S. Lang [L] conjectured in 1974 that a hyperbolic algebraic variety defined over a number field has only finitely many rational points, and its analogue over function fields. We discuss the Nevanlinna-Cartan theory over function fields of arbitrary dimension and apply it for Diophantine property of hyperbolic projective hypersurfaces (homogeneous Diophantine equations) constructed by Masuda-Nog...

In this paper, we investigate the system of compressible Navier-Stokes equations with hyperbolic heat conduction, i.e., replacing the Fourier’s law by Cattaneo’s law. First, by using Kawashima’s condition on general hyperbolic parabolic systems, we show that for small relaxation time τ , global smooth solution exists for small initial data. Moreover, as τ goes to zero, we obtain the uniform con...

Dispersive and Strichartz estimates for solutions to general strictly hyperbolic partial differential equations with constant coefficients are considered. The global time decay estimates of Lp−Lq norms of propagators are obtained, and it is shown how the time decay rates depend on the geometry of the problem. The frequency space is separated in several zones each giving a certain decay rate. Ge...

This paper proposes a numerical method to the two-dimensional hyperbolic equations with nonlocal integral conditions. The nonlocal integral equation is of major challenge in the frame work of the numerical solutions of PDEs. The method benefits from collocation radial basis function method, the generalized thin plate splines radial basis functions are used.Therefore, it does not require any str...

In this paper, we present a four order unconditionally stable implicit scheme for hyperbolic equations. Based on the scheme a class of parallel alternating group explicit (AGE) iterative method is derived, and convergence analysis is given. Results of numerical experiments show the iterative method can fast converge to the exact solution. Mathematics Subject Classification: 65M06, 74S20

The discrete Nahm equations, a system of matrix valued difference equations, arose in the work of Braam and Austin on half-integral mass hyperbolic monopoles. We show that the discrete Nahm equations are completely integrable in a natural sense: to any solution we can associate a spectral curve and a holomorphic linebundle over the spectral curve, such that the discrete-time DN evolution corres...

In this note we give a description of the implementation of the discontinuous Galerkin finite element discretization of the depth-averaged two-phase flow equations as formulated in Rhebergen et al. [9]. The equations are of hyperbolic type (depending, however, on the parameters required if parameters are chosen such that the equations are not hyperbolic, this code does not work). Furthermore, t...

In the following paper we will consider Navier-Stokes problem and it's interpretation by hyperbolic waves, focusing on wave propagation. We will begin with solution for linear waves, then present problem for non-linear waves. Later we will derive for numerical solution using PDE's. Also we will design a Matlab program to solve and simulate wave propagation. Keywords1: Navier-Stokes equations, F...

We present a new derivation of the equations governing the oscillations of slowly rotating relativistic stars. Previous investigations have been mostly carried out in the Regge–Wheeler gauge. However, in this gauge the process of linearizing the Einstein field equations leads to perturbation equations which as such cannot be used to perform numerical time evolutions. It is only through the tedi...