Numerical treatment of tempered space-fractional Zeldovich-Frank-Kamenetskii equation

This paper develops a spectral numerical scheme for the tempered space-fractional Zeldovich-Frank-Kamenetskii equation, which generalizes the classical combustion model by incorporating spatially tempered fractional derivatives to capture anomalous diffusion with exponentially weighted singularities. The method combines a Legendre polynomial-based spectral collocation approach for spatial discretization, using the optimal convergence properties and the exact boundary enforcement, with an implicit Runge-Kutta scheme for temporal integration. By deriving tempered fractional differentiation matrices tailored to shifted Legendre bases, the framework preserves spectral accuracy. As the first application of Legendre spectral methods to this tempered fractional system, the approach demonstrates superior accuracy and computational efficiency, validated through rigorous benchmark tests.