Extremely low order time-fractional differential equation and application in combustion process

•A time-fractional combustion model with exponential source term is proposed.•The blow-up phenomenon is theoretical proved and its time is calculated accurately.•An adaptive-finite-difference/discontinuous-Galerkin mixed method is implemented.•Extremely low order fractional reaction diffusion equati...

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Bibliographic Details
Published in:Communications in nonlinear science & numerical simulation 2018-11, Vol.64, p.135-148
Main Authors: Xu, Qinwu, Xu, Yufeng
Format: Article
Language:English
Subjects:
Adaptive finite difference method
Blow-up phenomenon
Cauchy problems
Combustion chambers
Combustion process
Computer simulation
Differential equations
Diffusion
Finite difference method
Finite element analysis
Fractional reaction–diffusion equation
Fractions
Galerkin method
Mathematical models
Numerical methods
Simulation
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Summary:•A time-fractional combustion model with exponential source term is proposed.•The blow-up phenomenon is theoretical proved and its time is calculated accurately.•An adaptive-finite-difference/discontinuous-Galerkin mixed method is implemented.•Extremely low order fractional reaction diffusion equation is simulated. Fractional blow-up model, especially which is of very low order of fractional derivative, plays a significant role in combustion process. The order of time-fractional derivative in diffusion model essentially distinguishes the super-diffusion and sub-diffusion processes when it is relatively high or low accordingly. In this paper, the blow-up phenomenon and condition of its appearance are theoretically proved. The blow-up moment is estimated by using differential inequalities. To numerically study the behavior around blow-up point, a mixed numerical method based on adaptive finite difference on temporal direction and highly effective discontinuous Galerkin method on spatial direction is proposed. The time of blow-up is calculated accurately. In simulation, we analyze the dynamics of fractional blow-up model under different orders of fractional derivative. It is found that the lower the order, the earlier the blow-up comes, by fixing the other parameters in the model. Our results confirm the physical truth that a combustor for explosion cannot be too small.
ISSN:1007-5704
1878-7274
DOI:10.1016/j.cnsns.2018.04.021