Solving Fractional Differential Equations By Orthogonal Polynomials

Fractional calculus is a generalization of ordinary calculus.Compared with integer-order differential equations,fractional differential equations are powerful mathematical tools to describe systems and processes with nonlocality and memory.Therefore,fractional differential equations are widely used in materials science,engineering,biomedical sciences and other fields.Nowdays,due to its wide application background,the study of fractional differential equations has become one of the hot spots of scientific research.It is very important to obtain the numerical solutions of fractional differential equations for studying the models described by the equations.Because of the complexity and particularity of fractional differential equations,the numerical methods for solving the integral-order differential equations are no longer suitable for studying fractional differential equations.Therefore,many scholars are devoted to the numerical methods of fractional differential equations.At present,the commonly used numerical methods are: collocation method,wavelet method,finite difference method and finite element method.This thesis is based on Alternative Legendre Polynomials(Hereinafter referred to as ALPs).We propose a numerical calculation method for solving fractional differential equations.Firstly,the operation matrix of the fractional differential equation is obtained by using the properties of ALPs.Then the fractional differential equation is transformed into the system of algebraic equations.Finally,the numerical solution of the original equation can be obtained by solving the algebraic system.In this thesis,the ALPs method is used to solve fractional differential equations with proportional delay,fractional pantograph equations of neutral type,fractional integral-differential equations and fractional Volterra integral-differential equations.The details are as follows:Chapter Ⅰ introduces the research background and current situation of fractional differential equations,and briefly summarizes the main content of this paper.Someconcepts of fractional calculus and the preliminary knowledge such as the definition of ALPs are given in Chapter II.In Chapter Ⅲ,the ALPs method is used to solve the fractional differential equations with proportional delay,and the error analysis is also given.Numerical examples show that the method can be easily used to obtain high-precision numerical solutions.Secondly,the ALPs method is used to solve fractional pantograph equations of neutral type.At the end of this chapter,the numerical examples are used to illustrate the feasibility of the ALPs method for solving such equations.In Chapter Ⅳ,the ALPs method is employed to solve the fractional integro-differential equations.In the numerical experiments,we not only compare the absolute errors with theses of other methods,but also illustrate the errors at some points.The results show the feasibility and accuracy of the method.Unlike the previous two chapters,which consider a single equation,Chapter V presents the numerical method for solving fractional Volterra integral-differential equations.The feasibility of the method for solving fractional Volterra integro-differential equations is shown through numerical experiments.Chapter Ⅵ provides a comprehensive summary of this thesis.