Related papers: Apery, Bessel, Calabi-Yau and Verrill
Two elementary and classical results about the Bessel quotient $y_\nu = \frac{I_{\nu+1}}{I_\nu}$ state that on the half-line $(0,\infty)$ one has for $\nu\ge -1/2$: \begin{itemize} \item[(i)] $0 < y_\nu< 1$; \item[(ii)] $y_\nu$ is strictly…
It is well known that the Bernoulli polynomials $\mathbf{B}_n(x)$ have nonintegral coefficients for $n \geq 1$. However, ten cases are known so far in which the derivative $\mathbf{B}'_n(x)$ has only integral coefficients. One may assume…
In this paper we study the Palais-Smale sequences of the conformal Dirac-Einstein problem. After we characterize the bubbling phenomena, we prove an Aubin type result leading to the existence of a positive solution. Then we show the…
A new representation for a regular solution of the radial Dirac system of a special form is obtained. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to the spectral parameter. For the…
We consider the size of the nodal set of the solution of the second order parabolic-type equation with Gevrey regular coefficients. We provide an upper bound as a function of time. The dependence agrees with a sharp upper bound when the…
In this paper, we provide suitable characterisations of pairs of weights $(V,W),$ known as Bessel pairs, that ensure the validity of weighted Hardy-type inequalities. The abstract approach adopted here makes it possible to establish such…
We establish the local well-posedness for a new nonlinearly dispersive wave equation and we show that the equation has solutions that exist for indefinite times as well as solutions which blowup in finite times. Furthermore, we derive an…
We introduce a numerical method to obtain approximate eigenvalues for some problems of Sturm-Liouville type. As an application, we consider an infinite square well in one dimension in which the mass is a function of the position. Two…
In their seminal 1983 paper, M. Maxey and J. Riley introduced an equation for the motion of a sphere through a fluid. Since this equation features the Basset history integral, the popularity of this equation has broadened the use of a…
Under different assumptions on the potential functions $b$ and $c$, we study the fractional equation $\left( I-\Delta \right)^{\alpha} u = \lambda b(x) |u|^{p-2}u+c(x)|u|^{q-2}u$ in $\mathbb{R}^N$. Our existence results are based on compact…
Building on the works of Mordell (1952) and Schinzel (2015), we prove that cubic diophantine equations $xyz=G(x,y)$ have infinitely many solutions.
In this paper we discuss some general properties of viscoelastic models defined in terms of constitutive equations involving infinitely many derivatives (of integer and fractional order). In particular, we consider as a working example the…
We prove explicit uniform two-sided bounds for the phase functions of Bessel functions and of their derivatives. As a consequence, we obtain new enclosures for the zeros of Bessel functions and their derivatives in terms of inverse values…
We consider time fractional parabolic equations in both divergence and non-divergence form when the leading coefficients $a^{ij}$ are measurable functions of $(t,x_1)$ except for $a^{11}$ which is a measurable function of either $t$ or…
It has been known since work of Lichtenstein [42] and Gunther [29] in the 1920's that the $3D$ incompressible Euler equation is locally well-posed in the class of velocity fields with H\"older continuous gradient and suitable decay at…
One dimensional Dirac equation is analysed with regard to the existence of exact (or closed-form) solutions for polynomial potentials. The notion of Liouvillian functions is used to define solvability, and it is shown that except for the…
Algorithms and underlying mathematics are presented for numerical computation with periodic functions via approximations to machine precision by trigonometric polynomials, including the solution of linear and nonlinear periodic ordinary…
The Airy integral and Bessel functions are of significant in mathematical description of spectral distribution of different types of radiation produced by relativistic charged particles moving in synchrotron and in periodical macro- and…
Spectral decomposition of dynamical equations using curl-eigenfunctions has been extensively used in fluid and plasma dynamics problems using their orthogonality and completeness properties for both linear and non-linear cases. Coefficients…
The Abel differential equations play a significant role in various fields of mathematics and applied sciences and are classified into two types: the first kind and the second kind. A novel derivative condition for the general solution of…