On an Inverse Hyperbolic Problem
نویسنده
چکیده
We consider the inverse problem of recovery of an initial condition of a hyperbolic PDE. This problem is also called sometimes “thermoacoustic tomography”. In the past publications both stability estimates and convergent numerical methods for this problem were obtained only under some restrictive conditions imposed on the principal part of the elliptic operator. In this paper logarithmic stability estimates are obatined for an arbitrary variable principal part of that operator. Convergence of the Quasi-Reversibility Method to the exact solution is also established for this case. Both complete and incomplete data collection cases are considered. This preprint is available online at http://www.ma.utexas.edu/mp arc/, posted on December 30, 2011.
منابع مشابه
Analytic solutions for the Stephen's inverse problem with local boundary conditions including Elliptic and hyperbolic equations
In this paper, two inverse problems of Stephen kind with local (Dirichlet) boundary conditions are investigated. In the first problem only a part of boundary is unknown and in the second problem, the whole of boundary is unknown. For the both of problems, at first, analytic expressions for unknown boundary are presented, then by using these analytic expressions for unknown boundaries and bounda...
متن کاملFekete-Szegö Problem of Functions Associated with Hyperbolic Domains
In the field of Geometric Function Theory, one can not deny the importance of analytic and univalent functions. The characteristics of these functions including their taylor series expansion, their coefficients in these representations as well as their associated functional inequalities have always attracted the researchers. In particular, Fekete-Szegö inequality is one of such vastly studied a...
متن کاملA meshless technique for nonlinear Volterra-Fredholm integral equations via hybrid of radial basis functions
In this paper, an effective technique is proposed to determine thenumerical solution of nonlinear Volterra-Fredholm integralequations (VFIEs) which is based on interpolation by the hybrid ofradial basis functions (RBFs) including both inverse multiquadrics(IMQs), hyperbolic secant (Sechs) and strictly positive definitefunctions. Zeros of the shifted Legendre polynomial are used asthe collocatio...
متن کاملOptical Aharonov - Bohm effect : an inverse hyperbolic problems approach
Optical Aharonov-Bohm effect: an inverse hyperbolic problems approach. Abstract We describe the general setting for the optical Aharonov-Bohm effect based on the inverse problem of the identification of the coefficients of the governing hyperbolic equation by the boundary meas-rements. We interpret the inverse problem result as a possibility in principle to detect the optical Aharonov-Bohm effe...
متن کاملAharonov - Bohm effect : an inverse hyperbolic problems approach
Optical Aharonov-Bohm effect: an inverse hyperbolic problems approach. Abstract We describe the general setting for the optical Aharonov-Bohm effect based on the inverse problem of the identification of the coefficients of the governing hyperbolic equation by the boundary meas-rements. We interpret the inverse problem result as a possibility in principle to detect the optical Aharonov-Bohm effe...
متن کاملStability for an inverse problem for a two speed hyperbolic pde in one space dimension
Suppose A(x), B(x) are 2×2 matrices on an interval [0,∞) and C a constant diagonal matrix with distinct positive entries. Let U(x, t) be the matrix solution of the system of hyperbolic PDEs CUtt − Uxx − AUx − BU = 0 on [0,∞) × R with the initial condition U(·, t) = 0 for t < 0 and the boundary condition U(0, t) = δ(t)I2. We prove a stability result for the inverse problem of recovering A,B from...
متن کامل