Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion. (English) Zbl 1127.65100
Convergence of the Grünwald-Letnikov difference scheme for the fractional diffusion equation [cf. R. Gorenflo and F. Mainardi, Oper. Theory, Adv. Appl. 121, 120–145 (2001; Zbl 1007.60082)], with and without central linear drift, is proved in the Fourier-Laplace domain, implying weak convergence of the discrete solution towards the probability of sojourn of a diffusing particle.
For the proof of convergence the authors distinguish between several cases, including classical diffusion, space-time fractional diffusion and time-fractional diffusion. The paper focuses on time-fractional diffusion. Discretisation is discussed and convergence to the corresponding fundamental solution is proved. Generating functions are used in the Fourier-Laplace domain. Convergence is proved for classical diffusion with central linear drift and for time-fractional diffusion with central linear drift.
For the proof of convergence the authors distinguish between several cases, including classical diffusion, space-time fractional diffusion and time-fractional diffusion. The paper focuses on time-fractional diffusion. Discretisation is discussed and convergence to the corresponding fundamental solution is proved. Generating functions are used in the Fourier-Laplace domain. Convergence is proved for classical diffusion with central linear drift and for time-fractional diffusion with central linear drift.
Reviewer: Pat Lumb (Chester)
MSC:
| 65R20 | Numerical methods for integral equations |
| 60J60 | Diffusion processes |
| 26A33 | Fractional derivatives and integrals |
| 60G50 | Sums of independent random variables; random walks |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 45K05 | Integro-partial differential equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
bivariate generating functions; weak convergence; fractional derivative; Grünwald-Letnikov difference scheme; fractional diffusion equationCitations:
Zbl 1007.60082References:
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