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This paper aims to reconstruct the initial condition of a hyperbolic equationwith an unknown damping coefficient. Our approach involves approximating thehyperbolic equation's solution by its truncated Fourier expansion in the timedomain and using a polynomial-exponential basis. This truncation processfacilitates the elimination of the time variable, consequently, yielding asystem of quasi-linear elliptic equations. To globally solve the system withoutneeding an accurate initial guess, we employ the Carleman contractionprinciple. We provide several numerical examples to illustrate the efficacy ofour method. The method not only delivers precise solutions but also showcasesremarkable computational efficiency.

The time dimensional reduction method to determine the initial conditions without the knowledge of damping coefficients