The Riemann–Liouville–Caputo (RLC) derivative is a new class of derivative that is motivated by modelling considerations; it lies between the more familiar Riemann–Liouville and Caputo derivatives. The present paper studies a two-point boundary value problem on the interval [0, L ] whose highest-order derivative is an RLC derivative of order $$\alpha \in (1,2)$$ α ∈ ( 1 , 2 ) . It is shown that the choice of boundary condition at $$x=0$$ x = 0 strongly influences the regularity of the solution. For the case where the solution lies in $$C^1[0,L]\cap C^{q+1}(0,L]$$ C 1 [ 0 , L ] ∩ C q + 1 ( 0 , L ] for some positive integer q , a finite difference scheme is used to solve the problem numerically on a uniform mesh. In the error analysis of the scheme, the weakly singular behaviour of the solution at $$x=0$$ x = 0 is taken into account and it is shown that the method is almost first-order convergent. Numerical results are presented to illustrate the performance of the method.

中文翻译：

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具有 Riemann-Liouville-Caputo 分数阶导数的两点边值问题的有限差分格式的收敛分析

Riemann-Liouville-Caputo (RLC) 导数是一类新的导数，其动机是建模考虑；它介于更熟悉的 Riemann-Liouville 和 Caputo 导数之间。本文研究了区间 [0, L ] 上的两点边值问题，其最高阶导数为 $$\alpha \in (1,2)$$ α ∈ ( 1 , 2 ) 阶的 RLC 导数. 结果表明，在$$x=0$$x = 0 时边界条件的选择对解的规律性有很大影响。对于解位于 $$C^1[0,L]\cap C^{q+1}(0,L]$$ C 1 [ 0 , L ] ∩ C q + 1 ( 0 , L ] 对于某个正整数 q ，采用有限差分格式在均匀网格上数值求解该问题。在该格式的误差分析中，考虑了 $$x=0$$x = 0 处解的弱奇异行为，表明该方法几乎是一阶收敛的。给出了数值结果来说明该方法的性能。