Applications of Fractional Differentiation Matrices in Solving Caputo Fractional Differential Equations

Wu, Zhongshu; Zhang, Xinxia; Wang, Jihan; Zeng, Xiangfeng

doi:10.3390/fractalfract7050374

Cited by 7 publications

(5 citation statements)

References 38 publications

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“…The generally accepted notion of the fractional-order integral differential operator is defined as follows [50][51][52][53][54]:…”

Section: Definition Of Conformable Fractional Derivative and Its Char...mentioning

confidence: 99%

Extracting the Ultimate New Soliton Solutions of Some Nonlinear Time Fractional PDEs via the Conformable Fractional Derivative

Iqbal,

Ganie,

Miah

et al. 2024

Fractal Fract

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Nonlinear fractional-order differential equations have an important role in various branches of applied science and fractional engineering. This research paper shows the practical application of three such fractional mathematical models, which are the time-fractional Klein–Gordon equation (KGE), the time-fractional Sharma–Tasso–Olever equation (STOE), and the time-fractional Clannish Random Walker’s Parabolic equation (CRWPE). These models were investigated by using an expansion method for extracting new soliton solutions. Two types of results were found: one was trigonometric and the other one was an exponential form. For a profound explanation of the physical phenomena of the studied fractional models, some results were graphed in 2D, 3D, and contour plots by imposing the distinctive results for some parameters under the oblige conditions. From the numerical investigation, it was noticed that the obtained results referred smooth kink-shaped soliton, ant-kink-shaped soliton, bright kink-shaped soliton, singular periodic solution, and multiple singular periodic solutions. The results also showed that the amplitude of the wave augmented with the pulsation in time, which derived the order of time fractional coefficient, remarkably enhanced the wave propagation, and influenced the nonlinearity impacts.

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“…The generally accepted notion of the fractional-order integral differential operator is defined as follows [50][51][52][53][54]:…”

Section: Definition Of Conformable Fractional Derivative and Its Char...mentioning

confidence: 99%

Extracting the Ultimate New Soliton Solutions of Some Nonlinear Time Fractional PDEs via the Conformable Fractional Derivative

Iqbal,

Ganie,

Miah

et al. 2024

Fractal Fract

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“…These are all integer-order derivatives in those forms. Since the initial values are known and the calculation of complicated fractional-order derivatives is not necessary, this greatly simplifies the problem's solution [26]. (3) In comparison to the other two definitions of fractional order, the Caputo fractional-order derivative in the application of neural networks greatly reduces the amount of calculation.…”

Section: Fractional Parameter Updatementioning

confidence: 99%

An Integer-Fractional Gradient Algorithm for Back Propagation Neural Networks

Zhang,

Xu,

et al. 2024

Algorithms

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This paper proposes a new optimization algorithm for backpropagation (BP) neural networks by fusing integer-order differentiation and fractional-order differentiation, while fractional-order differentiation has significant advantages in describing complex phenomena with long-term memory effects and nonlocality, its application in neural networks is often limited by a lack of physical interpretability and inconsistencies with traditional models. To address these challenges, we propose a mixed integer-fractional (MIF) gradient descent algorithm for the training of neural networks. Furthermore, a detailed convergence analysis of the proposed algorithm is provided. Finally, numerical experiments illustrate that the new gradient descent algorithm not only speeds up the convergence of the BP neural networks but also increases their classification accuracy.

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“…Gholami et al [19] derived a new pseudospectral integration matrix to solve fractional differential equations. Wu et al [20] applied fractional differentiation matrices in solving Caputo fractional differential equations. Moreover, the spectral collocation method has been applied to solve tempered fractional differential equations [21][22][23], variable-order Fokker-Planck equations [24] and Caputo-Hadamard fractional differential equations [25].…”

Section: Introductionmentioning

confidence: 99%

“…10 −7 6.17 1.109 × 10 −6 6 20. 1.968 × 10 −6 6.48 28 4.373 × 10 −8 5.01 1.055 × 10 −7 5.01 2.055 × 10 −7 5.01 3.648 × 10 −7 5.01 36 1.244 × 10 −8 5.00 2.998 × 10 −8 5.01 5.840 × 10 −8 5.01 1.037 × 10 −7 5.01 44 4.559 × 10 −9 5.00 1.099 × 10 −8 5.00 2.140 × 10 −8 5.00 3.800 × 10 −8 5.00 52 1.977 × 10 −9 5.00 4.764 × 10 −9 5.00 9.280 × 10 −9 5.00 1.648 × 10 −8 5.00 60 9.664 × 10 −10 5.00 2.328 × 10 −9 5.00 4.536 × 10 −9 5.00 8.054 × 10 −9 5.00 68 5.168 × 10 −10 5.00 1.245 × 10 −9 5.00 2.426 × 10 −9 5.00 4.307 × 10 −9 5.00…”

mentioning

confidence: 99%

Construction of Fractional Pseudospectral Differentiation Matrices with Applications

Li,

Ma,

Zhao

2024

Axioms

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Differentiation matrices are an important tool in the implementation of the spectral collocation method to solve various types of problems involving differential operators. Fractional differentiation of Jacobi orthogonal polynomials can be expressed explicitly through Jacobi–Jacobi transformations between two indexes. In the current paper, an algorithm is presented to construct a fractional differentiation matrix with a matrix representation for Riemann–Liouville, Caputo and Riesz derivatives, which makes the computation stable and efficient. Applications of the fractional differentiation matrix with the spectral collocation method to various problems, including fractional eigenvalue problems and fractional ordinary and partial differential equations, are presented to show the effectiveness of the presented method.

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Applications of Fractional Differentiation Matrices in Solving Caputo Fractional Differential Equations

Cited by 7 publications

References 38 publications

Extracting the Ultimate New Soliton Solutions of Some Nonlinear Time Fractional PDEs via the Conformable Fractional Derivative

Extracting the Ultimate New Soliton Solutions of Some Nonlinear Time Fractional PDEs via the Conformable Fractional Derivative

An Integer-Fractional Gradient Algorithm for Back Propagation Neural Networks

Construction of Fractional Pseudospectral Differentiation Matrices with Applications

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