Numerical solution of the variable-order time fractional advection reaction–diffusion equation via combination of a Newton’s polynomial and Cubic B-spline method

This work presents a numerical technique based on the cubic B-spline function for solving the variable-order time fractional advection reaction–diffusion equation in the sense of the Caputo derivative. Newton’s interpolation formulation has been employed to approximate the variable-order time-fractional derivative, while the cubic B-spline functions are utilized for spatial discretization. The proposed methodology demonstrated unconditionally stable and convergence of order $(\Delta r^{4-\vartheta (\varsigma ,r)}+h^2)$ through the Von Neumann analysis. Numerical investigations that confirm theoretical conclusions using data visualizations and tables to illustrate efficiency and accuracy. Furthermore, the comparative findings demonstrate that the novel discretization methodology outperforms the other techniques present in the literature in terms of accuracy.