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Theory and Modern Applications

Table 1 Advantage of using the Hilfer fractional derivatives

From: Existence results for infinite systems of the Hilfer fractional boundary value problems in Banach sequence spaces

α,β

Corresponding reduced BVPs

β: = 1

\((\mathcal{D}_{a^{+}}^{\alpha,\beta }u )(t)=-h(t), u(a)=u(b)=0\longrightarrow ^{c}\mathcal{D}_{a^{+}}^{\alpha }u(t)=-h(t), u(a)=u(b)=0\)

β: = 0

\((\mathcal{D}_{a^{+}}^{\alpha,\beta }u )(t)=-h(t), u(a)=u(b)=0\longrightarrow \mathcal{D}_{a^{+}}^{\alpha }u(t)=-h(t), u(a)=u(b)=0\)

α: = 2

\((\mathcal{D}_{a^{+}}^{\alpha,\beta }u )(t)=-h(t), u(a)=u(b)=0\longrightarrow u^{\prime \prime }(t)=-h(t), u(a)=u(b)=0\)