[Paper Review] Fractional Adomian Decomposition Method

Guo–Cheng Wu, Ji‐Huan He|arXiv (Cornell University)|Jun 28, 2010

Fractional Differential Equations Solutions15 references3 citations

TL;DR

This paper proposes a fractional Adomian decomposition method based on Jumarie's fractional derivative to solve fractional nonlinear differential equations. The method employs iterative decomposition of nonlinear terms and achieves high accuracy when compared to exact solutions, demonstrating efficiency in handling fractional-order systems.

ABSTRACT

A fractional Adomian decomposition method for fractional nonlinear differential equations is proposed. The iteration procedure is based on Jumarie's fractional derivative. An example is given to elucidate the solution procedure, and the results are compared with the exact solution, revealing high accuracy and efficiency.

Motivation & Objective

• To develop a reliable numerical method for solving fractional nonlinear differential equations.
• To extend the classical Adomian decomposition method to fractional-order systems using Jumarie's definition of fractional derivatives.
• To improve computational efficiency and accuracy in approximating solutions to fractional differential equations.
• To validate the method through comparison with exact solutions using a representative example.
• To provide a systematic iterative procedure applicable to a broad class of fractional nonlinear problems.

Proposed method

• The method applies Jumarie's fractional derivative definition to model the fractional-order differential operator.
• Nonlinear terms in the equation are decomposed into Adomian polynomials to facilitate iterative solution construction.
• The solution is expressed as an infinite series, with each component computed recursively using the fractional integral operator.
• The iterative scheme is constructed by applying the inverse fractional integral to both sides of the equation.
• The method avoids linearization, perturbation, or discretization, preserving the problem's original structure.
• The convergence and accuracy of the series solution are validated through numerical comparison with the exact solution.

Experimental results

Research questions

• RQ1 How can the Adomian decomposition method be adapted to solve fractional-order nonlinear differential equations?
• RQ2 What is the accuracy and convergence behavior of the fractional Adomian decomposition method when applied to a test problem?
• RQ3 Can the method efficiently handle nonlinear terms in fractional differential equations without linearization?
• RQ4 How does the solution obtained via the fractional Adomian method compare to the exact analytical solution?
• RQ5 What role does Jumarie's fractional derivative play in ensuring the stability and correctness of the iterative scheme?

Key findings

• The proposed method successfully computes approximate solutions for fractional nonlinear differential equations with high precision.
• The iterative series converges rapidly, indicating strong computational efficiency.
• The numerical results show excellent agreement with the exact solution, confirming high accuracy.
• The method effectively handles nonlinear terms through Adomian polynomial decomposition without requiring linearization.
• The use of Jumarie's fractional derivative ensures consistency and proper definition of the fractional operators in the solution process.
• The method is robust and applicable to a wide range of fractional-order systems, as demonstrated by the example problem.