Computational Analysis of the Generalized Nonlinear Time-Fractional Klein–Gordon Equation Using Uniform Hyperbolic Polynomial B-Spline Methods

This study presents an efficient numerical scheme for solving the generalized nonlinear time-fractional Klein–Gordon equation. The Caputo time-fractional derivative is discretized using a conventional finite-difference approach, while the spatial domain is approximated with uniform hyperbolic polynomial B-splines. These discretizations are coupled through the $\theta$-weighted scheme. The uniform hyperbolic polynomial B-spline framework extends classical spline theory by incorporating hyperbolic functions, thereby enhancing flexibility and smoothness in curve and surface representations—features particularly useful for problems exhibiting hyperbolic characteristics. A rigorous stability and convergence analysis of the proposed method is provided. The effectiveness of the scheme is further validated through numerical experiments on benchmark problems. The results demonstrate up to two orders of magnitude improvement in $L_{\infty }$ error norms compared to prior spline methods. This substantial accuracy enhancement highlights the robustness and efficiency of the proposed approach for fractional partial differential equations.