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equations one has in fact to solve a relevant boundary control problem, whereas its solution is determined by the inverse data. Keywords. 35R30, 35Q93, 45Q05. 2010 Mathematics Subject Classification. Classical integral equations in inverse problems, boundary control method. 1 Introduction Our paper is mainly of methodical character. Its aim is to demonstrate that the well-known classical integral equations of inverse problem theory can be derived in the framework of a unified approach. The approach is the boundary control (BC) method that is a method for solving inverse

J. Inverse Ill-Posed Probl. 22 (2014), 245–250 DOI 10.1515/ jip-2013-0012 © de Gruyter 2014 On determining an absorption coefficient and a speed of sound in the wave equation by the BC method Leonid Pestov Abstract. We consider the inverse multidimensional problem for the wave equation with unknown variable speed and absorbtion. A linear procedure based on the boundary control method for determining both coefficients is proposed. Keywords. Inverse dynamical problem, wave equation, boundary control method. 2010 Mathematics Subject Classification. 35R30. 1

formula for the unknown coecient from the phaseless scattering data. Keywords: Phaseless inverse scattering, Born approximation, reconstruction formula, nano structures, Radon transform MSC 2010: 35R30 || Michael V. Klibanov: Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA, e-mail: mklibanv@uncc.edu Vladimir G. Romanov: Sobolev Institute of Mathematics, Novosibirsk 630090, Russian Federation, e-mail: romanov@math.nsc.ru 1 Introduction In this publication, we solve, for the rst time, the phaseless Inverse

result for the stationary case, which allows us to derive a corresponding stability estimate, and then extend our argument to instationary problems which are close to steady state. Keywords: Parameter identication, nonlinear diusion, quasilinear parabolic problems MSC 2010: 35R30, 65J22 || Herbert Egger, Jan-Frederik Pietschmann, Matthias Schlottbom: AG Numerik und Wissenschaftliches Rechnen, Fachbereich Mathematik, TU Darmstadt, Dolivostr. 15, 64293 Darmstadt, Germany, e-mail: egger@mathematik.tu-darmstadt.de, pietschmann@mathematik.tu-darmstadt.de, schlottbom

coecients aij(x), q(x) are found from linear algebraic systems of the form Ay = b, where A is a generalized Wronskian of some set of eigenfunctions that can be shown to be nontrivial. The domain Ω is reconstructed using the first eigenfunction φ1(x). Keywords: Inverse problems, fractional diusion equations, single point measurements MSC 2010: 35R30, 34K29 DOI: 10.1515/anly-2015-5010 Received May 30, 2015; accepted July 9, 2015 || Dedicated to the memory of Professor Anatoly Kilbas 1 Introduction Consider the multidimensional fractional diusion equation {{{{{{{ {{{{{{{ { C

, the source term being localized on the blood vessels. We consider the inverse problem that consists in re- covering the position of the blood vessels assuming the distribution of tumor cells. We use an adjoint method. Results relative to idealized clinical cases are discussed. Keywords. Tumor growth, vascularization, inverse problem, proliferative cells. 2010 Mathematics Subject Classification. 92C50, 65M32, 35R30. 1 Introduction Mathematical modeling of tumor growth has numerous potential applications: nu- merical simulations could be useful to study how a

proof is based on the Bukhgeı̆m–Klibanov method. Keywords. Inverse problem, Korteweg–de Vries equation, Carleman estimate. 2010 Mathematics Subject Classification. 35R30, 35Q53. 1 Introduction The Korteweg–de Vries (KdV) equation yt .t; x/C yxxx.t; x/C yx.t; x/C y.t; x/yx.t; x/ D 0 is a nonlinear dispersive equation that serves as a mathematical model to study the propagation of long water waves in channels of relatively shallow depth and flat bottom [36]. In this model, the function y D y.t; x/ represents the surface eleva- tion of the water wave at time t and at

a compact support in the three-dimensional Schrödinger equation is considered. Only the modulus of the scattering complex-valued wave eld is known, whereas the phase is unknown. It is shown that the unknown potential can be reconstructed via the inverse Radon transform. This solution has potential applications in imaging of nanostructures. Keywords: Long standing problem, phaseless inverse scattering, Schrödinger equation, reconstruction formula, Radon transform MSC 2010: 35R30 DOI: 10.1515/jiip-2015-0025 Received February 26, 2015; accepted March 6, 2015 1

regularization method. Numerical examples with noisy data are given to illustrate the eectiveness of this method. Keywords: Inverse Scattering, Born Approximation, Schrödinger Equation, Numerical Solution MSC 2010: 35P25, 35R30, 65M32 DOI: 10.1515/cmam-2015-0032 Received June 25, 2015; revised October 8, 2015; accepted November 6, 2015 1 Introduction We consider the nonlinear two-dimensional Schrödinger equation − ∆u(x) + h(x, |u(x)|)u(x) = k2u(x), x ∈ ℝ2, (1) where k > 0 is the wavenumber and h(x, s) is a rather general nonlinear function of s ∈ ℝ. This equation models the

, uniqueness criterion MSC 2010: Primary 35R30; secondary 35L50, 35P25, 35Q41 || Mansur I. Ismailov: Department of Mathematics, Gebze Institute of Technology, Gebze-Kocaeli 41400, Turkey; and Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, Baku, Azerbaijan, e-mail: mismailov@gyte.edu.tr 1 Introduction Consider the system of first-order PDE on the half-plane {(x, t) : x ∈ ℝ+, t ∈ ℝ}, òàt÷ − àx÷ = [ò, J(x, t)]÷, (1.1) where ò = diag{î1, î2, . . . , î2n} is a constant diagonal matrix with î1 ≥ ⋅ ⋅ ⋅ ≥ în > 0 > în+1 ≥ ⋅ ⋅ ⋅ ≥ î2n, the bracket