In this paper, a direct rigorous mathematical proof of the Gregory–Laflamme instability for five-dimensional Schwarzschild black string is presented. Under choice ansatz perturbation and gauge choice, linearized vacuum Einstein equation reduces to an ordinary differential (ODE) problem single function. work, suitable rescaling change variables applied, which casts ODE into Schrödinger eigenvalu...

This paper is concerned with the geometric structure of transmission eigenvalue problem associated a general conductive condition. We prove that under mild regularity condition in terms Herglotz approximations one pair eigenfunctions, eigenfunctions must be vanishing around corner on boundary. The approximation Fourier extension eigenfunction, and growth rate density function can used to charac...

In a very interesting Letter, Alt et al. [1] have analyzed measurements on resonance widths of different eigen-modes of a superconducting cavity shaped like a stadium billiard, and have found agreement with a universal Porter-Thomas distribution (PT) [2]. However, one expects that the distribution in a stadium billiard is nonuniversal and should not obey Porter-Thomas due to the presence of bou...

We study the asymptotic behavior for nonlocal diffusion models of the form ut = J ∗ u − u in the whole R or in a bounded smooth domain with Dirichlet or Neumann boundary conditions. In R we obtain that the long time behavior of the solutions is determined by the behavior of the Fourier transform of J near the origin, which is linked to the behavior of J at infinity. If Ĵ(ξ) = 1 − A|ξ| + o(|ξ|) ...

Abstract The hot spots conjecture is only known to be true for special geometries. This paper shows numerically that the can fail easy construct bounded domains with one hole. underlying eigenvalue problem Laplace equation Neumann boundary condition solved integral equations yielding a non-linear problem. Its discretization via element collocation method in combination algorithm by Beyn yields ...

‎Radial basis functions (RBFs) are a powerful tool for approximating the solution of high-dimensional problems‎. ‎They are often referred to as a meshfree method and can be spectrally accurate‎. ‎In this paper, we analyze a new stable method for evaluating Gaussian radial basis function interpolants based on the eigenfunction expansion‎. ‎We develop our approach in two-dimensional spaces for so...

Structured epidemic models can be formulated as first-order hyperbolic PDEs, where the “spatial” variables represent individual traits, called structures. For with two structures, we propose a numerical technique to approximate R0, which measures transmissibility of an infectious disease and, rigorously, is defined dominant eigenvalue next generation operator. Via bivariate collocation and cuba...

where P is a real even polynomial with positive leading coefficient, which is called a potential. The boundary condition is equivalent to y ∈ L(R) in this case. It is well-known that the spectrum is discrete, and all eigenvalues λ are real and simple, see, for example [3, 14]. The spectrum can be arranged in an increasing sequence λ0 < λ1 < . . .. Eigenfunctions y are real entire functions of o...

For compact self-adjoint operators in Hilbert spaces, two algorithms are proposed to provide fully computable a posteriori error estimate for eigenfunction approximation. Both apply well the case of tight clusters and multiple eigenvalues, under settings target eigenvalue problems. Algorithm I is based on Rayleigh quotient min-max principle that characterizes The formula provided by easy comput...

Abstract In this paper, we are concerned with the eigenvalue gap and ratio of Dirichlet conformable fractional Sturm–Liouville problems. We show that kind differential equation satisfies property by Prüfer substitution. That is, n th eigenfunction has $n-1$ n − 1 zero in $( 0...