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

Abstract

We consider a wide class of integral and ordinary differential equations of fractional multi-orders (1/ ρ 1,1/ ρ 2,…,1/ ρ m ), depending on arbitrary parameters ρ i >0, μ i∈ R , i=1,…, m. Denoting the “differentiation” operators by D=D (ρ i),(μ i) , and by L=L (ρ i),(μ i) 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/ ρ i ),( μ i ) ( λz). We find a Poisson-type integral transformation P (generalizing the classical Poisson integral formula) that maps the cos m -function into the multi-index Mittag-Leffler function, and also transforms the simpler differentiation and integration operators of integer order m>1: D m =( d/ dz) m and l m (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 l m , via P as a transformation (transmutation) operator, we find the corresponding solution of the integral equation y(z)−λ L(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.