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.

翻译：本文旨在重构具有未知阻尼系数的双曲型方程的初始条件。我们的方法通过时间域上的截断傅里叶展开，并采用多项式-指数基来近似双曲型方程的解。这一截断过程有助于消除时间变量，从而得到一个拟线性椭圆型方程组。为了在无需精确初始猜测的情况下全局求解该系统，我们采用了Carleman收缩原理。我们提供多个数值算例来验证该方法的有效性。该方法不仅能提供精确的解，还展现出显著的计算效率。