An Efficient Iterative Method for Nonlinear Fractional Integro‐Differential Equations of Variable Order with Delay

Abstract Nonlinear variable‐order fractional integro‐differential equations with delay have become powerful tools for describing complex memory‐dependent phenomena. This study introduces an iterative numerical method for solving nonlinear variable‐order fractional Volterra‐type integro differential equations with delay, which frequently arise in the modeling of memory‐dependent phenomena in diverse engineering and scientific applications. The method utilizes the Caputo‐type variable‐order fractional derivative to capture the system's non‐local characteristics effectively. A uniform mesh is used for spatial discretization, and L1 scheme is applied to approximate the fractional derivative, leading to a nonlinear difference equation that accurately represents the underlying dynamics of the original problem. To solve this equation, the Daftardar‐Gejji and Jafari (DGJ) iterative technique is utilized. The convergence analysis and corresponding error bounds of the proposed scheme are established, demonstrating that the approximate solution converges to the exact solution with an order of . Also, comparative analysis with the Iterative Laplace Transform method results highlights the improved accuracy of the proposed approach. Additionally, the paper establishes sufficient conditions ensuring the existence and uniqueness of the solution. Numerical results are provided to validate the proposed method's accuracy, efficiency, and robustness.