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We investigate a variably distributed-order time-fractional wave partial differential equation, which could accurately model,  e.g.,  the viscoelastic behavior in vibrations in complex surroundings with uncertainties or strong heterogeneity in the data. A standard composite rectangle formula of mesh size  σ  is firstly used to discretize the variably distributed-order integral and then the L-1 formula of degree of freedom  N  is applied for the resulting fractional derivatives. Optimal error estimates of the corresponding fully-discrete finite element method are proved based only on the smoothness assumptions of the data. To maintain the accuracy, setting  σ  = O( N  −1 ) leads to  O ( N  3 ) operations of evaluating the temporal discretization coefficients. To improve the computational efficiency, we develop a novel time-stepping scheme by expanding the fractional kernel at a fixed fractional order to decouple the fractional operator from the variably distributed-order integral. Only  O (log N ) terms are needed for the expansion without loss of accuracy, which consequently reduce the computational cost of generating coefficients from  O ( N  3 ) to  O ( N  2  log N ). Optimal-order error estimates of this time-stepping scheme are rigorously proved via novel and different techniques from the standard analysis procedure of the L-1 methods. Numerical experiments are presented to substantiate the theoretical results.

Numerical discretization and fast approximation of a variably distributed-order fractional wave equation