On the Various Formulations of the One-Dimensional Inverse Problem for the Telegraph Equation

Abstract

In this note we establish the equivalence of two formulations of the inverse problem on finding the unknown coefficient q(z) in the telegraph equation

$$ {u_{tt}} = \Delta u + q(z)u,\;(x,y) \in {R_2},\;z > $$

((1))

if it is known that u(x, y, z, t) is a solution of equation (1) satisfying the conditions

$$ u{|_{t > 0}} \equiv 0$$

((2))

$$ {u_z}{|_{z = 0}} = \partial (x,y,t)$$

((3))

. Suppose that for z = 0 u is the function f (x, y, t)

$$ u{|_{z = 0}} = f(x,y,t)$$

((4))

certain properties of which are given.