Related papers: Continuum and Discrete Initial-Boundary-Value Prob…
Outer boundary conditions for strongly and symmetric hyperbolic formulations of 3D Einstein's field equations with a live gauge condition are discussed. The boundary conditions have the property that they ensure constraint propagation and…
Instabilities in finite difference codes due to the singularity of spherical coordinates at the center are studied. In typical Numerical Relativity applications, standard regularization techniques by themselves do not ensure long term…
Differential equations of infinite order are an increasingly important class of equations in theoretical physics. Such equations are ubiquitous in string field theory and have recently attracted considerable interest also from cosmologists.…
This article begins with a brief introduction to numerical relativity aimed at readers who have a background in applied mathematics but not necessarily in general relativity. I then introduce and summarise my work on the problem of treating…
In the 3+1 framework of the Einstein equations for the case of vanishing shift vector and arbitrary lapse, we calculate explicitly the four boundary equations arising from the vanishing of the projection of the Einstein tensor along the…
We make use of an improved existence result for the characteristic initial value problem for the conformal Einstein equations to show that given initial data on two null hypersurfaces $\mathcal{N}_\star$ and $\mathcal{N}'_\star$ such that…
A new method is introduced for studying boundary value problems for a class of linear PDEs with {\it variable} coefficients. This method is based on ideas recently introduced by the author for the study of boundary value problems for PDEs…
We develop numerical algorithms to approximate positive solutions of elliptic boundary value problems with superlinear subcritical nonlinearity on the boundary of the form $-\Delta u + u = 0$ in $\Omega$ with $\frac{\partial u}{\partial…
We revisit Winicour's affine-null metric initial value formulation of General Relativity, where the characteristic initial value formulation is set up with a null metric having two affine parameters. In comparison to past work, where the…
The regular finite initial value problem at infinity is used to obtain regularity conditions on the freely specifiable parts of initial data for the vacuum Einstein equations with non-vanishing second fundamental form. These conditions…
In this article we are concerned with an inverse initial boundary value problem for a non-linear wave equation in space dimension $n\geq 2$. In particular we consider the so called interior determination problem. This non-linear wave…
We consider the Einstein-Maxwell system as a Cauchy initial value problem taking the electric and magnetic fields as independent variables. Maxwell's equations in curved spacetimes are derived in detail using a 3+1 formalism and their…
Differential equations with infinitely many derivatives, sometimes also referred to as ``nonlocal'' differential equations, appear frequently in branches of modern physics such as string theory, gravitation and cosmology. The goal of this…
We investigate the initial-boundary value problem for linearized gravitational theory in harmonic coordinates. Rigorous techniques for hyperbolic systems are applied to establish well-posedness for various reductions of the system into a…
The demand to obtain answers to highly complex problems within strong-field gravity has been met with significant progress in the numerical solution of Einstein's equations - along with some spectacular results - in various setups. We…
In this paper the concept of measure of noncompactness is applied to prove the existence of solution for a boundary value problem for an infinite system of second order differential equations in $\ell_{p}$ space. We change the boundary…
In the last paper \cite{R7}, it was studied Hilbert, Poincare and Neumann boundary-value problems with arbitrary measurable data for generalized analytic functions and generalized harmonic functions with applications to the relevant…
We discuss the initial value problem of general relativity in its recently unified Lagrangian and Hamiltonian pictures and present a multi-domain pseudo-spectral collocation method to solve the resulting coupled nonlinear partial…
Einstein field equations are notoriously challenging to solve due to their complex mathematical form, with few analytical solutions available in the absence of highly symmetric systems or ideal matter distribution. However, accurate…
This paper deals with numerical solutions to a partial differential equation of fractional order. Generally this type of equation describes a transition from anomalous diffusion to transport processes. From a phenomenological point of view,…