Fractional Relaxation Equations and a Cauchy Formula for Repeated Integration of the Resolvent
Cauchy’s formula for repeated integration is shown to be valid for the function R(t) = λΓ(q)t q−1Eq,q(−λΓ(q)t q ) where λ and q are given positive constants with q ∈ (0, 1), Γ is the Gamma function, and Eq,q is a MittagLeffler function. The function R is important in the study of Volterra integ...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
ATNAA
2018-01-01
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| Series: | Advances in the Theory of Nonlinear Analysis and its Applications |
| Subjects: | |
| Online Access: | http://dergipark.gov.tr/download/article-file/596100 |
| Summary: | Cauchy’s formula for repeated integration is shown to be valid for the function
R(t) = λΓ(q)t
q−1Eq,q(−λΓ(q)t
q
)
where λ and q are given positive constants with q ∈ (0, 1), Γ is the Gamma function, and Eq,q is a MittagLeffler 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) = λtq−1 − λ
Z t
0
(t − s)
q−1R(s) ds
on the interval (0, ∞). 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, ∞) 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 |
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| ISSN: | 2587-2648 2587-2648 |