Variational formulation and efficient implementation for solving the tempered fractional problems. (English) Zbl 1407.82055
Summary: Because of the finiteness of the life span and boundedness of the physical space, the more reasonable or physical choice is the tempered power-law instead of pure power-law for the CTRW model in characterizing the waiting time and jump length of the motion of particles. This paper focuses on providing the variational formulation and efficient implementation for solving the corresponding deterministic/macroscopic models, including the space tempered fractional equation and time tempered fractional equation. The convergence, numerical stability, and a series of variational equalities are theoretically proved. And the theoretical results are confirmed by numerical experiments.
MSC:
| 82C80 | Numerical methods of time-dependent statistical mechanics (MSC2010) |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 82C41 | Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics |
| 35Q82 | PDEs in connection with statistical mechanics |
| 35R11 | Fractional partial differential equations |
| 35A15 | Variational methods applied to PDEs |
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