A High-Accuracy Fast Algorithm For Space Fractional Diffusion Equation And Its Theoretical Analysis

Fractional calculus is a mathematical extension of the standard-order integer-order integro-differential equations.It deals with arbitrary order differential and integral non-standard operator theory and its applications.It is an important branch of mathematical analysis.In recent years,the theory of fractional calculus and fractional differential equation has been continuously developed and improved.In many fields,they have been widely used,such as physics,mathematics and engineering.The study of fractional calculus and fractional differential equations has very important theoretical significance and practical application value.In particular,fractional differential equations abstracted from practical problems have become a hot topic for many mathematicians.As we all know,the range of fractional derivatives is increasingly used,but there are still drawbacks in the solution of its numerical solutions,mainly as follows: These numerical calculation difficulties are essentially due to the non-local of the fractional differential algorithm.In particular,for the high-dimensional problem of it difficult to measure the storage and computational costs.Therefore,both the direct solution and the Krylov subspace iteration method for such problems lose their advantages in solving integer differential equations.Therefore,this article will propose an efficient and quick solution.The main contents of this paper are:In the first chapter,the background of the fractional diffusion equation,the significance of the research and the analysis of the research status at home and abroad are briefly introduced.In the second chapter,the theoretical knowledge of fractional diffusion equations is introduced.For the one-dimensional and two-dimensional two-sided spatial diffusion equations,the related finite difference scheme is briefly reviewed.In the third chapter,according to the model of one-dimensional fractional differential equations,a corresponding quadratic spline discretization scheme is constructed.In addition to this,we use numerical experiments to give numerical errors,operation times,and average iteration times for Gaussian elimination and Bi-CGSTAB in the implicit Euler format.In contrast,it can be seen that Bi-CGSTAB is better than Gaussian under the same split.In the fourth chapter,based on the Bi-CGSTAB algorithm,we can obtain the fast quadratic spline configuration method(fast Bi-CGSTAB)combined with the fast matrix-vector product and cyclic precondition,and the secondary of T.Chan preconditioner.The T.Chan preconditioned fast Bi-CGSTAB method is more effective in dealing with the fractional diffusion equation.In the fifth chapter,we apply fast Bi-CGSTAB and T.Chan preconditioned fast Bi-CGSTAB to two-dimensional fractional diffusion equations.Through numerical experiments,Gaussian,Bi-CGSTAB and fast Bi-CGSTAB in implicit Euler format are given.The numerical results of Bi-CGSTAB and T.Chan preconditioned fast Bi-CGSTAB are compared with the numerical error,operation time and average number of iterations.