Regularity theory for time-fractional advection–diffusion–reaction equations

William McLean; Kassem Mustapha; Raed Ali; Omar Knio

doi:10.1016/j.camwa.2019.08.008

Regularity theory for time-fractional advection–diffusion–reaction equations

William McLean, Kassem Mustapha, Raed Ali, Omar Knio

Research output: Contribution to journal › Article › peer-review

33 Scopus citations

Abstract

We investigate the behavior of the time derivatives of the solution to a linear time-fractional, advection–diffusion–reaction equation, allowing space- and time-dependent coefficients as well as initial data that may have low regularity. Our focus is on proving estimates that are needed for the error analysis of numerical methods. The nonlocal nature of the fractional derivative creates substantial difficulties compared with the case of a classical parabolic PDE. In our analysis, we rely on novel energy methods in combination with a fractional Gronwall inequality and certain properties of fractional integrals.

Original language	English (US)
Pages (from-to)	947-961
Number of pages	15
Journal	Computers and Mathematics with Applications
Volume	79
Issue number	4
DOIs	https://doi.org/10.1016/j.camwa.2019.08.008
State	Published - Aug 27 2019

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KAUST005
Acknowledgements: The authors thank the University of New South Wales, Australia (Faculty Research Grant “Efficient numerical simulation of anomalous transport phenomena”), the King Fahd University of Petroleum and Minerals, Saudi Arabia (project No. KAUST005) and the King Abdullah University of Science and Technology, Saudi Arabia. ☆ The authors thank the University of New South Wales, Australia (Faculty Research Grant “Efficient numerical simulation of anomalous transport phenomena”), the King Fahd University of Petroleum and Minerals, Saudi Arabia (project No. KAUST005) and the King Abdullah University of Science and Technology, Saudi Arabia.

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title = "Regularity theory for time-fractional advection–diffusion–reaction equations",

abstract = "We investigate the behavior of the time derivatives of the solution to a linear time-fractional, advection–diffusion–reaction equation, allowing space- and time-dependent coefficients as well as initial data that may have low regularity. Our focus is on proving estimates that are needed for the error analysis of numerical methods. The nonlocal nature of the fractional derivative creates substantial difficulties compared with the case of a classical parabolic PDE. In our analysis, we rely on novel energy methods in combination with a fractional Gronwall inequality and certain properties of fractional integrals.",

author = "William McLean and Kassem Mustapha and Raed Ali and Omar Knio",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KAUST005 Acknowledgements: The authors thank the University of New South Wales, Australia (Faculty Research Grant “Efficient numerical simulation of anomalous transport phenomena”), the King Fahd University of Petroleum and Minerals, Saudi Arabia (project No. KAUST005) and the King Abdullah University of Science and Technology, Saudi Arabia. ☆ The authors thank the University of New South Wales, Australia (Faculty Research Grant “Efficient numerical simulation of anomalous transport phenomena”), the King Fahd University of Petroleum and Minerals, Saudi Arabia (project No. KAUST005) and the King Abdullah University of Science and Technology, Saudi Arabia.",

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doi = "10.1016/j.camwa.2019.08.008",

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AU - Ali, Raed

AU - Knio, Omar

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N2 - We investigate the behavior of the time derivatives of the solution to a linear time-fractional, advection–diffusion–reaction equation, allowing space- and time-dependent coefficients as well as initial data that may have low regularity. Our focus is on proving estimates that are needed for the error analysis of numerical methods. The nonlocal nature of the fractional derivative creates substantial difficulties compared with the case of a classical parabolic PDE. In our analysis, we rely on novel energy methods in combination with a fractional Gronwall inequality and certain properties of fractional integrals.

AB - We investigate the behavior of the time derivatives of the solution to a linear time-fractional, advection–diffusion–reaction equation, allowing space- and time-dependent coefficients as well as initial data that may have low regularity. Our focus is on proving estimates that are needed for the error analysis of numerical methods. The nonlocal nature of the fractional derivative creates substantial difficulties compared with the case of a classical parabolic PDE. In our analysis, we rely on novel energy methods in combination with a fractional Gronwall inequality and certain properties of fractional integrals.

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JO - Computers and Mathematics with Applications

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IS - 4

ER -

Regularity theory for time-fractional advection–diffusion–reaction equations

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