Numerical and analytical study of fractional hyperbolic equations with  <i>ψ</i> -Caputo derivatives

Abstract This paper presents a novel approach to address the Cauchy problem associated with fractional semilinear hyperbolic equations of order α ∈ (1, 2), encompassing various fractional derivatives. We introduce a priori assumptions on the solution and propose the Fourier truncation method to address the ill-posedness typically associated with such problems. Additionally, we establish a stability estimate characterized by logarithmic behavior. The theoretical findings are substantiated through numerical simulations, demonstrating the effectiveness of the proposed method.