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 function
R(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 the
unique continuous solution of the so-called resolvent equation
R(t) = tq􀀀1 􀀀
Z t
0
(t 􀀀 s)q􀀀1R(s) ds
on the interval (0;1). This solution, commonly called the resolvent, is used to derive a formula for the
unique continuous solution of the Riemann-Liouville fractional relaxation equation
Dqx(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 the
equation of motion of a falling body. The last example shows that the solution of a fractional relaxation
equation may be quite elementary despite the complexity of the resolvent.

Keywords

• Cauchy’s formula for repeated integration,fractional differential equations,Mittag-Leffler functions,relaxation equations,resolvents,Riemann-Liouville operators,Volterra integral equations

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