Publication Date:
2017-06-18
Description:
Compared to the classical first-order Grünwald–Letnikov formula at time $t_{k+1}; (ext{or}; t_{k})$, we firstly propose a second-order numerical approximate formula for discretizing the Riemann–Liouvile derivative at time $t_{k+frac{1}{2}}$, 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 RLD0,tαu(t)| t=tk+12=τ−α∑ℓ=0kϖℓ(α)u(tk−ℓτ)+O(τ2),k=0,1,…,α∈(0,1), where the coefficients $varpi_{ell}^{(alpha)}$$(ell=0,1,ldots,k)$ can be determined via the following generating function G(z)=(3α+12α−2α+1αz+α+12αz2)α,|z|
Print ISSN:
0272-4960
Electronic ISSN:
1464-3634
Topics:
Mathematics
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