cranknicholson加權(quán)與位移gr252;nwald差分格式的精度與穩(wěn)定性

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3、tFractional differential equations (FDEs) emerge from numerous topics suchas turbulent flow, chaotic dynamics of classical conservative systems, ground-water contaminant transport, and applications in physics, finance, b

4、iology, andimage processing. Unlike the second-order diffusion equations, there are fewFDEs whose analytical closed-form solutions are available. As a result, numer-ical methods for FDEs have been developed intensively i

5、n the last decades.In this thesis, we are concerned with numerical methods of the solution ofthe following fractional initial-boundary partial differential equation:?u(x, t)?t = d+(x, t)?αu(x, t)?+xα + d?(x, t)?αu(x, t)?

6、?xα + f(x, t),x ∈ (xL, xR), t ∈ (0, T],u(xL, t) = u(xR, t) = 0, 0 ≤ t ≤ T,u(x, 0) = u0(x), x ∈ [xL, xR],where 1 < α < 2, f(x, t) is the source term, and d±(x, t) ≥ 0.We apply the Crank-Nicholson weighted and s

7、hifted Gr¨ unwald difference(CN-WSGD) scheme proposed in [Mathematics of Computation, 84 (2015),pp. 1703–1727] to discretize the above equation. The CN-WSGD scheme hassecond order accuracy. Moreover, it is unconditi

8、oned stable when the diffusioncoefficients d±(x, t) are constant. In this thesis, we mainly study the case wherethe diffusion coefficients d±(x, t) are not constant. We discuss how to choosesuitable weights suc

9、h that the CN-WSGD scheme is optimal about the accuracyand stability. Numerical results are present to compare the CN-WSGD schemeswith different weights.Key words: Fractional differential equation; Gr¨ unwald-Letnik