### Abstract

Singular source terms represented by the Dirac delta function are found in various applications modeling natural problems. Solutions to differential equations perturbed by such singular source terms have jump discontinuity and their high order numerical approximations suffer from the Gibbs phenomenon. We use the Schwartz duality to approximate the Dirac delta function existent in fractional differential equations. The singular source term is approximated by the fractional derivative of the Heaviside function. We provide a Chebyshev spectral collocation method for solving the fractional advection equation with the singular source term and show that the Schwartz duality yields the consistent formulation resulting in vanishing Gibbs phenomenon. The numerical results show that the proposed approximation of the Dirac delta function is efficient and accurate, particularly for linear problems.

Original language | English (US) |
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Pages (from-to) | 205-212 |

Number of pages | 8 |

Journal | Applied Mathematics Letters |

Volume | 64 |

DOIs | |

State | Published - Feb 1 2017 |

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### Keywords

- Chebyshev spectral collocation method
- Dirac delta function
- Fractional derivative
- Schwartz duality

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

**Schwartz duality of the Dirac delta function for the Chebyshev collocation approximation to the fractional advection equation.** / Yang, He; Guo, Jingyang; Jung, Jae Hun.

Research output: Contribution to journal › Article

*Applied Mathematics Letters*, vol. 64, pp. 205-212. https://doi.org/10.1016/j.aml.2016.09.009

}

TY - JOUR

T1 - Schwartz duality of the Dirac delta function for the Chebyshev collocation approximation to the fractional advection equation

AU - Yang, He

AU - Guo, Jingyang

AU - Jung, Jae Hun

PY - 2017/2/1

Y1 - 2017/2/1

N2 - Singular source terms represented by the Dirac delta function are found in various applications modeling natural problems. Solutions to differential equations perturbed by such singular source terms have jump discontinuity and their high order numerical approximations suffer from the Gibbs phenomenon. We use the Schwartz duality to approximate the Dirac delta function existent in fractional differential equations. The singular source term is approximated by the fractional derivative of the Heaviside function. We provide a Chebyshev spectral collocation method for solving the fractional advection equation with the singular source term and show that the Schwartz duality yields the consistent formulation resulting in vanishing Gibbs phenomenon. The numerical results show that the proposed approximation of the Dirac delta function is efficient and accurate, particularly for linear problems.

AB - Singular source terms represented by the Dirac delta function are found in various applications modeling natural problems. Solutions to differential equations perturbed by such singular source terms have jump discontinuity and their high order numerical approximations suffer from the Gibbs phenomenon. We use the Schwartz duality to approximate the Dirac delta function existent in fractional differential equations. The singular source term is approximated by the fractional derivative of the Heaviside function. We provide a Chebyshev spectral collocation method for solving the fractional advection equation with the singular source term and show that the Schwartz duality yields the consistent formulation resulting in vanishing Gibbs phenomenon. The numerical results show that the proposed approximation of the Dirac delta function is efficient and accurate, particularly for linear problems.

KW - Chebyshev spectral collocation method

KW - Dirac delta function

KW - Fractional derivative

KW - Schwartz duality

UR - http://www.scopus.com/inward/record.url?scp=84989837131&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84989837131&partnerID=8YFLogxK

U2 - 10.1016/j.aml.2016.09.009

DO - 10.1016/j.aml.2016.09.009

M3 - Article

AN - SCOPUS:84989837131

VL - 64

SP - 205

EP - 212

JO - Applied Mathematics Letters

JF - Applied Mathematics Letters

SN - 0893-9659

ER -