Spectral approximations to the fractional integral and derivative

By Changpin Li, Fanhai Zeng, Fawang Liu. 2012

165 citations recorded by the source corpus.

In this paper, the spectral approximations are used to compute the fractional integral and the Caputo derivative. The effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the fractional integral. And the succinct scheme for approximating the Caputo derivative is also derived. The collocation method is proposed to solve the fractional initial value problems and boundary value problems. Numerical examples are also provided to illustrate the effectiveness of the derived methods.

Subfields: Mechanics of Materials, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/65a22e9d57ddab89fd53c90fced2bd81faf41c79

How researchers see this paper

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In Mechanics of Materials

1. Minhao Cai
2. Changpin Li
3. Bartłlomiej Dyda
4. Fawang Liu
5. Ian L Turner
6. Vo Anh
7. Kevin Burrage
8. Fanhai Zeng

In Modeling and Simulation

1. Changpin Li
2. Fawang Liu
3. Ian L Turner
4. Vo Anh
5. Minhao Cai
6. Fanhai Zeng
7. Kevin Burrage
8. Bartłlomiej Dyda

In Numerical Analysis

1. Changpin Li
2. Fawang Liu
3. Ian L Turner
4. Vo Anh
5. Minhao Cai
6. Fanhai Zeng
7. Kevin Burrage
8. Bartłlomiej Dyda

Citations in the corpus

• Numerical Approaches to Fractional Integrals and Derivatives: A Review (2020, methodology): “Generalizing the above approaches, we can derive spectral approximations [42].”
• Fractional calculus for power functions and eigenvalues of the fractional Laplacian (2012, result): “For results similar to Theorem 1 concerning fractional derivatives, we refer the reader to [35], [23], [31].”
• An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data (2017, unknown): “1), see [1], [5], [13], [14], [34], [27], [59], [63], [67].”
• Fractional calculus for power functions and eigenvalues of the fractional Laplacian (2012, result): “For results similar to Theorem 1 concerning fractional derivatives, we refer the reader to [35], [23], [31].”
• A Crank-Nicolson ADI Spectral Method for a Two-Dimensional Riesz Space Fractional Nonlinear Reaction-Diffusion Equation (2014, methodology): “Up to now, there has been some work on numerical methods for FDEs, such as finite difference methods [12, 20, 26, 47], finite element methods [5, 6, 11, 14, 42], spectral methods [13, 15, 16], and so on [28, 29, 30].”