科研之友
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摘要
Compared to the classical first-order Grunwald-Letnikov formula at time t(k+1) (or t(k)), we firstly propose a second-order numerical approximate formula for discretizing the Riemann-Liouvile derivative at time t(k+1/2), which is very suitable for constructing the Crank-Nicolson scheme for the fractional differential equations with time fractional derivatives. The established formula has the following form (RL)D(0,t)(alpha)u(t)vertical bar(t=tk+1/2) = tau(-alpha)Sigma(k)(l=0) pi((alpha))(l) u (t(k) - l(tau)) + O(tau(2)), k = 0, 1, ..., alpha is an element of (0,1), where the coefficients pi((alpha))(l) (l = 0, 1, ..., k) can be determined via the following generating function G(z) = (3 alpha + 1/2 alpha - 2 alpha + 1/alpha z + alpha + 1/2 alpha z(2))(alpha), vertical bar z vertical bar < 1. Next, applying the formula to the time fractional Cable equations with Riemann-Liouville derivative in one and two space dimensions. Then the high-order compact finite difference schemes are obtained. The solvability, stability and convergence with orders O(tau(2) + h(4)) and O(tau(2) + h(x)(4) + h(y)(4)) are shown, where t is the temporal stepsize and h, h(x), h(y) are the spatial stepsizes, respectively. Finally, numerical experiments are provided to support the theoretical analysis.
