A new nonlinear compact difference scheme for a fourth-order nonlinear Burgers type equation with a weakly singular kernel

In this study, we propose a new nonlinear compact difference scheme (NCDS) for a fourth-order nonlinear Burgers type equation (FO-NBTE) with a weakly singular kernel. The Crank–Nicolson method and product-integration rule are utilized for discretizing the time derivative and Riemann–Liouville (R–L) time fractional integral term on the graded meshes, respectively. A nonlinear operator is constructed to discretize the nonlinear convective term, while the double reduced-order method is employed to handle fourth-order spatial derivative, then the FO-NBTE is transformed into three coupled nonlinear equations. The main advantage of the proposed NCDS is that it achieves second-order convergence in time and simultaneously fourth-order convergence in space, addressing the limitation observed in Tian et al. (Comput Appl Math 41:328, 2022) where the order of spatial convergence was restricted to second-order. In addition, a series of theoretical proofs for the proposed NCDS are presented, including the existence, stability, convergence and uniqueness. Finally, two numerical examples are given which consistent with the theoretical analysis.