A Double Reduction Order Nonlinear Fourth‐Order Difference Method for the Nonlinear Nonlocal Fourth‐Order PIDEs With a Weakly Singular Kernel

ABSTRACT In the paper, a nonlinear fourth‐order difference and a linear compact difference operator are used to discretize the nonlinear fourth‐order nonlocal partial integro‐differential equations (PIDEs) with a weakly singular kernel. The first‐order convolution quadrature is used to approximate the Riemann–Liouville (R‐L) fractional integral terms. The Caputo derivative term in temporal direction is deal with L1 formula. We construct a new nonlinear fourth‐order difference operator to discretize the nonlinear convection term. The fourth‐order term is treated by a linear fourth‐order compact difference method. Then, we further prove the existence, stability, convergence, and uniqueness of the solution of the proposed fourth‐order nonlinear difference scheme. Finally, two examples are given to verify the correctness of the theory.