A Finite Difference Approach for Solving the Time-Fractional Klein–Gordon Equation in The Caputo Sense

This work presents a finite difference numerical scheme for solving the time-fractional Klein–Gordon equation (FKGE) in the Caputo derivative sense. The method employs the L1 approximation for the Caputo fractional derivative in time and central finite differences for the spatial second derivative. The resulting implicit nonlinear system is solved iteratively at each time step. The scheme is theoretically proven to converge with order O (τ2-α + h2). To validate accuracy and efficiency, the same benchmark problem used in Ojada and Akhigbe (2025)—with exact solution u(x,t)=(1-x)5/2 t3/2 is solved for α=1.4 and α=1.6 at times t=.7,0.8,0.9. Numerical results are compared directly with those obtained via the Chebyshev Spectral Collocation Method (SCM) and the Variational Iteration Method (VIM) as reported in the original study. The finite difference approach demonstrates superior accuracy, with maximum errors on the order of 10-6 to 10-4 , significantly outperforming both SCM and VIM. All computations were implemented in MAPLE 18.