A nonuniform L2 formula of Caputo derivative and its application to a fractional Benjamin–Bona–Mahony‐type equation with nonsmooth solutions

To recover the full accuracy of discretized fractional derivatives, nonuniform mesh technique is a natural and simple approach to efficiently resolve the initial singularities that always appear in the solutions of time‐fractional linear and nonlinear differential equations. We first construct a nonuniform L2 approximation for the fractional Caputo's derivative of order 1 <  α  < 2 and present a global consistency analysis under some reasonable regularity assumptions. The temporal nonuniform L2 formula is then utilized to develop a linearized difference scheme for a time‐fractional Benjamin–Bona–Mahony‐type equation. The unconditional convergence of our scheme on both uniform and nonuniform (graded) time meshes are proven with respect to the discrete H 1 ‐norm. Numerical examples are provided to justify the accuracy.