Crank-Nicolson ADI Finite Difference Method for Three-Dimensional Nonlinear Partial Integro-Differential Equations with Weak Singular Kernels

The main objective of this study is to present a fast and efficient numerical scheme for solving nonlinear integral differential equations with weak singular kernels in three-dimensional domain. First, the temporal derivative and integral term are approximated by the Crank-Nicolson (CN) method and the second-order fractional quadrature rule. After that the spatial discretization is carried out by combining the finite difference method and the alternating direction implicit (ADI) method, and the nonlinear term are approximated using the Taylor expansion. The stability and convergence of the proposed scheme are analyzed, followed by the verification of the theoretical results through numerical experiments.