Fast Algorithms For Two Kinds Of Fractional Partial Differential Equations

Fractional calculus studies the non-standard operator theory of differential and integral of arbitrary order and its application.It is a natural extension of integer-order standard calculus.It develops very slowly due to the many definitions of fractional calculus and the lack of practical application background and clear geometric and physical interpretation.With the development of science and technology,people have found that fractional-order models are more accurate than integer-order models in describing some phenomena and reflecting some properties of objects when studying practical problems.Therefore,the theory of fractional calculus and fractional order equations has been widely concerned and developed,and has been applied to many different fields.It has unique significance for the study of fractional calculus and differential equations.In particular,the fractional differential equations abstracted from practical problems have become the hots pot of current research.Since the fractional derivative is a non-local operator,it is dependent on the historical state of the initial moment when it is characterized.This makes the solution of fractional-order equations very difficult,especially for high-dimensional problems,the storage and computational cost are even more unimaginable.Therefore,for this kind of problem,whether it is direct solution or traditional iterative method,there is no advantage of solving.This paper proposes a fast calculation method for two kinds of space fractional diffusion equations.The main contents of this paper are:The first chapter briefly introduces the background of the fractional diffusion equation,the significance of the research,the basis of the topic selection and the analysis of the research status at home and abroad.The second chapter briefly introduces some basic concepts,basic properties and and several conclusions involved in format derivation and theoretical proof.In the third chapter,we construct a finite volume format for a one-dimensional spatial fractional diffusion equations and combine the Toeplitz matrix,fast Fourier transform(FFT)and circulant preconditions.We further propose a fast BICGSTAB method,which shows that it can effectively reduce the amount of calculation and storage and give the corresponding numerical experimental results.In the fourth chapter,we construct the finite volume format and combine the Toeplitz matrix and the circulant preconditions for the two-dimensional spatial fractional diffusion equation.We also propose the corresponding fast BICGSTAB method and give the corresponding numerical experimental results.The fifth chapter summarizes the work done and looks forward to the future.