Jacobi Spectral-collocation Methods For One Kind Of Fractional Volterra Integro-differential Equations With Weakly Singular Kernels

The use of fractional Volterra integro-differential equations in mathematical models, which is considered to better describe the behavior of some materials with memory properties, has become increasingly popular inrecent years. Because the exact solution of some fractional Volterra integro-differential equations is not obtained easily, or only need the approximate solution of high order accuracy in a real world application, so solving the numerical solution of high accuracy becomes very important and practical value.In this paper, our work is focused on the numerical computation for one kind of fractional Volterra integro-differential equations with weakly singular kernels by used of Jacobi spectral-collocation method. We convert fractional integro-differential equations to one type of second kind Volterra integral equations with weakly singular kernels, because fractional differential equations are equivalent to the second kind Volterra integral equations. Then we use some transformations to change the equation into a new Volterra integro equation, so that the solution of the new equation possesses beter regularity. Jacobi spectral-collocation method is used to slove the new Volterra integro equation. The spectral rate of convergence for the proposed method is established in the L_? norm and the weighted L_?~2 norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.