Approximate Solution of a Kind of Time-Fractional Evolution Equations Based on Fast L1 Formula and Barycentric Lagrange Interpolation

In this paper, an effective numerical approach that combines the fast L1 formula and barycentric Lagrange interpolation is proposed for solving a kind of time-fractional evolution equations. This type of equation contains a nonlocal term involving the time variable, resulting in extremely high computational complexity of numerical discrete formats in general. To reduce the computational burden, the fast L1 technique based on the L1 formula and sum-of-exponentials approximation is employed to evaluate the Caputo time-fractional derivative. Meanwhile, a fast and unconditionally stable time semi-discrete format is obtained. Subsequently, we utilize the barycentric Lagrange interpolation and its differential matrices to achieve spatial discretizations so as to deduce fully discrete formats. Then error estimates of related fully discrete formats are explored. Eventually, some numerical experiments are simulated to testify to the effective and fast behavior of the presented method.