A High‐Order Compact Scheme for Nonlinear Integro‐Differential Equations With Weakly Singular Kernels and Tempered ψ‐Caputo Derivatives

ABSTRACT This paper investigates a nonlinear time‐fractional partial integro‐differential equation with tempered ‐Caputo derivatives. A high‐order numerical scheme is proposed, combining ‐discretization for fractional derivatives, a compact finite difference scheme for the nonlinear convective term, and piecewise linear interpolation for the singular kernel integral term. The method achieves fourth‐order spatial accuracy and ‐order temporal accuracy. Finally, stability and convergence analyses confirm its reliability, and numerical experiments validate its efficiency.