Fractional Explicit Iterative Method to Solve Fractional 
Differential Equations

Mahendar Pichkiya; Ekta Mittal

doi:https://doi.org/10.14445/22315373/IJMTT-V71I7P111

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 71 | Issue 7 | Year 2025 | Article Id. IJMTT-V71I7P111 | DOI : https://doi.org/10.14445/22315373/IJMTT-V71I7P111

Fractional Explicit Iterative Method to Solve Fractional Differential Equations

Mahendar Pichkiya, Ekta Mittal

Received	Revised	Accepted	Published
28 May 2025	30 Jun 2025	20 Jul 2025	29 Aug 2025

Citation :

Mahendar Pichkiya, Ekta Mittal, "Fractional Explicit Iterative Method to Solve Fractional Differential Equations," International Journal of Mathematics Trends and Technology (IJMTT), vol. 71, no. 7, pp. 107-113, 2025. Crossref, https://doi.org/10.14445/22315373/IJMTT-V71I7P111

Abstract

This paper presents the Fractional Explicit iterative Method (FEIM), a new numerical technique for solving fractional differential equations of order 0 < 𝜏 < 1. In order to increase accuracy and stability, the method incorporates an explicit iterative correction, augmenting the classical Euler Method (FEM) and the Modified Euler Method (MFEM) fractional type. Numerical experiments show that, particularly for larger 𝑥, IFEM yields solutions that are closer to the exact values than FEM and MFEM. The effectiveness of the proposed strategy in reducing truncation errors is confirmed by theoretical error analysis, which validates these results.

Keywords

Fractional differential equations (FDEs), Fractional Explicit iteration method, Fractional derivative operator, Caputo fractional derivative.

References

[1] Iqbal M. Batiha et al., “A Fractional Mathematical Examination on Breast Cancer Progression for the Healthcare System of Jordan,” Communication in Mathematical Biology and Neuroscience, 2025.
[Google Scholar] [Publisher Link]
[2] Mohamed Khader, “Using Modified Fractional Euler Formula for Solving the Fractional Smoking Model,” European Journal of Pure and Applied Mathematics, vol. 17, no. 4, pp. 2676-2691, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[3] Sania Qureshi et al., “On the Construction and Comparison of an Explicit Iterative Algorithm with Nonstandard Finite Difference Schemes,” Mathematical Theory and Modeling, vol. 3, no. 13, pp. 78-87, 2013.
[Google Scholar] [Publisher Link]
[4] Tuljaram Meghwar et al., “Development of an Explicit Iterative Numerical Scheme Over the Modified Euler’s Method,” VFAST Transactions on Mathematics, vol. 11, no. 1, pp. 107-120, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[5] Iqbal M. Batiha et al., “A Numerical Scheme for Dealing with Fractional Initial Value Problem,” International Journal of Innovative Computing, Information and Control, vol. 19, no. 3, pp. 763-774, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[6] Sania Qureshi et al., “On Error Bound for Local Truncation Error of an Explicit Iterative Algorithm in Ordinary Differential Equations,” Science International, vol. 26, no. 2, 2014.
[Google Scholar]
[7] Zaid M. Odibat, and Shaher Momani, “An Algorithm for the Numerical Solution of Differential Equations of Fractional Order,” Journal of Applied Mathematics & Informatics, vol. 26, no. 1-2, pp. 15-27, 2008.
[Google Scholar] [Publisher Link]
[8] Ramzi B. Albadarneh et al., “Numerical Approach for Approximating the Caputo Fractional-order Derivative Operator,” AIMS Mathematics, vol. 6, no. 11, pp. 12743-12756, 2021.
[CrossRef] [Google Scholar]
[9] T.A. Biala, and S.N. Jator, “Block Implicit Adams Methods for Fractional Differential Equations,” Chaos, Solitons & Fractals, vol. 81, pp. 365-377, 2015.
[CrossRef] [Google Scholar] [Publisher Link]
[10] Lina Song, and Weiguo Wang, “A New Improved Adomian Decomposition Method and Its Application to Fractional Differential Equations,” Applied Mathematical Modelling, vol. 37, no. 3, pp. 1590-1598, 2013.
[CrossRef] [Google Scholar] [Publisher Link]
[11] Mehdi Ganjiani, “Solution of Nonlinear Fractional Differential Equations using Homotopy Analysis Method,” Applied Mathematical Modelling, vol. 34, no. 6, pp. 1634-1641, 2010.
[CrossRef] [Google Scholar] [Publisher Link]
[12] Shaher Momani, and Zaid Odibat, “Homotopy Perturbation Method for Nonlinear Partial Differential Equations of Fractional Order,” Physics Letters A, vol. 365, no. 5-6, pp. 345-350, 2007.
[CrossRef] [Google Scholar] [Publisher Link]
[13] Saurabh Kumar, and Vikas Gupta, “An Application of Variational Iteration Method for Solving Fuzzy Time-fractional Diffusion Equations,” Neural Computing and Applications, vol. 33, pp. 17659-17668, 2021.
[CrossRef] [Google Scholar] [Publisher Link]