Nonlinear integral equation of coefficient inversion of acoustic wave equation and iteration (Q909141)

The authors consider the problem of determining the coefficient function k in the wave-equation \[ (k(x)u_ x)_ x=u_{tt},\quad x>0,\quad t>0;\quad u(x,t)=0,\quad x>0,\quad t\leq 0;\quad u_ x(0,t)=\delta (t),\quad t>0 \] from measurements of u(0,.). This problem is reduced to a nonlinear integro-differential equation for k. Then, the analogous problems in 2 and 3 space dimensions are treated in an analogous way. Iterative methods for solving the resulting nonlinear integro- differential equations are presented (without convergence proofs). The iteration steps involve solving ill-posed linear integral equations related to problems of integral geometry.