Quenching phenomenon in a fractional diffusion equation and its numerical simulation

ABSTRACT

In this paper, we introduced a fractional diffusion equation with degenerate source term. The fractional derivative, acting on the spatial variable, contains left-sided Caputo derivative and right-sided Riemann–Liouville derivative simultaneously. Due to the symmetry property of such fractional derivative pattern, the existence of quenching phenomenon is verified and the quenching time is estimated under certain settings of parameters. An numerical example is carried out by using a finite-difference scheme with uniform and non-unique mesh, which demonstrates the theoretical analysis of this model and presents several significant relation between order of derivatives and size of spatial domain.

Keywords:

• Fractional derivatives
• degenerate diffusion equation
• quenching phenomenon
• numerical simulation
• numerical combustion process

2010 MSC Subject Classifications:

• 26A33
• 34A08

Acknowledgments

The author is grateful to the referees for their constructive comments and suggestions which help to improve the quality of this paper. The author would like to sincerely thank Professor Qin Sheng (Baylor University, USA) and Professor Hai-Wei Sun (University of Macau, Macao) for their kind help and continuous encouragement in recent years.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 We sincerely thank the reviewer for providing many concrete applications of fractional derivative, which absolutely bring a more comprehensive understanding of fractional calculus.

2 In this case, Ψ is a single-valued function. We still use the same notation ${\mathrm{\partial }}_{x\pm }^{\alpha }$ to depict fractional derivative of Ψ with respective to x. Notice that the partial fractional derivative becomes the usual Caputo fractional derivative for single-variable function. This shall not arise any confusions later on.

Additional information

Funding

This work is partly supported by the National Natural Science Foundation of China [No. 11501581], the ‘2+6’ Program [No. 502042032] of CSU, and the Project funded by China Postdoctoral Science Foundation [No. 2015M570683].