A Picard–Integrating Factor Iterative Scheme for Nonlinear Fractional Differential Equations with Error and Convergence Analysis

Abstract

In this paper, an iterative algorithm, namely Picard–integrating factor, is developed and analyzed for a class of nonlinear fractional differential equations. Fractional models exhibit a nonlocal and memory-dependent structure, which typically does not have an exact closed-form solution, motivating their study. This proposed scheme is based on the Riemann–Liouville fractional integral and the use of an integrating factor in the equivalent fixed-point formulation, together with Picard-type recursive construction. The method is formulated in the presence of the Lipschitz condition on the nonlinear term, and a convergence result is presented in a weighted supremum norm to make the assumptions under which the iteration is a contraction clear. Three nonlinear fractional initial value problems are analyzed: one Riccati-type model, one Bernoulli-type model, and a fourth model, which is considered with a known closed-form solution, to see the true error directly. It presents the numerical results for two cases of  Alfa = 1 and Alfa = 0.5. In all examples, the absolute value of the difference between the two Picard approximations and the Picard – integrating factor approximation is tabulated and plotted to visualize their accuracy; residual diagnostics are calculated for Riccati and Bernoulli-type examples, and the true absolute errors are reported for all the examples for which the solution is known. On a quantitative level, for the  case, the mean absolute deviation of the respective approximations is minimized when using the Picard–integrating factor approximation, being about 54.4% lower than that of the Picard approximation, while the residual diagnostics present lower endpoint residuals for the nonlinear Riccati and Bernoulli-type tests. The results suggest that the formulation with the integration factor gives a semi-analytical approximation method that is straightforward, structured, and convenient, without requiring the calculation of Adomian polynomials or correction functionals.