Numerical Solutions Of Fractional Differential Equations Based On Operational Matrix

With the rapid development of society,both engineering issues and physical issues become more and more complicated.As a result,more models need to be described by fractional differential equations.Therefore,fractional differential equations has become one of the important research topics in recent years.Many scholars pay more attention to this kind of question.However,it is difficult to obtain the exact solutions for most of fractional differential equations.Therefore,how to construct numerical solutions is particularly important.Through the unremitting efforts of scholars,a number of numerical methods have been proposed.For example,there are finite element method,collocation method,finite difference method,etc..This paper is mainly composed of two parts.In the first part,the numerical solutions of multi-term fractional differential equations are studied by Bernoulli polynomials collocation method.First,the required operational matrixes of fractional derivative and product can be obtained by using the related concepts of Bernoulli polynomials.Then,the equations are simplified to a system of algebraic equations with unknown Bernoulli coefficients by using the operational matrix and collocation method.Finally,the numerical solutions of the equations can be obtained.In the second part,we apply least squares approximation method based on fractional-order Boubaker polynomials to solve different types of fractional differential equations.We first obtain the required operational matrixes by using the properties of fractional-order Boubaker polynomials.After that,we employ the obtained operational matrixes and least squares approximation method to reorganize the problem into a system of algebraic equations.Finally,the numerical solution of the equation can be obtained.The details are as follows:The first chapter introduces the research background and current situation of fractional differential equations and gives the chapter arrangement of this article.The second chapter gives the related concepts of fractional calculus and the preliminary knowledges of Bernoulli polynomials and fractional-order Boubaker polynomials.Chapter ? applies the Bernoulli polynomials collocation method to solve multiterm fractional differential equations and gives the corresponding error analysis.Finally,numerical examples are conduct to verify the efficiency and accuracy of the approach.In the fourth chapter,a class of fractional differential equations,fractional integrodifferential equations with weakly singular kernels and a class of fractional integrodifferential equations are solved by least squares approximation method based on fractional-order Boubaker polynomials.By using the proposed method to different types of equations,the numerical solutions of the corresponding equations are obtained,respectively.The results show the feasibility and accuracy of the method.Chapter ? provides a comprehensive summary of this article.