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

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• 作者：He Yang^a ; ^{yang.1671@osu.edu" class="auth_mail" title="E-mail the corresponding author} ; Jingyang Guo^b ; ^{jguo4@buffalo.edu" class="auth_mail" title="E-mail the corresponding author} ; Jae-Hun Jung^b ; ^{jaehun@buffalo.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Fractional derivative ; Schwartz duality ; Chebyshev spectral collocation method ; Dirac delta function
• 刊名：Applied Mathematics Letters
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：64
• 期：Complete
• 页码：205-212
• 全文大小：897 K

文摘

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.