Numerical approximation for solving time-fractional Benjamin-Bona-Mahony-Burger model via cubic B-spline functions

Many years of research have gone into spline functions, and they are now used in countless computational tasks. Splines have a lot of useful properties that make them an excellent tool for numerical problem solving, which account for their never-ending applications. The piecewise continuous functions known as spline functions yield smooth outcomes. The numerical solution to the nonhomogeneous time-fractional Banjamin-Bona-Mahony-Burger problem was presented in this study. The objective of the study was to obtain accurate numerical results by applying the Atangana-Baleanu fractional derivative with the help of the forward difference scheme for integer-order time derivative while the $\theta$-weighted scheme with the collaboration of cubic B-spline functions was used for the spatial derivatives. The stability of the proposed scheme was analyzed and proved to be unconditionally stable. The convergence analysis was also studied, and it was of the second order $O(h^2+(\mathrm{\Delta }s{)}^2)$. The proposed scheme was applicable and accurate, as demonstrated by numerical examples and their conceivable outcomes. The proposed scheme provided accuracy compared to other numerical techniques because it yielded numerical solutions in $C^2$ continuous piecewise form at each knot in the domain.