The Time–Fractional Wave Equation with Variable Coefficients

In this paper, we primarily use the inverse operator method to find a unique series solution to a time–fractional wave equation with variable coefficients based on the Mittag–Leffler function. In addition, we also derive the series and integral convolution solutions to the Klein–Gordon equation using the Fourier transform and Green’s functions. Furthermore, our series solutions significantly simplify the process of finding solutions with several illustrative examples, avoiding the need for complicated integral computations.