Computational modeling of wave propagation phenomena via a fractional Klein-Gordon equation with modified Atangana-Baleanu-Caputo derivative

This article presents a robust numerical method for solving the time fractional Klein-Gordon equation in one, two, and three dimensions. The proposed Laplace-transformed Chebyshev spectral collocation scheme efficiently handles the fractional order dynamics, which are crucial for modeling memory effects and nonlocal interactions in wave propagation phenomena. By first employing the Laplace transform, the time fractional derivative is eliminated, reducing the problem to a parameter-dependent elliptic equation in Laplace space. This transformation inherently incorporates initial conditions, avoiding time stepping restrictions and improving numerical stability. Spatial discretization is then performed using a Chebyshev spectral collocation method, which achieves exponential convergence with relatively few spatial nodes, ensuring high accuracy. Finally, the time domain solution is accurately retrieved using the improved Talbot’s method, a contour integration method that offers efficient and stable numerical inversion of the Laplace transform. Numerical experiments conducted on 1D, 2D, and 3D fractional Klein-Gordon problems demonstrate that the proposed numerical scheme delivers superior accuracy and significantly reduced computational cost compared to classical time stepping methods. The results confirm that the method is not only computationally efficient but also a versatile and reliable tool for solving a broad class of time-fractional partial differential equations.