The Solution Of The Fractional Diffusion Equation Based On The Shifted Chebyshev Collocation Method

Fractional calculus is used to describe complex systems,especially when describing materials with memory and heredity,and has good properties.However,the research on fractional differential equations is still in its infancy.Most fractional differential equations have no exact solutions,and the solution methods are also limited.They can only be solved by approximation theory and numerical calculation.The Chebyshev polynomials have good approximation properties,especially the interpolation polynomials obtained using the roots of the first kind of Chebyshev polynomials,as interpolation nodes play an important role in approximation theory.This study is mainly divided into the following six parts:The first chapter mainly introduces the background and significance of fractional calculus,fractional differential equations,and the research status at home and abroad.The second chapter briefly introduces the basic knowledge used in the research,including the definition and properties of three kinds of fractional calculus,the concept and properties of three special functions,the Gamma function,Beta function,Mittag-Leffler function,and also introduces the Picard iterative method used later,as well as the derivation,properties and series expression of the first kind of the shifted Chebyshev polynomials.The third chapter is mainly about the error analysis of the first kind of Chebyshev polynomials collocation method.We discuss the boundedness of coefficients and error norms in the Chebyshev series expression of the function and further deduce the boundedness of the first kind of the shifted Chebyshev series expression of the function.In the fourth Chapter,we introduce the shifted Chebyshev collocation method for solving the fractional ordinary differential equation under the mixed boundary value condition.The solution process has two main steps.First,the boundary value problem is transformed into an equivalent fractional differential equation.The second step is to write the solution of the equation into the shifted Chebyshev series,and then solve the converted fractional differential equation with the collocation method.At this time,the fractional differential equation becomes an easily solved equation group,the equations are solved by Mathematica to obtain the values of coefficients and further obtain the approximate solutions of the equations.In Chapter 5,we consider the solution of the spatiotemporal fractional diffusion equation under Dirichlet boundary value conditions.At this time,the coefficients in the shifted Chebyshev polynomial series are variables.Therefore,based on Chapter 4,we obtain the series expression of the coefficients by using the relationship between the integral of the Caputo fractional derivative and the original function and the Picard iteration method and further obtain the approximate solution of the equation,we take the variable of the coefficient as a fixed value and compare it with the exact solution of the equation to prove the effectiveness of this method.Chapter six is the discussion and prospect of the deficiency of the research content of this paper.