A third-order numerical method for solving fractional ordinary differential equations

Xiaopeng Yi; Chongyang Liu; Huey Tyng Cheong; Kok Lay Teo; Song Wang; Xiaopeng Yi; Chongyang Liu; Huey Tyng Cheong; Kok Lay Teo; Song Wang

doi:10.3934/math.20241026

AIMS Mathematics

2024, Volume 9, Issue 8: 21125-21143. doi: 10.3934/math.20241026

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Research article

A third-order numerical method for solving fractional ordinary differential equations

1.
School of Mathematical Sciences, Sunway University, Kuala Lumpur 47500, Malaysia
2.
School of Mathematics and Information Science, Shandong Technology and Business University, Yantai 264005, China
3.
Yantai Key Laboratory of Big Data Modeling and Intelligent Computing, Yantai 264005, China
4.
School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Perth 6845, Australia

Received: 19 March 2024 Revised: 10 June 2024 Accepted: 14 June 2024 Published: 01 July 2024

In this paper, we developed a novel numerical method for solving general nonlinear fractional ordinary differential equations (FODEs). First, we transformed the nonlinear FODEs into the equivalent Volterra integral equations. We then developed a time-stepping algorithm for the numerical solution of the Volterra integral equations based on the third-order Taylor expansion for approximating the integrands in the Volterra integral equations on a chosen mesh with the mesh parameter $ h $. This approximation led to implicit nonlinear algebraic equations in the unknowns at each given mesh point, and an iterative algorithm based on Newton's method was developed to solve the resulting implicit equations. A convergence analysis of this numerical scheme showed that the error between the exact solution and numerical solution at each mesh point is $ \mathcal{O}(h^{3}) $, independent of the fractional order. Finally, four numerical examples were solved to verify the theoretical results and demonstrate the effectiveness of the proposed method.
- fractional ordinary differential equations,
- time-stepping discretization,
- implicit scheme,
- Newton's method,
- convergence analysis
Citation: Xiaopeng Yi, Chongyang Liu, Huey Tyng Cheong, Kok Lay Teo, Song Wang. A third-order numerical method for solving fractional ordinary differential equations[J]. AIMS Mathematics, 2024, 9(8): 21125-21143. doi: 10.3934/math.20241026

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Abstract

In this paper, we developed a novel numerical method for solving general nonlinear fractional ordinary differential equations (FODEs). First, we transformed the nonlinear FODEs into the equivalent Volterra integral equations. We then developed a time-stepping algorithm for the numerical solution of the Volterra integral equations based on the third-order Taylor expansion for approximating the integrands in the Volterra integral equations on a chosen mesh with the mesh parameter $ h $. This approximation led to implicit nonlinear algebraic equations in the unknowns at each given mesh point, and an iterative algorithm based on Newton's method was developed to solve the resulting implicit equations. A convergence analysis of this numerical scheme showed that the error between the exact solution and numerical solution at each mesh point is $ \mathcal{O}(h^{3}) $, independent of the fractional order. Finally, four numerical examples were solved to verify the theoretical results and demonstrate the effectiveness of the proposed method.

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