The objective of this paper is to compute initial conditions for quasi-linear hyperbolic equations. Our proposed approach involves approximating the solution of the hyperbolic equation by truncating its Fourier expansion in the time domain using the polynomial-exponential basis. This truncation enables the elimination of the time variable, resulting in a system of quasi-linear elliptic equations. Thus, we refer to our approach as the "time dimensional reduction method." To solve this system globally without requesting a good initial guess, we employ the Carleman contraction principle. To demonstrate the effectiveness of our method, we provide several numerical examples. The time dimensional reduction method not only provides accurate solutions but also exhibits exceptional computational speed.

翻译：本文旨在计算拟线性双曲型方程的初值条件。我们提出的方法通过在时间域内利用多项式-指数基截断其傅里叶展开，从而近似双曲型方程的解。这种截断使时间变量得以消除，进而得到一个拟线性椭圆型方程组。因此，我们将该方法称为“时间维数约化方法”。为无需良好初始猜测即可全局求解该方程组，我们采用了Carleman压缩映射原理。通过若干数值算例验证了该方法的有效性。时间维数约化方法不仅提供精确解，还展现出极高的计算效率。