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Justification of the steepest descent method for the integral statement of an inverse problem for a hyperbolic equation. (Russian, English) Zbl 0981.35097
Sib. Mat. Zh. 42, No. 3, 567-584 (2001); translation in Sib. Math. J. 42, No. 3, 478-494 (2001).
Consider the inverse problem of finding a solution \(u(x,t)\) and a coefficient \(q(x)\) such that \[ \begin{gathered} u_{tt}=u_{xx}-q(x)u, \quad x\in {\mathbb R},\;t>0, \\ u(x,0)=q(x), \quad u_t(x,0)=0,\quad x\in {\mathbb R}, \\ u(0,t)=f(t), \quad u_x(0,t)=0,\quad t\geq 0. \end{gathered} \] Using the d’Alembert formula, the authors transform this problem to the problem of solving a nonlinear integral equation. The solution to this equation, in turn, is defined as the function minimizing some generally unavailable objective functional. The authors study this optimization problem by the steepest descent method and estimate the rate of convergence in the mean.

35R30 Inverse problems for PDEs
35L15 Initial value problems for second-order hyperbolic equations
49M05 Numerical methods based on necessary conditions
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