Abstract:
We investigate a convolution group that is closely related to fractional integrals.Using this group, we are able to define a fractional calculus on a certain class of distributions. When acting on on causal functions, this definition agrees with the traditional Riemann-Liouville definition for t > 0 but includes some singularities at t = 0 so that the group property holds. Using this group, we are able to extend the definition of Caputo derivatives of order in (0,1) to a certain class of locally integrable functions without using the first derivative. The group property allows us to de-convolve the fractional differential equations to integral equations with completely monotone kernels, which then enables us to prove the general Gronwall inequality (or comparison principle) with the most general conditions. Some other fundamental results for fractional ODEs are also established within this frame under very weak conditions. The theory here may be useful for modelling memory effects and dissipation effects in nonequilibrium statistical mechanics, complex fluids and kinetic process in cellular flows. This is a joint work with Jianguo Liu. |