Quenching of combustion explosion model with balanced space‐fractional derivative

摘要

>Balanced space‐fractional derivative is usually applied in modelling the state‐dependence, isotropy, and anisotropy in diffusion phenomena. In this paper, we introduce a class of space‐fractional reaction‐diffusion model with singular source term arising in combustion process. The fractional derivative employed in this model is defined in the sum of left‐sided and right‐sided Riemann‐Liouville fractional derivatives. With assistance of Kaplan's first eigenvalue method, we prove that the classic solution of this model may not be globally well‐defined, and the heat conduction governed by this model depends on the order of fractional derivative, the parameters in the equation, and the length of spatial interval. Finite difference method is implemented to solve this model, and an adaptive strategy is applied to improve the computational efficiency. The positivity, monotonicity, and stability of the numerical scheme are discussed. Numerical simulation and observation of the quenching and stationary solutions coincide the theoretical studies.