An Expansion Formula with Higher-Order Derivatives for Fractional Operators of Variable Order

Figure 2

Exact (46) and numerical approximations of the left Riemann-Liouville derivative ₀𝐷_𝑡^𝛼(𝑡)𝑥(𝑡) with 𝑥(𝑡)=𝑡⁴ and 𝛼(𝑡)=(𝑡+1)/4 obtained from Theorem 4 with 𝑛=2 and 𝑁∈{3,5}. The error (48) is 𝐸≈0.03294 for 𝑁=3 and 𝐸≈0.003976 for 𝑁=5.