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

We consider a wide class of integral and ordinary differential equations of fractional multi-orders (1/ρ1, 1/ρ2, . . . , 1/ρm), depending on arbitrary parameters ρi > 0, μi ∈ R, i = 1, . . . , m. Denoting the “differentiation” operators by D = D(ρi ),(μi ), and by L = L(ρi ),(μi ) the corresponding “integrations” (operators right inverse to D), we first observe that D and L can be considered as operators of the generalized fractional calculus, respectively—as generalized fractional “derivatives” and “integrals.” A solution of the homogeneous ODE of this kind, Dy(z) = λy(z), λ = 0, 0 < |z| <∞, is the recently introduced “multi-index Mittag-Leffler function” E(1/ρi),(μi )(λz). We find a Poisson-type integral transformation P (generalizing the classical Poisson integral formula) that maps the cosm-function into the multi-indexMittag-Leffler function, and also transforms the simpler differentiation and integration operators of integer order m > 1: Dm = (d/dz)m and lm (the m-fold integration) into the operators D and L. Thus, from the known solution of the Volterra-type integral equation with the m-fold integration lm, via P as a transformation (transmutation) operator, we find the corresponding solution ofthe integral equation y(z) − λL(z) = f (z). Then, a solution of the fractional multi-order differential equation Dy(z) − λy(z) = f (z) comes out, in an explicit form, as a series of integrals involving Fox’s H-functions. For each particularly chosen R.H.S. function f (z), such a solution can be evaluated as an H-function. Special cases of the equations considered here, lead to solutions in terms of the Mittag-Leffler, Bessel, Struve, Lommel and hyper-Bessel functions, and some other known generalized hypergeometric functions.  2002 Elsevier Science (USA). All rights reserved.