An Iterative Approach for Solving Fractional Differential Equations Using the α-Generalized Daftardar–Jafari Method

Background Fractional calculus has gained significant attention due to its applications in modeling complex systems. This paper introduces several fundamental fractional operators, including the Riemann–Liouville, Caputo, and Hadamard operators. From these, a novel α-generalized fractional operator is derived, which plays a crucial role in solving fractional partial differential equations (FPDEs). Methods The α-generalized operator is analytically constructed and its properties are rigorously proven. It is then integrated with the Daftardar–Jafari iterative method (DJM) to solve both linear and nonlinear FPDEs. The convergence of the proposed method is established using the Lipschitz condition. Two benchmark equations—the linear fractional Burger’s equation and the nonlinear fractional Heat-type equation—are used to demonstrate the method’s applicability. MATLAB is employed to implement the numerical schemes and visualize the results. Results The DJM combined with the α-generalized operator yields superior numerical results compared to the Adomian decomposition method. This superiority is evident in both accuracy and convergence, especially for the nonlinear fractional Heat-type equation. Tabulated results and graphical comparisons confirm the effectiveness of the proposed approach across different values of α. Conclusions The integration of the α-generalized operator with the Daftardar–Jafari method provides a powerful tool for solving FPDEs. Its numerical advantages make it a promising alternative to existing methods. The study recommends extending this framework to other fractional operators, iterative schemes, and potential transformation techniques, with applications in quantum physics, fluid dynamics, and beyond.