Numerical Methods For Normalized Tempered Fractional Diffusion Equations

In this paper,we mainly study the numerical methods of space normalized tempered fractional diffusion equations developed from space fractional diffusion equations.In the first chapter,we introduce the definitions of the fractional derivatives that appear in this article,the mathematical model studied and its research status.In the second chapter,we study the third-order numerical scheme of the one-sided space normalized tempered fractional diffusion equations with drift.Based on the existing third-order quasi-compact algorithm for the one-sided space normalized tempered fractional diffusion equations,the idea of this algorithm is applied to solve the one-sided space normalized tempered fractional diffusion equations with drift,and the numerical scheme is derived.The energy method is used to obtain the stability and convergence results of the numerical scheme in the sense of the L₂ norm.The numerical scheme is validated by numerical experiments.In the third chapter,we study the Crank-Nicolson quasi-compact scheme of the one-sided space normalized tempered fractional diffusion equations with drift.Applying the idea of the third-order quasi-compact algorithm and using the Crank-Nicolson method to discrete time partial derivative,the numerical scheme for solving the one-sided space normalized tempered fractional diffusion equations with drift is derived.The energy method is used to obtain the stability and convergence results of the numerical scheme in the sense of the L₂ norm.The validity of the numerical scheme is demonstrated through numerical experiments.In the fourth chapter,we study the second-order numerical methods of the two-sided space normalized tempered fractional diffusion equations.First,a class of second-order tempered difference operators for the left and right Riemann-Liouville tempered fractional derivative are constructed.Then the second-order tempered dif-ference operators are used to approximate the space tempered fractional derivatives,and the Crank-Nicolson method is used to discrete time partial derivative,thereby a class of second-order numerical schemes for solving the two-sided space normalized tempered fractional diffusion equations are derived.The stability results of numer-ical schemes are obtained by the matrix method,and the convergence results of numerical schemes in the sense of the L₂ norm are obtained by the energy method.The validity of numerical schemes are confirmed by numerical experiments.