Numerical Study For Nonsmooth Solutions Of Two Classes Of Variable-order Fractional Differential Equations

In recent decades,it is found that some complex social and physical phenomena are difficult to be modeled by general fractional differential equations.Hence,variable-order fractional differential equations are introduced.Variable-order fractional differential equations can be used to deal with problems with historical dependence and global correlation,but it is almost impossible for us to get the exact solutions of the equations.Therefore scholars in related fields try to find their numerical solutions.In this paper,two kinds of variable-order fractional differential equations are selected and solved by minimal search method based on reproducing kernel theory.The general order of this paper is as follows:In the first chapter,the background and significance of variable-order fractional differential equations and variable-order time fractional Mobile-Immobile advection-diffusion equations are introduced.In consideration of the situation that smooth solutions are mainly studied at home and abroad recently,this paper aims to study non-smooth numerical solutions.Besides,the preliminary knowledge required for this thesis is given and the main contents of this paper are briefly summarized.In the second chapter,the numerical method of a class of variable-order fractional ordinary differential equations is considered.Firstly,the model equation is homogenized so that its boundary value is 0.Then the homogenized equation is linearized by means of F derivative and Newton iterative formula.The linear operator derived from the resulting equation is proved to be bounded.In order to obtain non-smooth solutions,a solution space with weighted inner product is defined,whose basis is derived from multi-wavelet basis.The minimal search method of the equation is obtained by normal equations.Next,the convergence order and stability of the algorithm are analyzed.Two effective numerical examples verify the feasibility of the proposed method.In the third chapter,a class of variable-order time fractional Mobile-Immobile advecti on-diffusion equations is discussed.After homogenizing the model equation,we integrate x of the equation twice to make it become an integral equation,then prove the boundedness of the linear operator as in the previous chapter.The binary reproducing kernel space and its base are constructed on the basis of the space and base of t and x,respectively.In this chapter,we also use minimal search method to solve the equation,and the convergence analysis is given.The final numerical example illustrates the effectiveness and stability of the given method.