A Generalized Definition of Caputo Derivatives and Its Application to Fractional ODEs

We extend in this paper the definition of Caputo derivatives of order in $(0,1)$ to a certain class of locally integrable functions using a convolution group. Our strategy is to define a fractional calculus for a certain class of distributions using the convolution group. When acting on causal functions, this fractional calculus agrees with the traditional Riemann-Liouville definition for $t>0$ but includes some singularities at $t=0$ so that the group property holds. Then, making use of this fractional calculus, we introduce the generalized definition of Caputo derivatives. The new definition is consistent with various definitions in literature while reveals the underlying group structure. Since the new definition is valid for a class of locally integrable functions that can blow up in finite time, it provides a framework for solutions to fractional ODEs and fractional PDEs with very weak conditions. The underlying group property makes many properties of Caputo derivatives natural. In particular, it allows us to de-convolve the fractional differential equations to integral equations with completely monotone kernels, which then enables us to prove the general Gr\"onwall inequality (or comparison principle) with the most general conditions. This then provides the essential tools for {\it a priori} energy estimates of fractional PDEs. Some other fundamental results for fractional ODEs are also established within this frame under very weak conditions.