Extension of Colombeau algebra to derivatives of arbitrary order Dα, α∈R+∪{0}. Application to ODEs and PDEs with entire and fractional derivatives

We give an extension of Colombeau algebra of generalized functions to fractional derivatives. We apply it in solving ODEs and PDEs with entire and fractional derivatives with respect to temporal and spatial variables. We give applications to ODEs and PDEs driven by fractional derivatives of delta distribution.