Solutions of fractional multi-order integral and differential equations using a Poisson-type transform (Q1604599)

The authors consider fractional multi-order integral equations of second kind \[ y(z)-\lambda\mathcal{L}y(z)=f(z), \] where \(\mathcal{L}\)\ is a generalized fractional integral operator with a special \(H\)-function in the kernel, as well as initial value problems for the corresponding fractional multi-order differential equation of the form \[ \mathcal{D}y(z)-\lambda y(z)=f(z), \] where \(\mathcal{D}y(z)=z^{-1}\prod\nolimits_{i=1}^{m}( z^{1+( 1-\mu_{i}) \rho_{i}}D_{z^{\rho_{i}}}^{1/\rho_{i}}z^{( \mu _{i}-1) \rho_{i}}) y(z).\) The unique solution of the first problem and a particular one for the second are given in terms of series of integrals involving \(H\)-functions. Examples are presented for special choices of the right-hand side function \(f\).