Related papers: Inverse problem for Einstein-scalar field equation…
We review discoveries in the nonlinear dynamics of curved spacetime, largely made possible by numerical solutions of Einstein's equations. We discuss critical phenomena and self-similarity in gravitational collapse, the behavior of…
The inverse linear-quadratic optimal control problem is a system identification problem whose aim is to recover the quadratic cost function and hence the closed-loop system matrices based on observations of optimal trajectories. In this…
One way to understand more about spacetime singularities is to construct solutions of the Einstein equations containing singularities with prescribed properties. The heuristic ideas of the BKL picture suggest that oscillatory singularities…
We prove that the Einstein equations can be solved in a very general form for arbitrary spacetime dimensions and various types of vacuum and non-vacuum cases following a geometric method of anholonomic frame deformations for constructing…
Two main aims of this paper are to develop a numerical method to solve an inverse source problem for parabolic equations and apply it to solve a nonlinear coefficient inverse problem. The inverse source problem in this paper is the problem…
This work presents and analyzes space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of parabolic optimal control problems. Using Babu\v{s}ka's theorem, we show well-posedness of…
The integration of the Einstein equations split into the solution of constraints on an initial space like 3 - manifold, an essentially elliptic system, and a system which will describe the dynamical evolution, modulo a choice of gauge. We…
We propose a numerical method to solve general hyperbolic systems in any space dimension using forward Euler time stepping and continuous finite elements on non-uniform grids. The properties of the method are based on the introduction of an…
This paper develops a method for solving Einstein's equation numerically on multi-cube representations of manifolds with arbitrary spatial topologies. This method is designed to provide a set of flexible, easy to use computational…
Four-dimensional cylindrically symmetric spacetimes with homothetic self-similarity are studied in the context of Einstein's Theory of Gravity, and a class of exact solutions to the Einstein-massless scalar field equations is found. Their…
The general stationary cylindrically symmetric solution of Einstein-massless scalar field system with a non-positive cosmological constant is presented. It is shown that the general solution is characterized by four integration constants.…
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…
The scaling of the exact solution of a hyperbolic balance law generates a family of scaled problems in which the source term does not depend on the current solution. These problems are used to construct a sequence of solutions whose…
In this paper we consider the single patch pseudo-spectral scheme for tensorial and spinorial evolution problems on the 2-sphere presented in [3,4] which is based on the spin-weighted spherical harmonics transform. We apply and extend this…
The Universe is homogeneous and isotropic on large scales, so on those scales it is usually modelled as a Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) space-time. The non-linearity of the Einstein field equations raises concern over…
The Bianchi identities for the Weyl curvature tensor of a spacetime $(M, g)$ solving the vacuum Einstein equations in a double null foliation exhibit a hyperbolic structure, which can be used to obtain detailed nonlinear estimates on the…
A uniqueness result in the inverse problem for an inhomogeneous hyperbolic system on a real vector bundle over a smooth compact manifold, based on energy measurements for improperly known sources, is established.
We study nonnegative, measure-valued solutions to nonlinear drift type equations modelling concentration phenomena related to Bose-Einstein particles. In one spatial dimension, we prove existence and uniqueness for measure solutions.…
In the first part of this work we show a uniqueness result for globally hyperbolic spacetimes with a spacelike conformal boundary satisfying the vacuum Einstein equations with positive cosmological constant. Then we present applications of…
The paper derived differential equations which solve the problem of restoration the motion parameters for a rigid reference frame from the known proper acceleration and angular velocity of its origin as functions of proper time. These…