  This paper aims to reconstruct the initial condition of a hyperbolic equation
with an unknown damping coefficient. Our approach involves approximating the
hyperbolic equation's solution by its truncated Fourier expansion in the time
domain and using a polynomial-exponential basis. This truncation process
facilitates the elimination of the time variable, consequently, yielding a
system of quasi-linear elliptic equations. To globally solve the system without
needing an accurate initial guess, we employ the Carleman contraction
principle. We provide several numerical examples to illustrate the efficacy of
our method. The method not only delivers precise solutions but also showcases
remarkable computational efficiency.
