In this work, we propose a fourth-order accurate extrapolation nonlinear difference method for the fourth-order nonlinear partial integrodifferential equations (FON-PIDEs) with a weakly singular kernel. In space direction, we define a operator to handle the nonlinear convection term \(uu_x\) , and use the fourth order central difference formula to discretize the fourth-order derivative term. In time direction, we use product-integration rule and Crank–Nicolson method to deal with Riemanm–Liouville fractional order integral (RL-FOI) terms on the graded meshes. Then a difference scheme with second-order accurate in both time and space is obtained. A series of theory analysis are proved, including the existence, stability, convergence and uniqueness. Then we apply the extrapolation method to improve the spatial second-order convergence to fourth-order. Moreover, we provide the theory proof of the extrapolation method. Finally, two numerical examples are given, and the results of examples are in agreement with the theoretical expectations. This proves the effectiveness of our constructed scheme.
