The Numerical Algorithms For Tempered Fractional Advection-diffusion Equation

This article discusses the numerical algorithm and theory of two kinds of tempered frac-tional advection-diffusion equations.In the second chapter,the Riesz tempered fractional partial derivative is approximated by second-order modified Lubich tempered difference op-erator,the advection term is discretized by central difference formula and the one-order temporal partial derivative is approached by implicit midpoint formula,thus a numeri-cal algorithm is obtained for solving space-Riesz tempered fractional advection-diffusion equation.The energy method is used to give the stability and convergence analysis of the numerical algorithm,and the effectiveness of numerical algorithm is demonstrated by numerical experiments.In the third chapter,we mainly point at the time Caputo tem-pered fractional advection-diffusion equation.According to the relationship between Ca-puto fractional derivative and Riemann-Liouville fractional derivative,we transform Caputo tempered fractional advection-diffusion equation into Riemann-Liouville tempered fractional advection-diffusion equation.The Riemann-Liouville tempered fractional partial derivative is discretized by fractional-compact Gr¨unwald-Letnikov tempered difference formula,the Riesz fractional partial derivative is approximated by fractional central formula and the ad-vection term is discretized by central difference formula,thus,a new numerical algorithm is constructed.The stability and convergence analysis of the numerical algorithm in the sense of_?norm are analyzed,and the availability of the numerical algorithm is verified by numerical experiments.