REGULARIZED ITERATIVE FDEM: A NUMERICAL APPROACH FOR NONLINEAR CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper, we propose a novel numerical technique-Regularized Iterative Fractional Differential Equation Method (FDEM)-for solving nonlinear fractional differential equations (FDEs) involving the Caputo derivative. The method begins by reformulating the fractional differential equation as a Volterra integral equation and addresses the weak singularity in the kernel by a regularization strategy that decomposes the nonlinear term. This decomposition enables a stable and accurate numerical integration using the composite trapezoidal rule. To handle the nonlinearity, a fixed-point iteration is employed at each time step. The resulting algorithm is simple to implement, computationally effi-cient with O(m²) complexity, and adaptable to various types of nonlinear FDEs. The method’s stability, accuracy, and flexibility make it suitable for practical applications, including systems of fractional equations and higher-dimensional problems. For validating the robustness and effec-tiveness of the established approach, numerous numerical problems are addressed.