BSSN-type evolution equations are discussed. The name refers to the Baumgarte, Shapiro, Shibata, and Nakamura version of the Einstein evolution equations, without introducing the conformal-traceless decomposition but keeping the three connection functions and including a densitized lapse. It is proved that a pseudodifferential first order reduction of these equations is strongly hyperbolic. In ...

In this paper, a nonconforming mixed finite element method is studied for semilinear pseudo-hyperbolic partial integrodifferential equations. By use of the interpolation technique instead of the generalized elliptic projection, the optimal error estimates of the corresponding unknown function are given. Keywords—Pseudo-hyperbolic partial integro-differential equations; Nonconforming mixed eleme...

This paper discusses hyperbolic behavior of Jacobi fields along billiard flows on multidimensional Reimannian manifolds. A class of generalized differential operators associated with the impulsive equations (generalized Jacobi equations) are defined using a new Radon measure. We investigate the hyperbolic behavior of functions in the null-space of the operator by applying operator theory.

We develop the error analysis for the hp-version of a discontinuous nite element approximation to second-order partial diierential equations with non-negative characteristic form. This class of equations includes classical examples of second-order elliptic and parabolic equations, rst-order hyperbolic equations , as well as equations of mixed type. We establish an a priori error bound for the m...

This dissertation presents a step towards high-order methods for continuum-transition flows. In order to achieve maximum accuracy and efficiency for numerical methods on a distorted mesh, it is desirable that both governing equations and corresponding numerical methods are in some sense compact. We argue our preference for a physical model described solely by first-order partial differential eq...

We introduce an operator splitting based stochastic Galerkin method for the one-dimensional compressible Euler equations with random inputs. The method uses a generalized polynomial chaos approximation in the stochastic Galerkin framework (referred to as the gPC-SG method). It is well-known that such approximations for nonlinear system of hyperbolic conservation laws do not necessarily yield gl...

In this article, we study a free boundary problem modeling the growth of tumors with drug application. The model consists of two nonlinear second-order parabolic equations describing the diffusion of nutrient and drug concentration, and three nonlinear first-order hyperbolic equations describing the evolution of proliferative cells, quiescent cells and dead cells. We deal with the radially symm...

Nonlinear degenerate parabolic-hyperbolic equations are one of the most important classes of nonlinear partial differential equations. Nonlinearity and degeneracy are two main features of these equations and yield several striking phenomena that require new mathematical ideas, approaches, and theories. On the other hand, because of the importance of these equations in applications, there is a l...

We investigate the behavior of five hyperbolic reaction-diffusion equations most commonly employed to describe systems of interacting organisms or reacting particles where dispersal displays inertia. We first discuss the macroscopic or mesoscopic foundation, or lack thereof, of these reaction-transport equations. This is followed by an analysis of the temporal evolution of spatially uniform sta...

We show how partner symmetries of the elliptic and hyperbolic complex Monge-Ampère equations (CMA and HCMA) provide a lift of non-invariant solutions of threeand two-dimensional reduced equations, i.e., a lift of invariant solutions of the original CMA and HCMA equations, to non-invariant solutions of the latter four-dimensional equations. The lift is applied to non-invariant solutions of the t...