Fractional Relaxation Equations and a Cauchy Formula for Repeated Integration of the Resolvent

Abstract

Cauchy’s formula for repeated integration is shown to be valid for the functionR(t) = 􀀀(q)tq􀀀1Eq;q(􀀀􀀀(q)tq)where and q are given positive constants with q 2 (0; 1), 􀀀 is the Gamma function, and Eq;q is a Mittag-Leffler function. The function R is important in the study of Volterra integral equations because it is theunique continuous solution of the so-called resolvent equationR(t) = tq􀀀1 􀀀 Z t0(t 􀀀 s)q􀀀1R(s) dson the interval (0;1). This solution, commonly called the resolvent, is used to derive a formula for theunique continuous solution of the Riemann-Liouville fractional relaxation equationDqx(t) = 􀀀ax(t) + g(t) (a > 0)on the interval [0;1) when g is a given polynomial. This formula is used to solve a generalization of theequation of motion of a falling body. The last example shows that the solution of a fractional relaxationequation may be quite elementary despite the complexity of the resolvent.