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

Muayyad Mahmood Khalil — 2026-06-24

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.

Response of Chinese Cabbage (Brassica rapa var. pekinensis) to Zinc Sulfate and Nano Zinc Oxide Fertilization and Their Effects on Growth and Yield

Abrar A. Naser — 2026-06-23

The objective of the study was to evaluate field growth and yield response of ZnO-NPs and ZnSO₄ applied at various concentrations on Chinese cabbage during 2024/25. A randomized block design with three replications (8 treatments: selections including 25, 50, 75 and 100 mg/L zinc source concentrations) was employed to evaluate the increasing effect of ZnSO₄ treatment as compared to the control in different traits. Among all the treatments, the treatment consisted of CA and MA maintained good vegetative growth on all growth parameters such as average bulb weight (2.86 kg/plant), marketable bulb weight (2.54 kg/plant), bulb circumference (48.7 cm) and number of leaves (19.6 leaves/plant). In addition, it also produced highest total yield (82.4 t·ha⁻¹) and marketable bulb yield (73.2 t·ha⁻¹) compared to the other treatments. This is due to the importance of zinc ions for enzyme activation in chlorophyll biosynthesis. This improves photosynthesis and protein synthesis as a positive effect on vegetative growth and headformation. the other hand, the efficiency of nano-zinc oxide slacking off at higher amounts d which there is probability in aggregation of nanoparticles and thereby obstacle it absorption on the other vital elements. It has been reported that 75 mg·L⁻¹ zinc sulfate might be the best concentration for growth characteristics and yield of Chinese cabbage. This further highlights the significance of choosing an appropriate source of zinc for promoting greater absorption and bestowing positive physiological effects.

Jurnal MIPA dan Pembelajarannya

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

Response of Chinese Cabbage (Brassica rapa var. pekinensis) to Zinc Sulfate and Nano Zinc Oxide Fertilization and Their Effects on Growth and Yield