I've been contemplating the traditional differential operator ( D ) used in calculus, and I'm interested in a potentially broader application. Instead of having a fixed real or fractional order for differentiation, what if the order itself could be governed by a function ( \alpha(x) )? Specifically, for a given value ( x ), the function ( \alpha ) dictates the order in which another function ( f ) should be differentiated to obtain the value ( y ).
Has there been any work or discussion about this concept, where the order of differentiation is not a fixed constant but a function itself?
How would one formally define such an operator? I imagine it could look something like: ( D^{\alpha(x)} f(x) ), but the semantics of applying a variable order operator like this is unclear.
Are there potential problems or pitfalls with this concept? Continuity and well-definedness come to mind.
Would measure theory play a role in defining or working with such an operator, given that the differential order is variable and could be determined by sets of ( x ) values?
Could this idea be integrated into the existing frameworks of fractional calculus, or would it necessitate a wholly new mathematical framework?
Additionally, building on the concept above:
If we are given two functions, ( f(x) ) and ( g(x) ), is there a systematic way to determine a function ( \alpha(x) ) such that ( g(x) = D^{\alpha(x)} f(x) )? In essence, given the input and output, can we find the 'order function' that maps one to the other through our modified differentiation process?
What would the properties or constraints on ( f(x) ) and ( g(x) ) be to ensure that such an ( \alpha(x) ) exists? Are there specific classes of functions where this process is more feasible or well-defined?
Any insights, references, or suggestions for further exploration would be greatly appreciated!