Direct methods of solving multidimensional inverse hyperbolic problems.
type: Item
instance of: scholarly article
title: Direct methods of solving multidimensional inverse hyperbolic problems.
zbMATH Open document ID: 1069.65105
author: A. D. Satybaev
author: Sergey I. Kabanikhin
author: Maxim A. Shishlenin
published in: Inverse and Ill-Posed Problems Series
publication date: 2004-12-09T00:00:00Z
review text: The authors present the so-called ``direct methods'' for solving two inverse hyperbolic problems. It is known in the literature there are two groups of solution methods for inverse problems: the first one is based on Volterra operator equations, iterative or optimization procedures which require solutions of direct and adjoint problems at every step, whereas the second group consisting of the Gel'fand-Levitan-Krein method, the method of boundary control, the finite-difference scheme inversion and linearization methods does not require the multiple direct problem solution and allows one to find the solution in a specific point of the medium. The authors refer to the methods of the second group as the ``direct methods''. In this book the authors apply these methods to the following two inverse problems. Problem 1: Let \(\psi(x,y)\), \(\varphi(x,y)\) and \(f(y,t)\) \((x > 0\), \(y \in \mathbb{R}^n\), \(t> 0)\) be given. It is required to find functions \(u(x,y,t)\) and \(q(x,y)\) from the equations \[ \begin{aligned} \frac{\partial ^2 u}{\partial t^2}&=\Delta_{x,y}u - q(x,y)u, \quad x>0, y \in \mathbb{R}^n, t> 0,\\ u(x,y,0)&=\psi(x,y), \quad \frac{\partial u}{\partial t}(x,y,0) =\varphi(x,y),\quad x>0, y \in \mathbb{R}^n,\\ u(0,y,t)&=f(y,t), \quad y \in \mathbb{R}^n, t > 0. \end{aligned} \] Here, \(\Delta_{x,y}u := \partial ^2 u/\partial x^2 + \partial ^2 u/\partial y_1^2 + \dots + \partial ^2 u/\partial y_n^2.\) Problem 2: Let \(H(y,t)\), \(f(y,t)\), \(y \in \mathbb{R}^n\), \(t > 0\) be given. It is required to determine \(\rho(x,y)\) or \(c(x,y)\) or \(\rho(x,y)\) and \(c(x,y)\) simultaneously from the equations \[ \begin{aligned} c^{-2}(x,y)\frac{\partial ^2 v}{\partial t^2}&= \Delta_{x,y}v - \nabla_{x,y} \ln \rho(x,y)\nabla_{x,y}v, \quad x>0, y \in \mathbb{R}^n, t> 0,\\ v|_{t<0} &\equiv 0,\\ \frac{\partial v}{\partial x}(+0,y,t) &=H(y,t), \quad y \in \mathbb{R}^n, t> 0,\\ u|_{x=+0} &= f(y,t), \quad y \in \mathbb{R}^n, t> 0. \end{aligned} \] Here, \(\nabla_{x,y} v=(…
Mathematics Subject Classification ID: 65M32
Mathematics Subject Classification ID: 65-02
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65M12
zbMATH DE Number: 2121579
zbMATH Keywords: inverse coefficient problems
zbMATH Keywords: ill-posed problems
zbMATH Keywords: textbook
zbMATH Keywords: direct methods
zbMATH Keywords: inverse hyperbolic problems
zbMATH Keywords: Gel'fand-Levitan-Krein method
zbMATH Keywords: method of boundary control
zbMATH Keywords: finite difference scheme inversion
zbMATH Keywords: linearization methods
zbMATH Keywords: stability
zbMATH Keywords: Fredholm integral equations
zbMATH Keywords: wave equation
zbMATH Keywords: numerical examples
zbMATH Keywords: system of nonlinear Volterra integral equations
zbMATH Keywords: convergence
zbMATH Keywords: projection method
reviewed by: Dinh Nho Hào
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2121579