A note on generalized Abel equations with constant coefficients

Abstract

The generalized Abel equation (GAE) with constant coefficients, ${\gamma }_1{}_a{D}_x^{-\sigma }u+{\gamma }_2{}_x{D}_b^{-\sigma }u=f$, , has renewed interest since it has appeared as a component in double-sided fractional diffusion equations. Motivated by direct applications in modeling, we seek the necessary and sufficient conditions for interchanging the order of differentiation and fractional integration in GAEs. Put another way, assume functions $u,v,f$ lie in Hölder spaces that admit integrable singularities at the endpoints and

$${\gamma }_1{}_a{D}_x^{-\sigma }u+{\gamma }_2{}_x{D}_b^{-\sigma }u=f\text{ and}{\gamma }_1{}_a{D}_x^{-\sigma }v+{\gamma }_2{}_x{D}_b^{-\sigma }v=Df,$$

then we prove that $Du=v$ if and only if $u\left(a\right)=u\left(b\right)=0$.