An implicit difference scheme for the fourth-order nonlinear non-local PIDEs with a weakly singular kernel

In this paper, an implicit difference scheme is constructed for the fourth-order nonlinear non-local partial integro-differential equations (PIDEs) with a weakly singular kernel. The Caputo derivative term in temporal direction is approximated by L1-discretization formula. And the first-order convolution quadrature is used to deal with the Riemann–Liouville (R–L) fractional integral terms. By combining the standard central difference approximation, a fully discrete difference scheme is established. For the nonlinear convection term uu_x , the Galerkin method based on piecewise linear test functions is used to handle it implicitly and attain a system of nonlinear algebraic equations. For the fourth-order term u_xxxx , we use the Taylor expansion with integral remainder to deal with it. Then, the existence of the numerical solutions is proved. The stability and the convergence of the numerical solutions are strictly proved in the L^2 -norm and L^∞ -norm, respectively. The uniqueness is also proved for the nonlinear system. Finally, we introduce and compare two iterative algorithms of solving the implicit difference scheme. And the numerical results are consistent with the theoretical analysis.