A Finite Block Method Framework for Nonlinear Fractional Integro‐Differential Equations

ABSTRACT This paper introduces a robust and efficient numerical framework for solving nonlinear time‐fractional partial integro‐differential equations (NLTFPIDEs). Temporal discretization is performed using the weighted and shifted Grünwald–Letnikov formula, incorporating the fractional trapezoidal rule and the second‐order backward differentiation formula (BDF2). Spatial discretization leverages Chebyshev nodes as discretization points, with the Lagrange‐collocation method applied to approximate partial derivatives. For irregular computational domains, the framework utilizes the finite block method (FBM) in two dimensions. Nonlinearities in the equations are handled through the quasilinearization technique. A comprehensive stability and convergence analysis using the energy method confirms the reliability of the proposed schemes. Numerical experiments validate the theoretical findings, highlighting the accuracy and efficiency of the approach.