It is well known that solutions of time-fractional ODEs/PDEs exhibit weak singularities at $t=0$ so that usual approximation techniques do not yield accurate results. 

In this talk, we present two classes of numerical methods to address this problem: The first one is based on the generalized Jacobi functions, while the second one is based on a new set of log orthogonal functions (LOFs). Space-time Petrov-Galerkin approach will be used in both cases. We shall show rigorous error analysis with high-order convergence rate for both methods despite the weak singularities at $t=0$, and present ample numerical results  to demonstrate the effectiveness of our methods.

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