| Fractional calculus,which includes fractional integral and fractional differential,is usually regarded as an extension of traditional integral calculus theory.For quite a long time,fractional derivative theory,as a pure mathematical theory,developed slowly and was rarely used in science and engineering.Until the 1970s,mathematician Mandelbrot put forward the fractal theory and used RiemannLiouville fractional calculus to study Brownian motion,and the fractional differential equations became particularly noticeable.The fractional derivative provided an accurate description of the memory properties and genetic effects of various materials and processes.For example,the time fractional derivative is related to particle adhesion and capture,and the space fractional derivative can be used to simulate long particle jumps.Nowadays,differential equations with fractional order operators are often encountered in various scientific and engineering fields,such as physics,finance,hydrology,etc.,which has attracted extensive attention.Fractional derivatives and integrals are convolutions with a power law,which can be multiplied by an exponential factor to obtain tempered fractional derivatives and integrals.Because of the nonlocality of fractional operator,it is difficult to obtain the analytical solution of such equation,we need to develop effective numerical methods to solve the tempered fractional differential equations.This paper mainly study the numerical solutions of two kinds of Caputo-tempered time fractional differential equations.The research in this paper is divided into four parts:The first part is the introduction,which introduces the research status of tempered fractional differential equation at home and abroad,and briefly describes the research contents and structure of this paper.In the second part,the one-dimensional time tempered fractional sub-diffusion equation is studied.The Caputo-tempered time fractional derivative is discretized by using the tempered L1 scheme,and the compact operator is used to act on the spatial derivative.Then the compact difference scheme of one-dimensional time tempered fractional sub-diffusion equation is obtained.The stability and convergence of the difference scheme are proved by energy method.Finally,the validity of the numerical scheme is verified by constructing the analytical solution of the differential equation and taking different tempered parameters and the order of fractional derivatives.In the third part,the one-dimensional time tempered fractional order convectiondiffusion equation is studied.The convection term and diffusion term are approximated by the compact exponential finite difference scheme,and the Caputotempered time fractional derivative is discretized by using the tempered L1 scheme.Then the high order compact exponential difference scheme of one-dimensional time tempered fractional order convection-diffusion equation is obtained,and the stability of the difference scheme is proved by Fourier analysis method,the convergence of the difference scheme is proved by matrix analysis method.Finally,the validity of the numerical scheme is verified by constructing the analytical solution of the differential equation and taking different tempered parameters and the order of fractional derivatives.In the fourth part,the two-dimensional time tempered fractional sub-diffusion equation is studied.The Caputo-tempered time fractional derivative is discretized by using the tempered L1 scheme,and the compact operator is used to act on the spatial derivative.Then the compact ADI difference scheme of the twodimensional time tempered fractional sub-diffusion equation is obtained.The stability and convergence of the difference scheme are proved by using the energy method.The fifth part is the summary of the this paper and the prospect of the future work. |
