Some Numerical Methods For The Time Fractional Subdiffusion Equations With Nonsmooth Data

As an important part of differential equation theory,fractional differential equations have important research significance.Because of the non-locality and memory,fractional differential equations can better simulate some natural phenomena and movement process of object.Therefore,fractional differential equations are widely used in many fields.In recent years,the numerical solution of fractional differential equations has been developed rapidly,but the high-order numerical methods generally rely on the smoothness of exact solution and initial data.When the exact solution is not smooth or the initial data is not smooth,the accuracy of the numerical method will decrease.To solve this problem,the numerical solution of the one-dimensional Capfuto time fractional diffusion equation is studied in this paper.The main work is as follows:In Chapter one,the research significance of fractional differential,the research status and motivation of fractional differential equation numerical solution and the research content of this article are given.In chapter two,the preliminary knowledge related to this article is given,includ-ing several fractional derivatives,the definition and properties of the Mittag-Leffler function and the Laplace transformation,and related theorems.In Chapter three,the one-dimensional homogeneous time fractional subdiffusion e-quation is studied.Based on the weighted C-N scheme,the second order modified weighted C-N scheme that does not require smoothness for the initial data and the exact solution is proposed.Then the convergence and stability analysis of the modified weighted C-N scheme is given.Finally,several numerical experiments confirm that when the initial data and the exact solution are not smooth,the weighted C-N scheme has only first-order accuracy in time,but the modified weighted C-N scheme still maintains second-order accuracy in time,which verifies the reliability,effectiveness and accuracy of the method.In Chapter four,the one-dimensional nonhomogeneous time fractional subdiffusion equation is studied.Based on the nonhomogeneous weighted C-N scheme,the nonhomo-geneous modified weighted C-N scheme is proposed.The convergence analysis is given.Finally,several numerical experiments verifies the reliability,effectiveness and accuracy of the method.In Chapter five,the one-dimensional homogeneous and nonhomogeneous time frac-tional partial differential equations are studied.Based on the modified weighted C-N scheme and the nonhomogeneous modified weighted C-N scheme,the new type of mod-ified weighted C-N schemes are proposed.The convergence analysis shows that the new type of modified weighted C-N scheme maintains second-order accuracy in time.Finally,several numerical experiments verifies the reliability,effectiveness and accuracy of the method.