Numerical Methods With High Order Temporal Accuracy For Fractional Diffusion Type Equations

Fractional calculus theory is an important branch of mathematics,it was born almost at the same time as the classical calculus theory and can be regarded as an important extension of classical calculus.In the past two decades,due to the non-locality of fractional differential operators and their applicability in the characterization of materials with memory and genetic prop-erties,in the modeling process in many fields including natural sciences and engineering,fractional differential operators has received extensive attention and has been widely used.Since most fractional differential equations cannot obtain the analytical solutions,and even if the analytical solutions can be obtained,but the analytical solutions usually contain special functions that are extremely complex and difficult to compute,such as Mittag-Leffler functions,Wright functions,etc.Therefore,numerically solving fractional differential equations becomes an imminent task for numerical computing workers.For the fractional diffusion type equations studied in this paper,in the past ten years,numerical workers have proposed some numerical methods,and have made certain contributing.So far,how to construct high order accuracy numerical method for solv-ing fractional diffusion type equations,compared with the spatial accuracy,the key of the problem is to improve the temporal accuracy.However,before 2015,the numerical methods proposed have at most second order temporal accuracy,and has not seen the construction method based on the second order compact approximation formula based of first derivative.In the Chapter 1 of this paper,we described the development history of fractional calculus,expound the research content and motivation of this paper,the concept and some properties of common fractional derivative are introduced;In Chapter 2 and Chapter 4 of this paper,based on the numerical integration method,we propose the numerical method for solving modified fractional diffusion equation and the numerical method for solving fractional anomalous subdiffusion equation,respectively;In Chapter 3,Chapter 5 and Chapter 6 of this paper,based on the second order compact approximation formula of the first derivative,we propose the numerical method for solving modified fractional diffusion equation,the numerical method for solving fractional reaction-subdiffusion equation and the numerical method for solving fractional Cable equation,respectively.The five numer-ical methods presented in this paper all have second order temporal accuracy and fourth order spatial accuracy.Applying algebraic theory,the solvability of the five numerical methods are verified,respectively.Through Fourier analysis,we respectively analyzed the stability and convergence of the five numerical methods.For the five numerical meth-ods,we also performed numerical experiments,and the numerical experiments strongly support our theoretical analysis results.