The High-precision Collocation Method Of Fractional Integral-differential Equation

In the initial development period of fractional calculus,theory and application developed slowly due to the lack of actual background.The situation did not improve until the late 1970 s.Since then,fractional calculus developed rapidly and was successfully applied to the research of many complex systems.The fractional integral-differential equation is applied widely in physics and engineering.However,analytical solution of it either does not exist or is difficult to find.It is because of this fact that many numerical methods for solving fractional integral-differential equation have emerged.At the same time,because the definition of fractional calculus is different from the definition of integer calculus,the theoretical analysis applicable to fractional integral-differential equation needs further research.In this paper,nonlinear fractional Volterra integral-differential equation,fractional Volterra-Fredholm integral-differential equation and the system of fractional VolterraFredholm integral-differential equations are solved based on the shifted Legendre polynomials combined with the collocation method.Firstly the shifted Legendre polynomials are applied to the nonlinear fractional Volterra integral-differential equation,which first of all,according to the Gronwall inequality and Lipschitz condition the existence and uniqueness of equation solution are achieved,and owing to the property of shifted Legendre polynomials,the nonlinear part can be converted to some particular form,then the approximate equation is obtained,subsequently we apply the projection operator theory to prove the existence and uniqueness of discrete equation and obtain convergence analysis,and at last,numerical examples are given to illustrate the accuracy of the method.And then the collocation method of fractional integral-differential is investigated,which firstly we prove the existence and uniqueness of equation due to that the linear integral operator is compact and the Fredholm chooses certain principles,then the numerical solution via collocation method and convergence analysis are obtained,meanwhile some numerical examples are shown illustrating the validity of the method.Moreover the numerical method based on the shifted Legendre polynomial to solve the system of fractional Volterra-Fredholm integro-differential equations is given,first of all,we prove the existence and uniqueness of analytic equations depending on the use of Gronwall inequality and Laplace transform,then by using shifted Legendre polynomials,the system of approximation equations is obtained,and finally the theoretical analysis of the system of discrete equations and convergence analysis are achieved on the basis of the projection operator theory and numerical examples are given to illustrate the effectiveness of the method.