An accurate and unconditionally stable cubic B-splines method for the time-fractional KdV–Burgers’ equation: comparative analysis with existing schemes

In this work, we introduce a numerical method that utilizes a fourth-order cubic B-splines to solve the time-fractional Korteweg−de Vries Burgers’ (TFKdVB) equation, which involves the Caputo fractional order derivative. A θ-weighted finite difference method (FDM) is utilized for the spatial discretization, providing flexibility and improved accuracy in handling the convective and dispersive effects. The temporal fractional derivative in the Caputo sense is discretized using a finite difference approximation suited for non-integer orders. A comprehensive stability analysis is conducted, and the system is found to be unconditionally stable. To evaluate the effectiveness of the proposed method, we compare its numerical results with those in the existing literature. The results show that the proposed scheme yields significantly higher accuracy while maintaining computational efficiency. Numerical experiments confirm the reliability and robustness of the method in capturing the complex dynamics governed by time-fractional partial differential equations.