This manuscript deals with a space-fractional generalized wave problem involving the fractional Laplacian operator of order \(\alpha\) for \(1<\alpha \le 2\) . We propose an accurate numerical method to solve the mentioned fractional wave problem. The problem is discretized in spatial direction by the Fourier spectral method, and in temporal direction by using the fourth-order exponential time-differencing Runge–Kutta method. One of the main features of this method is reducing the mentioned fractional wave model to an ODE by using the Fourier transform. Then the fourth-order exponential time-differencing Runge–Kutta method is used to solve this ODE. We define the discrete energy function and check the energy-conserving properties. The convergence of this method is proved. Various numerical experiments are conducted to confirm the accuracy and dependability of the suggested approach.
