Related papers: Global Uniqueness for an Inverse Stochastic Hyperb…
This paper aims to determine the initial conditions for quasi-linear hyperbolic equations that include nonlocal elements. We suggest a method where we approximate the solution of the hyperbolic equation by truncating its Fourier series in…
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…
In this paper we continue the study of non-diagonalisable hyperbolic systems with variable multiplicity started by the authors in \cite{Garetto2018}. In the case of space dependent coefficients, we prove a representation formula for…
We consider an inverse problem of recovering a parameter appearing in all levels in a second-order hyperbolic equation from a single boundary measurement. The model is motivated from applications in photoacoustic tomography when one seeks…
In this article, we investigate the determination of the spatial component in the time-dependent second order coefficient of a hyperbolic equation from both theoretical and numerical aspects. By the Carleman estimates for general hyperbolic…
This paper is concerned with an inverse source problem for the stochastic wave equation driven by a fractional Brownian motion. Given the random source, the direct problem is to study the solution of the stochastic wave equation. The…
We present a novel approach to determine the evolution of level sets under uncertainties in the velocity fields. This leads to a stochastic description of the level sets. To compute the quantiles of random level sets, we use the stochastic…
By means of a dynamical process we provide a characterization of the Goldbach Conjecture in an infinite set of even numbers that depends on time.
General hyperbolic systems of balance laws with inhomogeneous flux and source are studied. Global existence of entropy weak solutions to the Cauchy problem is established for small $BV$ data under appropriate assumptions on the decay of the…
In the present paper, we consider a non self adjoint hyperbolic operator with a vector field and an electric potential that depend not only on the space variable but also on the time variable. More precisely, we attempt to stably and…
Einstein's equations for general relativity, when viewed as a dynamical system for evolving initial data, have a serious flaw: they cannot be proven to be well-posed (except in special coordinates). That is, they do not produce unique…
The three-dimensional, homogeneous, incompressible Navier-Stokes equations are studied in the absence of viscosity in one direction. It is shown that there are arbitrarily large initial data generating a unique global solution, the main…
Inverse problem to determine simultaneously a general space- and time-dependent source and an initial state in a fractional diffusion equation from an {\it a posteriori} measurement of the normal derivative of the state on a portion of a…
In this study, the existence and uniqueness of the unpredictable solution for a non-homogeneous linear system of ordinary differential equations is considered. The hyperbolic case is under discussion. New properties of unpredictable…
We consider an inverse problem for the Boltzmann equation on a globally hyperbolic Lorentzian spacetime $(M,g)$ with an unknown metric $g$. We consider measurements done in a neighbourhood $V\subset M$ of a timelike path $\mu$ that connects…
In this article we establish new inequalities, both conditional and unconditional, for the restriction problem associated to the hyperbolic, or one-sheeted, hyperboloid in three dimensions, endowed with a Lorentz-invariant measure. These…
We consider the inverse problem of identification of degenerate diffusion coefficient of the form $x^\alpha a(x)$ in a one dimensional parabolic equation by some extra data. We first prove by energy methods the uniqueness and Lipschitz…
In this paper, we establish the existence of solutions for a particular class of degenerate hyperbolic equations. Following this, we approximate these degenerate equations by employing a sequence of uniformly hyperbolic equations. Notably,…
We propose a global convergent numerical method to reconstruct the initial condition of a nonlinear parabolic equation from the measurement of both Dirichlet and Neumann data on the boundary of a bounded domain. The first step in our method…
In this paper we give a more geometrical formulation of the main theorem in [E1] on the inverse problem for the second order hyperbolic equation of general form with coefficients independent of the time variable. We apply this theorem to…