A general framework in which functions of differential operators, not restricted to powers, are given meaning is the theory of pseudodifferential operators.

It is not at all beyond the bounds of possibility that a unifying theory of fractional differintegration will emerge. If it does, how it will relate to other functions of the differential operator than powers is not known.

Are some of (the many definitions of the fractional derivative) "better" than the others in some sense?

I'm not sure about this. As well as Riemann-Liouville, Grunwald-Letnikov, and Caputo, the others have included some that have hardly been referred to after their announcement. The hottest definition in the FC field at the moment - by number of papers that refer to it - is the Atangana-Baleanu one, which I asked about here.

A general framework that gives meaning to functions of differential operators that are not restricted to powers is the theory of pseudodifferential operators.

It is not at all beyond the bounds of possibility that a unifying theory of fractional differintegration will emerge, either within the theory of pseudodifferential operators or outside it. If it does, how it will relate to other functions of the differential operator than powers is not known.

Are some of (the many definitions of the fractional derivative) "better" than the others in some sense?

I'm not sure about this. As well as Riemann-Liouville, Grunwald-Letnikov, and Caputo, the others have included some that have hardly been referred to after they were announced. The hottest definition in the fractional calculus field at the moment - by number of papers that refer to it - is the Atangana-Baleanu one, which I asked about here.

A general framework in which functions of differential operators, not restricted to powers, are given meaning is the theory of pseudodifferential operators.

It is not at all beyond the bounds of possibility that a unifying theory of fractional differintegration will emerge. If it does, how it will relate to other functions of the differential operator than powers is not known.

A general framework in which functions of differential operators, not restricted to powers, are given meaning is the theory of pseudodifferential operators.

It is not at all beyond the bounds of possibility that a unifying theory of fractional differintegration will emerge. If it does, how it will relate to other functions of the differential operator than powers is not known.

Are some of (the many definitions of the fractional derivative) "better" than the others in some sense?

I'm not sure about this. As well as Riemann-Liouville, Grunwald-Letnikov, and Caputo, the others have included some that have hardly been referred to after their announcement. The hottest definition in the FC field at the moment - by number of papers that refer to it - is the Atangana-Baleanu one, which I asked about here.