| Spectral methods play an important role in solving differential equations and are currently wide used in engineering problems.The spectral method has the advantage of high precision,that is,the convergence of ”infinite order”,its convergence speed increases with the smoothness of the true solution.Fractional integro-differential have memory properties and have many applications in viscoelasticity,electrochemistry,control and electromagnetics.People pay more and more attention to its research,and it has become a hot topic of research.Therefore,the main work of this paper is to study the piecewise Jacobi spectral collocation method of fractional integro-differential equations.Firstly,this paper uses the Riemann-Liouville fractional derivative to transform the equation with the initial value problem into an equivalent equation,in order to use the Gauss integral to discretely use the correlation transform to convert the integral interval to [-1,1];Secondly,divide the interval [-1,1] into M+1 subintervals and select the N+1 Legendre-Gauss-Lobatto points on the standard interval as the collocation points to enter the equation;Finally,the points between each interval discretely obtain discrete forms of fractional differential integral equations.In the text,the error analysis of the piecewise Jacobi spectral collocation method is carried out.The results show that the error between the numerical solution and the exact solution obtained by the fractional integro-differential equation is exponentially convergent.In addition,we can improve the accuracy of the numerical solution by mesh subdivision and increasing the number of approximation polynomials.At the end of the paper,the effectiveness of the method is illustrated by some numerical examples.The comparison between the piecewise Jacobi spectral collocation method and the Jacobi spectral collocation method shows that the convergence of the picecwise Jacobi spectral collocation method is more better. |
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