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Numerical approaches to the functional distribution of anomalous diffusion with both traps and flights

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Abstract

The functional distributions of particle trajectories have wide applications. This paper focuses on providing effective computation methods for the models, which characterize the distribution of the functionals of the paths of anomalous diffusion with both traps and flights. Two kinds of discretization schemes are proposed for the time fractional substantial derivatives. The Galerkin method with interval spline scaling bases is used for the space approximation; compared with the usual finite element or spectral polynomial bases, the spline scaling bases have the advantages of keeping the Toeplitz structure of the stiffness matrix, and being easy to generate the matrix elements and to perform preconditioning. The rigorous stability analyses for both the semi and the full discrete schemes are skillfully developed. Under the assumptions of the regularity of the exact solution, the convergence of the provided schemes is also theoretically proved and numerically verified. Moreover, the theoretical background of the selected basis function and the implementation details of the algorithms involved are described in detail.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Zhijiang Zhang & Weihua Deng

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  1. Zhijiang Zhang
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  2. Weihua Deng
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Correspondence to Weihua Deng.

Additional information

Communicated by: Jan Hesthaven

This work was supported by the National Natural Science Foundation of China under Grant No. 11271173 and Grant No. 11671182.

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Zhang, Z., Deng, W. Numerical approaches to the functional distribution of anomalous diffusion with both traps and flights. Adv Comput Math 43, 699–732 (2017). https://doi.org/10.1007/s10444-016-9503-y

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  • DOI: https://doi.org/10.1007/s10444-016-9503-y

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