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

He Yang, Jingyang Guo, Jae Hun Jung

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)205-212
Number of pages8
JournalApplied Mathematics Letters
Volume64
DOIs
StatePublished - Feb 1 2017

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Chebyshev approximation
Delta functions
Advection Equation
Dirac delta function
Advection
Source Terms
Chebyshev
Collocation
Duality
Fractional
Gibbs Phenomenon
Differential equations
Approximation
Heaviside step function
Higher Order Approximation
Fractional Differential Equation
Fractional Derivative
Spectral Methods
Collocation Method
Numerical Approximation

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.

In: Applied Mathematics Letters, Vol. 64, 01.02.2017, p. 205-212.

Research output: Contribution to journalArticle

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