Jacobi Spectral Collocation Method And Its Convergence Analysis For Fractional Integro-Differential Equations With Non-Smooth Solutions

The fractional operators are nonlocal with weakly singular kernels,which make the fractional equations more complicated than the integer-order counterparts.In recent years,although many numerical methods have been extended to fractional integro-differential equations,most of them ignore the case of non-smooth solutions.Due to the weak singularity of the kernel in the fractional operators of the fractional Integro differential equation,the solution of the equation usually shows weak singularity and the regularity of the solution is not high,so the numerical solution of the numerical method often fails to reach the optimal convergence effect.In this paper,based on the above problems,the smoothing transformation is introduced to make the solution has good regularity,and the spectral collocation method with global and high precision characteristics is used to solve the transformed equation discretely.Firstly,exploiting the natures and circumscriptions of fractional integral differential,the original fractional integro-differential equation is rewritten as an equivalent Volterra integral equation of the second kind,which has weak singular kernel.The smoothing transformation makes the solution has good regularity.We can maximize the convergence rate by adjusting the parameters in the auxiliary transformation.Then the transformed equation is solved discretely by the Jacobi spectral collocation method,and the theoretical analysis of the convergence of the method is strictly deduced in the sense of L?and Lw?,?2 norm.Finally,a numerical example is given to show the exponential decay with respect to the deviation of the true solution and the arithmetic solution,and then the feasibility and effectiveness of the Jacobi spectral collocation method to solve the fractional integro differential equation with non-smooth solution are verified.