A nonlinear compact method based on double reduction order scheme for the nonlocal fourth-order PDEs with Burgers’ type nonlinearity

In this article, a novel double reduction order technique and a newly constructed nonlinear compact difference operator are developed on graded meshes to simulate the nonlocal fourth-order partial differential equations with Burgers’ type nonlinearity. Based on a novel double reduction order approach, a triple coupled nonlinear system is obtained. The nonlinear term \(uu_{x}\) $u{u}_x$ is handled with novel fourth-order nonlinear compact difference operator, and the second and fourth derivatives of space are processed by a linear fourth-order compact difference method. Some theoretical analysis, including existence, uniqueness, \(\alpha \) $\alpha$ -robust stability and convergence are proved with considering the typical singularity of the solution near \(t=0\) $t=0$ . At last, some numerical results are presented to support our theory analysis.