In these notes we discuss obstructions to the existence of local invariant manifolds in some smoothness class, near hyperbolic fixed points of diffeomorphisms. We present an elementary construction for continuously differentiable invariant manifolds, that are not necessarily normally hyperbolic, near attracting fixed points. The analogous theory for invariant manifolds near hyperbolic equilbria...

We study conjugacy classes of solutions to systems of equations and inequations over torsion-free hyperbolic groups, and describe an algorithm to recognize whether or not there are finitely many conjugacy classes of solutions to such a system. The class of immutable subgroups of hyperbolic groups is introduced, which is fundamental to the study of equations in this context. We apply our results...

First-order hyperbolic systems are promising as a basis for numerical integration of Einstein’s equations. In previous work, the lapse and shift have typically not been considered part of the hyperbolic system and have been prescribed independently. This can be expensive computationally, especially if the prescription involves solving elliptic equations. Therefore, including the lapse and shift...

We introduce and study the Dunkl symmetric systems. We prove the wellposedness results for the Cauchy problem for these systems. Eventually we describe the finite speed of it. Next the semi-linear Dunkl-wave equations are also studied.

We consider di erent situations of blowup in sup norm for hyper bolic equations For scalar conservation laws with a source the asymptotic pro le of the solution close to a blowup point is described in detail Based on an example of Je rey we next show how blowup for ordinary di erential equations can be used to construct examples of blowup for systems of hy perbolic equations Finally we outline ...

This paper deals with the analysis of the asymptotic limit towards the derivation of hyperbolic macroscopic equations for a class of equations modeling complex multicellular systems. Cellular interactions generate both modification of biological functions and proliferating destructive events related to growth of tumor cells in competition with the immune system. The asymptotic analysis refers t...

In this paper, the Chebyshev spectral collocation method(CSCM) for one-dimensional linear hyperbolic telegraph equation is presented. Chebyshev spectral collocation method have become very useful in providing highly accurate solutions to partial differential equations. A straightforward implementation of these methods involves the use of spectral differentiation matrices. Firstly, we transform ...

Evolution equations associated with irreversible physical processes like diffusion and heat conduction lead to parabolic partial differential equations. When the equation is a model for a reversible physical process like propagation of acoustic or electromagnetic waves, then the evolution equation is generally hyperbolic. The mathematical models usually begin with a conservation statement that ...

Thermoelastic damping (TED) in a circular micro-plate resonator subjected to an electrostatic pressure is studied. The coupled thermo-elastic equations of a capacitive circular micro plate are derived considering hyperbolic heat conduction model and solved by applying Galerkin discretization method. Applying complex-frequency approach to the coupled thermo-elastic equations, TED is obtained for...

In 1773 Laplace obtained two fundamental semi-invariants, called Laplace invariants, for scalar linear hyperbolic partial differential equations PDEs in two independent variables. He utilized this in his integration theory for such equations. Recently, Tsaousi and Sophocleous studied semiinvariants for systems of two linear hyperbolic PDEs in two independent variables. Separately, by splitting ...