Volume 55, Number 5, September-October 2021
|Page(s)||2211 - 2232|
|Published online||13 October 2021|
Numerical discretization and fast approximation of a variably distributed-order fractional wave equation
School of Mathematics and Statistics, Shandong Normal University, Jinan, Shandong 250358, China
2 School of Mathematical Sciences, Peking University, Beijing 100871, China
3 Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA
* Corresponding author: firstname.lastname@example.org
Accepted: 17 August 2021
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(N3) 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(logN) terms are needed for the expansion without loss of accuracy, which consequently reduce the computational cost of generating coefficients from O(N3) to O(N2 logN). 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.
Mathematics Subject Classification: 35R11 / 65N30
Key words: Variably distributed-order time-fractional wave equation / viscoelastic problem / well-posedness and regularity / finite element method / optimal-order error estimate / fast algorithm
© The authors. Published by EDP Sciences, SMAI 2021
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