An improved Yuan–Agrawal method with rapid convergence rate for fractional differential equations

Liu, Q. X.; Chen, Y. M.; Liu, J. K.

doi:10.1007/s00466-018-1621-6

Cited by 6 publications

(5 citation statements)

References 28 publications

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“…1. The first suggestion is to propose some new modifications and improvements of the CDR and SDR methods to obtain some fast and accurate numerical methods to approximate the Caputo fractional derivative (See some improvements of the YA method by Diethelm and et al, [3,5,8,9,11,12,28]).…”

Section: Concluding Remarks and Future Workmentioning

confidence: 99%

The sine and cosine diffusive representations for Caputo fractional derivative: A matrix approach to multi-term fractional differential equations

Khosravian-Arab

Dehghan

2023

Preprint

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In the last decade, there has been a surge of interest in application of fractional calculus in various areas such as, mathematics, physics, engineering, mechanics and etc. So, numerical methods have rapidly been developed to handle problems containing fractional derivatives (or integrals). Due to the fact that all the operators which appear in fractional calculus are non-local, so, the classical linear multi-step methods have some diffculties from the point of view of (time/space) computational complexity. Recently, two newnon-classical methods or diffusive based methods have been proposed by the authors to approximate the Caputo fractional derivatives. Here, the main aim of this paper is to use these methods to solve linear multi-term fractional differential equations numerically. To reach our aim, an effcient matrix approach hasbeen provided to solve some well-known multi-term fractional differential equations. Mathematics Subject Classifications (2000): 26A33; 65D30; 65D25; 65D32.

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Section: Concluding Remarks and Future Workmentioning

confidence: 99%

The sine and cosine diffusive representations for Caputo fractional derivative: A matrix approach to multi-term fractional differential equations

Khosravian-Arab

Dehghan

2023

Preprint

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“…Definition 1. The Caputo fractional derivative of a function R(ζ, τ) with a fractional order α is expressed as [18]…”

Section: Preliminariesmentioning

confidence: 99%

“…In this section, we consider the most general type of a nonlinear fractional order differential equation [18]:…”

Section: General Procedures Of Oafmmentioning

confidence: 99%

“…This method introduces an auxiliary function that satisfies an ordinary differential equation optimized to yield approximate solutions for the original problem. The OAFM has been used effectively in various areas, including fluid dynamics, chemical reactions, and population dynamics [18][19][20][21]. In this paper, we aim to combine the optimal auxiliary function method with the Caputo operator in the context of the nonlinear system of Belousov-Zhabotinsky equation.…”

Section: Introductionmentioning

confidence: 99%

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Optimal Auxiliary Function Method for Analyzing Nonlinear System of Belousov–Zhabotinsky Equation with Caputo Operator

Alshehry,

Yasmin,

Ahmad

et al. 2023

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This paper introduces the optimal auxiliary function method (OAFM) for solving a nonlinear system of Belousov–Zhabotinsky equations. The system is characterized by its complex dynamics and is treated using the Caputo operator and concepts from fractional calculus. The OAFM provides a systematic approach to obtain approximate analytical solutions by constructing an auxiliary function. By optimizing the parameters of the auxiliary function, an approximate solution is derived that closely matches the behavior of the original system. The effectiveness and accuracy of the OAFM are demonstrated through numerical simulations and comparisons with existing methods. Fractional calculus enhances the understanding and modeling of the nonlinear dynamics in the Belousov–Zhabotinsky system. This study contributes to fractional calculus and nonlinear dynamics, offering a powerful tool for analyzing and solving complex systems such as the Belousov–Zhabotinsky equation.

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“…The principle is to transform the FD into an equivalent improper integral, and subsequently transform this improper integral into a system of ordinary differential equations by applying the Gauss-Laguerre or Gauss-Jacobi quadrature formulas. Although the memory-free method is further improved in Ref [20][21][22], the convergence of the method is not ideal. Therefore, a novel decomposition strategy is proposed to achieve exponential convergence, which solves poor accuracy caused by slow decay rate.…”

Section: Introductionmentioning

confidence: 99%

Fast Dynamical Analysis of High-dimensional Fractional-Order Memristive-Coupled Hopffeld Neural Networks utilizing a modiffed Non-Classical Methods

Liu,

Chen,

Liu

2023

Preprint

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Solving high-dimensional fractional differential equations has consistently remained one of the most signiffcant challenges within the realm of fractional calculus. Classical numerical methods often grapple with extensive history datasets, resulting in decreased computational efffciency. Existing non-classical methods frequently result in huge dimensional integer-order systems. To address these issues, an improved non-classical method is established in this paper. By introducing an integral interval decomposition strategy, the decay rate of the integrand in the non-classical method is signiffcantly accelerated, achieving a transition from algebraic decay to exponential decay. Consequently, there is a substantial reduction in the dimensionality of the approximate integerorder systems. Simultaneously, the introduction of the interpolation quadrature method further improves the computational accuracy. Theoretical analysis and numerical examples demonstrate that the proposed method offers substantial performance advantages in tackling fractional differential equations. In conclusion, the dynamic behavior of a high-dimensional memristive-coupled Hopffeld neural network was effectively investigated using the proposed method by phase diagrams and bifurcation diagrams. The proposed method demonstrates the capability to accurately identify both chaotic and periodic behavior within the system. Furthermore, it offers a substantial reduction in computation time for high-dimensional systems. This efffciency in capturing complex dynamics while decreasing computational overhead can be highly advantageous in various applications. It holds crucial theoretical implications for the numerical solution of fractional differential equation. Furthermore, it has the potential to foster and stimulate research and practical applications within the realm of fractional order systems.

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An improved Yuan–Agrawal method with rapid convergence rate for fractional differential equations

Cited by 6 publications

References 28 publications

The sine and cosine diffusive representations for Caputo fractional derivative: A matrix approach to multi-term fractional differential equations

The sine and cosine diffusive representations for Caputo fractional derivative: A matrix approach to multi-term fractional differential equations

Optimal Auxiliary Function Method for Analyzing Nonlinear System of Belousov–Zhabotinsky Equation with Caputo Operator

Fast Dynamical Analysis of High-dimensional Fractional-Order Memristive-Coupled Hopffeld Neural Networks utilizing a modiffed Non-Classical Methods

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