)\big)u=0$. We…
A higher order theory of gravitation is considered which is obtained by modifying Einstein field equations. The Lagrange used to modify this in the form a polynomial in (scalar curvature) R. In this equation we have studied spherical…
Solutions of nonlinear multi-component Euler-Monge partial differential equations are constructed in n spatial dimensions by dimension-doubling, a method that completely linearizes the problem. Nonlocal structures are an essential feature…
In this work we show how to get advantage from the Riemann--Hilbert analysis in order to obtain first and second order differential equations for the orthogonal polynomials and associated functions with a weight on the unit circle. We…
This paper is dedicated to the analysis and detailed study of a procedure to generate both the weighted arithmetic and harmonic means of $n$ positive real numbers. Together with this interpretation, we prove some relevant properties that…