Related papers: On the Bessel solution of Kepler's equation
In this paper, we proceed to develop a new approach which was formulated first in Ershkov (2017) for solving Poisson equations: a new type of the solving procedure for Euler-Poisson equations (rigid body rotation over the fixed point) is…
In this text matrix Volterra integral equation of the first kind is addressed. It is assumed that kernels of the equation have jump discontinuities on non-intersecting curves. Such equations appear in the theory of evolving dynamic systems.…
We consider a Kepler problem in dimension two or three, with a time-dependent $T$-periodic perturbation. We prove that for any prescribed positive integer $N$, there exist at least $N$ periodic solutions (with period $T$) as long as the…
In the late nineteenth century, Felix Klein revived the problem of solving the quintic equation from the moribund state into which Galois had placed it. Klein's approach was a mix of algebra and geometry built on the structure of the…
We prove the existence of classical solutions to the Dirichlet problem for a class of fully nonlinear elliptic equations of curvature type on Riemannian manifolds. We also derive new second derivative boundary estimates which allows us to…
A representation for the kernel of the transmutation operator relating the perturbed Bessel equation with the unperturbed one is obtained in the form of a functional series with coefficients calculated by a recurrent integration procedure.…
The Conte-Musette method has been modified for the search of only elliptic solutions to systems of differential equations. A key idea of this a priory restriction is to simplify calculations by means of the use of a few Laurent series…
Orthogonal polynomials of several variables have a vector-valued three-term recurrence relation, much like the corresponding one-dimensional relation. This relation requires only knowledge of certain recurrence matrices, and allows simple…
In this work we develop a theory of Stieltjes-analytic functions. We first define the Stieltjes monomials and polynomials and we study them exhaustively. Then, we introduce the Stieltjes analytic functions locally, as an infinite series of…
The $Kepler$ $orbits$ form a 3-parameter family of $unparametrized$ plane curves, consisting of all conics sharing a focus at a fixed point. We study the geometry and symmetry properties of this family, as well as natural 2-parameter…
This open problem, first posed by Erd{\omicron}s, was further explored by Terence Tao. Tao work shows that the series can converge conditionally, but only under a sufficiently strong form of the Hardy-Littlewood conjecture for k-primary…
Discrepancy between periodic orbit theory and numerical calculation of a modified Kepler problem is cleared by a quantum mechanical calculation. The diagonal approximation already gives a good fit for the numerical calculation. A better…
We present several results related to statistics for elliptic curves over a finite field $\mathbb{F}_p$ as corollaries of a general theorem about averages of Euler products that we demonstrate. In this general framework, we can reprove…
We investigate a family of integrals involving modified Bessel functions that arise in the context of neutrino scattering. Recursive formulas are derived for evaluating these integrals and their asymptotic expansions are computed. We prove…
It is shown in this paper that there is a connection between the Riemann zeta-function $\zf$ and the Bessel's functions. In this direction, a new class of the nonlinear integral equations is introduced.
In this work, we investigate the one-dimensional heat equation within the framework of Stieltjes calculus. We first consider the equation associated with two fixed derivators and develop a constructive approach to establish the existence of…
The invariance of the Lagrangian under time translations and rotations in Kepler's problem yields the conservation laws related to the energy and angular momentum. Noether's theorem reveals that these same symmetries furnish generalized…
The sum of elliptic integrals simultaneously determines orbits in thr Kepler problem and the addition of divisors on elliptic curves. Periodic motion of a body in physical space is defined by symmetries, whereas periodic motion of divisors…
A strict integer Laurent polynomial in a variable $x$ is 0 or a sum of one or more terms having integer coefficients times $x$ raised to a negative integer exponent. Equations that can be transformed to certain such polynomials times…
We obtain an approximate solution $\tilde{E}=\tilde{E}(e,M)$ of Kepler's equation $E-e\sin(E)=M$ for any $e\in[0,1)$ and $M\in[0,\pi]$. Our solution is guaranteed, via Smale's $\alpha$-theory, to converge to the actual solution $E$ through…