Related papers: On the Dirichlet to Neumann Problem for the 1-dime…
For the two versions of the KdV equation on the positive half-line an initial-boundary value problem is well posed if one prescribes an initial condition plus either one boundary condition if $q_{t}$ and $q_{xxx}$ have the same sign (KdVI)…
We consider the inverse Calder\'on problem consisting of determining the conductivity inside a medium by electrical measurements on its surface. Ideally, these measurements determine the Dirichlet-to-Neumann map and, therefore, one usually…
This article shows that knowledge of the Dirichlet-Neumann map on certain subsets of the boundary for input functions supported roughly on the rest of the boundary can be used to determine a magnetic Schr\"{o}dinger operator. With some…
A review of the author's results is given. Inversion formulas and stability estimates for the solutions to 3D inverse scattering problems with fixed-energy data are obtained. Inversions of exact and noisy data are stidied. The inverse…
Consider a real-analytic orientable connected complete Riemannian manifold $M$ with boundary of dimension $n\ge 2$ and let $k$ be an integer $1\le k\le n$. In the case when $M$ is compact of dimension $n\ge 3$, we show that the manifold and…
We establish the global existence of solutions to the Fokas-Lenells equation for any initial data in a weighted Sobolev space $H^{3}(\mathbb{R})\cap H^{2,1}(\mathbb{R})$.This result removes all spectral restrictions on the initial data…
In the context of the Cauchy problem for the Korteweg-de Vries equation we extend the inverse scattering transform to initial data that behave at plus infinity like a sum of Wigner-von Neumann type potentials with small coupling constants.…
We propose a globally convergent computational technique for the nonlinear inverse problem of reconstructing the zero-order coefficient in a parabolic equation using partial boundary data. This technique is called the "reduced dimensional…
This paper discusses the solvability (global in time) of the initial-boundary value problem of the Navier-stokes equations in the half space when the initial data $ h\in \dot{ B}_{q \sigma}^{\alpha-\frac{2}{q}}(\R_+)$ and the boundary data…
In this paper, we are interested to an inverse Cauchy problem governed by the Stokes equation, called the data completion problem. It consists in determining the unspecified fluid velocity, or one of its components over a part of its…
In this paper, we consider the inverse eigenvalue problem for the positive doubly stochastic matrices, which aims to construct a positive doubly stochastic matrix from the prescribed realizable spectral data. By using the real Schur…
In this article we are concerned with an inverse boundary value problem for a non-linear wave equation of divergence form with space dimension $n\geq 3$. In particular the so called the interior determination problem. This non-linear wave…
The paper aims to apply the inverse scattering transform to the defocusing Hirota equation with fully asymmetric non-zero boundary conditions (NZBCs), addressing scenarios in which the solution's limiting values at spatial infinities…
The paper investigates the sensitivity of the inverse problem of recovering the velocity field in a bounded domain from the boundary dynamic Dirichlet-to-Neumann map (DDtN) for the wave equation. Three main results are obtained: (1)…
In this paper, we explore the initial-boundary value (IBV) problem for an integrable spin-1 Gross-Pitaevskii system with a 4x4 Lax pair on the finite interval by extending the Fokas unified transform approach. The solution of this system…
We present a rigorous theory of the inverse scattering transform (IST) for the three-component defocusing nonlinear Schrodinger (NLS) equation with initial conditions approaching constant values with the same amplitude as $x\to\pm\infty$.…
We consider the linearized electrical impedance tomography problem in two dimensions on the unit disk. By a linearization around constant coefficients and using a trigonometric basis, we calculate the linearized Dirichlet-to-Neumann…
We show how to solve initial-boundary value problems for integrable nonlinear differential-difference equations on a finite set of integers. The method we employ is the discrete analogue of the unified transform (Fokas method). The…
Solving complex optimization problems in engineering and the physical sciences requires repetitive computation of multi-dimensional function derivatives. Commonly, this requires computationally-demanding numerical differentiation such as…
We study the wave equation in a bounded domain or on a compact Riemannian manifold with boundary. Assume that we are given the hyperbolic Neumann-to-Dirichlet map on the boundary corresponding to physical boundary measurements. We consider…