The Numerical Algorithms For Tempered Fractional Convection-diffusion Equation With Variable Coefficients

In this paper,two kinds of numerical algorithms for tempered fractional convection-diffusion equations with variable coefficients are studied.In the first chapter,we introduce some research achievements have been obtained and the research background,and give the definition of fractional derivatives needed in this paper.In the second chapter,we study the variable-coefficient two-sided space tempered frac-tional diffusion equation with a nonlinear source term,as for the space partial derivative,the central difference formula is used to discretize first order space partial derivative,the second order tempered and weighted and shifted difference operator is used to approximate the left and right Riemann-Liouville space tempered fractional partial derivative,as for the time partial derivative,the first order time partial derivative is discretized by implicit midpoint method,the numerical scheme for solving this kind equation is constructed,we prove the stability and convergence of the numerical algorithm in the sense of L₂norm,and the time and space convergence order of the numerical scheme is second order.The numerical exper-iments are consistent with the theoretical analysis,which shows that the numerical scheme is effective.In the third chapter,we study the variable coefficient time Caputo tempered frac-tional convection-diffusion equation with time delay,the central difference formula is used to discretize the first order space partial derivative,and the space Riesz fractional partial derivative is approximated by the second difference formula,then,the fractional-compact Gr¨nwald second order tempered difference operator is used to approximate the time tem-pered fractional partial derivatives.Thus,a numerical algorithm is constructed to solve this equation.The stability and convergence of the numerical algorithm in the sense of L_∞ norm are proved.Numerical examples show the effectiveness of the algorithm.