Fast Solutions Method For Several Kinds Of Fractional Partial Differential Equations And Applications

In scientific and engineering calculations,there are many large-scale application problems that require physics and mathematics workers to construct models for numerical simulation.By analyzing the model,designing numerical calculation methods,and quickly solving,so as to have a further understanding of the phenomenon under study.In many fields,Partial Differential Equation(PDE)is the most commonly used mathematical model to describe problems.In recent years,due to the nonlocality of fractional differential operators,it has become an important tool to describe models more accurately.Compared with integer-order partial differential equations,fractional-order partial differential equations are more accurate in simulating memory effects,diffusion motion,and genetic properties.Therefore,they are widely used in abnormal diffusion,signal processing,superconductivity,and fluid mechanics.Different from integer-order partial differential equations,the coefficient matrix of the linear system after discrete fractional partial differential equations is very dense,resulting in a huge amount of calculation and memory requirements,and a long calculation time,which creates many obstacles for in-depth research.How to use numerical methods to discretize,and through the study of the algebraic structure of the coefficient matrix of the discrete system,designing efficient,fast,and stable fast algorithms has become the first and most critical issue for simulation in practical engineering applications.This paper mainly studies two types of fast solving problems of fractional partial differential equations and two types of practical applications based on fractional models.By constructing the numerical scheme,and aiming at the structural characteristics of the linear system obtained by the discretization of different models,a series of efficient and economical fast solving strategies are designed using matrix splitting iterative method or preconditioning technology to numerically solve different types of fractional partial differential equations.The full text has seven chapters:The first chapter details the scientific background,research significance,and research status of the research topic,and briefly introduces the main research content and innovations of this article.The second chapter mainly studies the fast iterative solution of the fractional Ginzburg-Landau equation in one-dimensional space.Through finite difference discretization,a complex linear system of Toeplitz structure is obtained,and a new type of matrix splitting iterative method is designed according to the structural characteristics of the system,and fast calculation is realized through circulant preconditioner.The convergence of the iterative method proves the effectiveness of the method.Numerical experiments verify the efficiency of the proposed method.The third chapter mainly studies the fast numerical solution of the fractional Ginzburg-Landau equation in two-dimensional space.First of all,an alternate direction implicit(ADI)difference scheme is designed for the equation,the equation is discretized into two linear subsystems.Then,a matrix splitting iterative method with complex respective scaling is designed to solve it quickly,we also prove its convergence.Numerical experiments show the economy and efficiency of the proposed method.The fourth chapter mainly studies the two-dimensional time space fractional Stokes equations,and constructs a fast solution method for discrete linear systems.Through finite difference discretization,the linear system coefficient matrix contains the embedded block-Toeplitz-Toeplitz-block with Saddle point structure.This new solution method has computational advantages because circulant approximation and fast Fourier transform(FFT)can be used to solve the linear subsystems involved.Theoretical analysis shows that most of the eigenvalues of the preconditioned coefficient matrix are clustered around 1,which shows that the proposed preconditioner converges very quickly.Numerical examples confirm the effectiveness of the new method.Chapter 5 mainly studies the image restoration model based on the two-dimensional space fractional Cahn-Hilliard equation.Through finite difference discretization and JFNK(Jacobian-free-Newton-Krylov)iteration,the indefinite matrix are obtained which is of BTTB embed in 2 × 2 block-structured Jacobian matrix.We constructed a class of efficient preconditioners.The eigenvalues of the corresponding preconditioned system are gathered in the vicinity of 1.Further,through the circulant approximation of the BTTB structure,it can be quickly calculated by FFT,which reflects its more economical and efficient characteristics.Finally,numerical experiments verify that the proposed fast algorithm is efficient in the image inpainting robustness.The sixth chapter mainly studies the fast numerical solution of the time space fractional Black-Scholes pricing model based on the two European options.First,the model is discretized by the implicit difference method,and after studying the structure of the linear system obtained,we proposed a class of bi-diagonal-all-at-once preconditioners.Theoretical analysis and experiments show that the spectrum of the preconditioned system is concentrated around 1,which can effectively accelerate the Krylov subspace method to solve linear systems.Numerical experiments show the efficiency of the proposed method.In chapter 7,we briefly summarizes the whole paper and looks forward to future work.