In this paper hyperbolic partial differential equations with random coefficients are discussed. We consider the challenging problem of flux functions with coefficients modeled by spatiotemporal random fields. Those fields are given by correlated Gaussian random fields in space and Ornstein–Uhlenbeck processes in time. The resulting system of equations consists of a stochastic differential equat...

The initial data in the polygon approach to (2+1)D gravity coupled to point particles are constrained by the vertex equations and the particle equations. We establish the hyperbolic nature of the vertex equations and derive some consequences. In particular we are able to identify the hyperbolic group of motions as discrete analogues of the diffeomorphisms in the continuum theory. We show that p...

In this work we apply an extended hyperbolic function method to solve the nonlinear family of third order Korteweg de-Vries (KdV) equations, namely, the KdV equation, the modified KdV (mKdV) equation, the potential KdV (pKdV) equation, the generalized KdV (gKdV) equation and gKdV with two power nonlinearities equation. New exact travelling wave solutions are obtained for the KdV, mKdV and pKdV ...

The paper is a part of an ongoing program which aims to show that the existential theory in free groups (hyperbolic groups or even toral relatively hyperbolic) is NP-complete. For that we study compression of solutions with straight-line programs (SLPs) as suggested originally by Plandowski and Rytter in the context of a single word equation. We review some basic results on SLPs and give full p...

Discontinuous Galerkin (DG) methods exhibit ”hidden accuracy” that makes the superconvergence of this method an increasing popular topic to address. Previous work has implemented a convolution kernel approach that allows us to improve the order of accuracy from k+1 to order 2k+m for time-dependent linear convection-diffusion equations, where k is the highest degree polynomial used in the approx...

It is proved that the Dirichlet problem is correct in the characteristic rectangle Dab = [0, a] × [0, b] for the linear hyperbolic equation ∂4u ∂x2∂y2 = p0(x, y)u + p1(x, y) ∂u ∂x + p2(x, y) ∂u ∂y + +p3(x, y) ∂2u ∂x∂y + q(x, y) with the summable in Dab coefficients p0, p1, p2, p3 and q if and only if the corresponding homogeneous problem has only the trivial solution. The effective and optimal ...

Cotton-type invariants for a subclass of a system of two linear elliptic equations, obtainable from a complex base linear elliptic equation, are derived both by spliting of the corresponding complex Cotton invariants of the base complex equation and from the Laplace-type invariants of the system of linear hyperbolic equations equivalent to the system of linear elliptic equations via linear comp...

A new development of the “monodromy transform” method for analysis of hyperbolic as well as elliptic integrable reductions of Einstein equations is presented. Compatibility conditions for some alternative representations of the fundamental solutions of associated linear systems with spectral parameter in terms of a pair of dressing (“scattering”) matrices give rise to a new set of linear (quasi...

The gluing equations of a cusped hyperbolic 3-manifold M are a system of polynomial equations in the shapes of an ideal triangulation T of M that describe the complete hyperbolic structure of M and its deformations. Given a Neumann-Zagier datum (comprising the shapes together with the gluing equations in a particular canonical form) we define a formal power series with coefficients in the invar...

flood routing has many applications in engineering projects and helps designers in understanding the flood flow characteristics in river flows. floods are taken unsteady flows that vary by time and location. equations governing unsteady flows in waterways are continuity and momentum equations which in case of one-dimensional flow the saint-venant hypothesis is considered. dynamic wave model as ...