Solving a time-fractional semilinear hyperbolic equations by Fourier truncation with boundary conditions

In this paper, we discuss a pioneering contribution in the field by addressing, for the first time, the Cauchy problem associated with fractional semilinear hyperbolic equations of order alpha is an element of (1, 2), involving a general form of fractional derivative. Introducing an a priori assumption on the solution, we advocate the application of the Fourier truncation method to address the inherent ill -posed nature of the problem. Furthermore, we establish a stability estimate of logarithmic type.