Fast Method For Partial Differential Coupled Equations And Fractional Equations And Its Applications

This thesis mainly studies numerical schemes and fast methods for several partial differential coupled models and fractional equations,as well as their applications in magnetohydrodynamics and hemodynamics.The thesis creatively proposes various fast methods to solve partial differential coupled models by utilizing numerical methods such as finite element method and spectral method.This thesis applies the discrete Gronwall inequality to theoretical analysis,proves the stability and convergence of numerical schemes,and provides mechanism analysis,providing theoretical support and analytical framework for the development of fast method.A correction method is proposed to address the singularity problem of time fractional models at t=0.A multi-parameter estimation method is proposed based on the Bayesian technique.This thesis takes the lead in introducing fast method into the calculations of fractional magnetohydrodynamic coupled models and hemodynamic coupled models,promoting interdisciplinary cooperation in fields such as mathematics,physics,electromagnetics,and hemodynamics.Specifically:In Chapter 1,we provide a concise introduction to the research background of fractional calculus theory and coupled model.It explains the development and research significance of magnetohydrodynamics and hemodynamics,and outlines the main research objectives.In Chapter 2,a coupled model is established to describe the flow and heat transfer of fractional Oldroyd-B fluid between parallel plates by introducing fractional derivative into the constitutive equation.A numerical scheme and fast method for solving the coupled model are proposed based on the Ll-spectral collocation point method,and exponential sum rule.In Chapter 3,we investigate the fractional Maxwell fuid flow and heat transfer coupled model and the fast method.Creatively introducing the discrete Gronwall inequality into theoretical analysis,demonstrating the stability and convergence of numerical schemes,as well as the stability analysis of fast method.The singularity problem of the fractional model at t=0 is solved by adding correction terms.In Chapter 4,numerical research is conducted on the flow and heat transfer of space fractional magnetic fluid.A numerical scheme and fast method based on the matrix function vector product method of the spectral method are proposed.We successfully estimate the space fractional order parameters in the coupled model by using the Bayesian technique,and solve the problem of multi-parameter estimation in fractional coupled models.In Chapter 5,we investigate the flow and heat transfer problems of fractional Jeffrey fluid in a two-dimensional irregular straight channels,and build the numerical scheme and the fast method to solve the space-time fractional coupled model.By combining the L2-1α method with the unstructured mesh finite element method,a fully discrete scheme is obtained,and its stability and convergence are demonstrated.In Chapter 6,we study the blood flow in the human brain tissue blood vessels,establish a one-dimensional coupled model to describe blood flow,and propose a fast method based on reduced-order extrapolation algorithm to improve the computing efficiency.The effectiveness of the fast method is verified through numerical examples,and the changes of vascular cross-sectional area,volume flow,and pulsatile blood pressure within a cardiac cycle are elucidated.In Chapter 7,we conclude the content presented in this thesis and suggest the research proposal in the future.