Numerical Solution For Three Types Of Fractional Order And Variable Order Fractional Differential Equations

Nowadays,because fractional order calculus theory has good historical memory and better correlation and it can better solve some complicated practical problems,it develops very fast.The case of variable order fractional arises in the process of development about fractional order calculus theory and variable order fractional computing is becoming an effective computational tool.In addition,a lot of practical problems need the help of nonlinear equation model for instructions.So it is critical to study the related issues about nonlinear fractional order and variable order fractional differential equations.Based on this,the thesis will mainly solve the problems of numerical solutions about three types of fractional order and variable order fractional differential equations,which include the fractional order nonlinear Fisher equation,variable order fractional nonlinear Riccati equation and variable order fractional linear Cable equation.The thesis includes the following contents:Firstly,the background of the nonlinear fractional differential equation is introduced and the Fisher equation which will be solved in this chapter are put forward.Through using a general formulation for the Legendre polynomial and the natures of fractional differential operation,we can express each part of the equation in a form of matrix product in order to accomplish function approximation.Then,through discreting variables we can get the algebraic equations and solving the equations,we can get the numerical solution of the original problem.Secondly,based on the above contents,the thesis quotes the definition of onedimensional generalized Legendre polynomial to approach unknown function.The derivation process about the numerical algorithm is also given.Then the original problem is expressed as the form of matrix product.Through the collocation method to discrete variables and using MATLAB software combined with the least square method,we will get coefficient matrix of the polynomial.Then we can solve the original equation.In the end,by taking advantage of the definition about two-dimensional generalized Legendre polynomial,this thesis will solve the numerical solution of the variable order fractional linear Cable equation.We construct the error differential equation to receive the approximation error function,which can make the numerical solution be corrected.