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

McLean, William; Mustapha, Kassem; Ali, Raed; Knio, Omar

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2019-08-27

Online Publication Date

2019-08-27

Print Publication Date

2019-08

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http://hdl.handle.net/10754/656731

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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.

Citation

McLean, W., Mustapha, K., Ali, R., & Knio, O. M. (2020). Regularity theory for time-fractional advection–diffusion–reaction equations. Computers & Mathematics with Applications, 79(4), 947–961. doi:10.1016/j.camwa.2019.08.008

Sponsors

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.