An Adomian decomposition method with some orthogonal polynomials to solve nonhomogeneous fractional differential equations (FDEs)

Mariam Al-Mazmumy; Maryam Ahmed Alyami; Mona Alsulami; Asrar Saleh Alsulami; Saleh S. Redhwan; Mariam Al-Mazmumy; Maryam Ahmed Alyami; Mona Alsulami; Asrar Saleh Alsulami; Saleh S. Redhwan

doi:10.3934/math.20241475

AIMS Mathematics

2024, Volume 9, Issue 11: 30548-30571. doi: 10.3934/math.20241475

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An Adomian decomposition method with some orthogonal polynomials to solve nonhomogeneous fractional differential equations (FDEs)

1.
Department of Mathematics and Statistics, Faculty of Science, University of Jeddah, Jeddah 23218, Saudi Arabia
2.
Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China

Received: 23 August 2024 Revised: 11 October 2024 Accepted: 15 October 2024 Published: 28 October 2024
MSC : 26A33, 34A08, 34K37, 65L10

The present study introduced modifications to the standard Adomian decomposition method (ADM) by combining the Taylor series with orthogonal polynomials, such as Legendre polynomials and the first and second kinds of Chebyshev polynomials. These improvements can be applied to solve fractional differential equations with initial-value problems in the Caputo sense. The approaches are based on the use of orthogonal polynomials, which are essential components in approximation theories. The study carefully analyzed their respective absolute error differences, highlighting the computational benefits of the proposed modifications, which offer improved accuracy and require fewer computational steps. The effectiveness and accuracy of the approach were validated through numerical examples, confirming its efficiency and reliability.
- fractional differential equation,
- initial-value problem,
- Adomian decomposition method,
- Taylor series,
- Legendre polynomials,
- Chebyshev polynomials
Citation: Mariam Al-Mazmumy, Maryam Ahmed Alyami, Mona Alsulami, Asrar Saleh Alsulami, Saleh S. Redhwan. An Adomian decomposition method with some orthogonal polynomials to solve nonhomogeneous fractional differential equations (FDEs)[J]. AIMS Mathematics, 2024, 9(11): 30548-30571. doi: 10.3934/math.20241475

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Abstract

The present study introduced modifications to the standard Adomian decomposition method (ADM) by combining the Taylor series with orthogonal polynomials, such as Legendre polynomials and the first and second kinds of Chebyshev polynomials. These improvements can be applied to solve fractional differential equations with initial-value problems in the Caputo sense. The approaches are based on the use of orthogonal polynomials, which are essential components in approximation theories. The study carefully analyzed their respective absolute error differences, highlighting the computational benefits of the proposed modifications, which offer improved accuracy and require fewer computational steps. The effectiveness and accuracy of the approach were validated through numerical examples, confirming its efficiency and reliability.

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