This paper aims to determine the initial conditions for quasi-linear hyperbolic equations that include nonlocal elements. We suggest a method where we approximate the solution of the hyperbolic equation by truncating its Fourier series in the time domain with a polynomial-exponential basis. This truncation effectively removes the time variable, transforming the problem into a system of quasi-linear elliptic equations. We refer to this technique as the "time dimensional reduction method." To numerically solve this system comprehensively without the need for an accurate initial estimate, we used the newly developed Carleman contraction principle. We show the efficiency of our method through various numerical examples. The time dimensional reduction method stands out not only for its precise solutions but also for its remarkable speed in computation.

翻译：本文旨在确定包含非局部项的拟线性双曲方程的初始条件。我们提出一种方法，通过采用多项式-指数基在时域上截断其傅里叶级数来逼近双曲方程的解。这种截断有效地消除了时间变量，将原问题转化为拟线性椭圆方程组。我们将此技术称为"时间维度约简法"。为了在无需精确初始估计的情况下全面数值求解该方程组，我们采用了新近发展的Carleman压缩原理。通过多个数值算例，我们展示了该方法的有效性。时间维度约简法不仅以其精确解见长，其计算速度亦尤为显著。