A unified numerical scheme for the multi-term time fractional diffusion and diffusion-wave equations with variable coefficients (Q1675968)

The authors derive a numerical scheme based on the use of a Legendre spectral method in space and a finite difference method in time for multi-term time fractional diffusion and diffusion-wave equations with variable coefficients in a bounded domain. The fractional derivatives are given in Caputo sense. A detailed error analysis is given for the scheme. The convergence rate in \(L^\infty(L^2)\) norm is proved to be \(\mathrm{O}(\tau^2+N^{1-m})\), where \(\tau\), \(N\), and \(m\) are the time-step size, polynomial degree, and regularity in the space variable of the exact solution, respectively. Some numerical examples are presented to support the theoretical results.