The time dimensional reduction method to determine the initial conditions for nonlinear and nonlocal hyperbolic equations

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.