In recent times we have seen so much work being done related to fractional partial differential equations, especially in the field of Lie symmetry analysis and conservation laws. In case of partial differential equations where the conservation laws can be determined using various methods like; Noether's theorem and direct method of Bluman and Anco. For non-trivial conservation laws, the divergence expression of fluxes must annihilate on the solution space of PDE under consideration.
My query is, whether such annihilation of fluxes also holds for conservation laws of a fractional partial differential equation?
Dear Professor Singh,
Your question is quite interesting. Actually, the key point is: What is a conservation law for a fractional differential equation (FDE)?
As far as I understand (I´m not an specialist in the field of FDE), some definitions of the derivative operator does not necessarily satisfy the vanishing of "fractional" vector fields. Of course, this is a problem with our classical definition of conservation laws. Moreover, as far as I know (and hope, I am wrong), we do not have physical phenomena modelled by FDE yet, although many groups of researchers try to find one. This certainly would be helpful to understand better the situation, even more if the phenomena had something conserved.
I have seen some tentative of Noether type theory for constructing conservation laws for FDE, but I do not know how successful they have been.
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