Iterative Procedure for Non-Linear Fractional Integro-Differential Equations via Daftardar--Jafari Polynomials
Qasim Khan, Anthony Suen
TL;DR
The paper tackles the challenge of solving non-linear FPIDEs with Caputo time derivatives by introducing the Iterative Aboodh Transform Method (IATM), which represents the solution as $u=\sum_{j=0}^{\infty} u_j$ and models the nonlinear term using Daftardar--Jafari polynomials $\mathbf{J}_j$. The method leverages the Aboodh transform $\mathcal{A}$ to operate in the Laplace domain, then inverts back to obtain a convergent series in the time domain. Numerical experiments on several FPIDEs demonstrate superior accuracy and faster convergence than Laplace-Adomian Decomposition Method (LADM), particularly for $\alpha\in(0.5,1]$, with fractional-order solutions approaching integer-order results as $\alpha\to1$. Implemented in MAPLE with GPU support, the approach shows promise for applications in heat flow with memory and viscoelastic phenomena, and suggests extensions to stochastic nonlinear FPIDEs in future work.
Abstract
In this paper, we introduce a novel approach called the Iterative Aboodh Transform Method (IATM) which utilizes Daftardar--Jafari polynomials for solving non-linear problems. Such method is employed to derive solutions for non-linear fractional partial integro-differential equations (FPIDEs). The key novelty of the suggested method is that it can be used for handling solutions of non-linear FPIDEs in a very simple and effective way. {More precisely, we show that Daftardar--Jafari polynomials have simple calculations as compared to Adomian polynomials with higher accuracy}. The results obtained within the Daftardar--Jafari polynomials are demonstrated with graphs and tables, and the IATM's absolute error confirms the higher accuracy of the suggested method.