Hybrid Collocation And Adaptive Collocation Algorithm For Fractional Differential Equations

This paper studies hybrid collocation and adaptive collocation algorithm for frac-tional differential equations(FDEs).Hybrid collocation method is suitable for solving a class of FDEs, the derivative of the solution of which has singularity near the left end point of the interval consid-ered. The commonly graded polynomial collocation method, on the one hand could not reflect the singularity of the solution due to the use of piecewise polynomial ap-proximation, and on the other hand may cause seriously round-off error problem due to the use of extremely non-uniform mesh. To reflect the singularity of the solution, in the first sub-interval, we approximate the unknown functions by using non-polynomial functions, while in the remaining sub-intervals, we still use polynomial functions ap-proximate the unknowns. In addition, to avoid using extremely small sub-intervals, we correct the commonly used graded mesh and obtain a new mesh. Under the theoretical analysis and calculations, we obtain the global convergence order of the method, the numerical experiments demonstrate the effectiveness of the method.In order to obtain a user-oriented error estimates, we have established adaptive collocation algorithm for FDEs. For this purpose, a posteriori error estimates of collo-cation method is derived and the derived error indicators are used to adaptively adjust meshes when the error is larger than the user-defined error tolerance. Such method is much more efficient than that of the collocation methods on uniform meshes. Numeri-cal computations confirm the effectiveness of this algorithm via several examples.