Finite block method for nonlinear time-fractional partial integro-differential equations: Stability, convergence, and numerical analysis

Abstract

This paper investigates a nonlinear time-fractional partial integro-differential equation. For temporal discretization, the Caputo fractional derivative is approximated using the weighted and shifted Grünwald–Letnikov formula, while the Volterra integral operator is addressed using the fractional trapezoidal rule. Spatial discretization employs Chebyshev nodes as discretization points, and the spectral-collocation method is used to approximate the partial derivatives. To handle irregular computational domains in the two-dimensional nonlinear problem, the finite block method is adopted. The quasilinearization technique is implemented to manage the nonlinearity, transforming the problem into a linear form. Rigorous analysis of the stability and convergence of the proposed numerical schemes is conducted, and their effectiveness is demonstrated through numerical experiments, confirming both accuracy and efficiency.