Leigh C. BECKER, İoannis K. PURNARAS

8373

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

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 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.
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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Advances in the Theory of Nonlinear Analysis and its Application-Cover
  • Başlangıç: 2017
  • Yayıncı: Erdal KARAPINAR
Arşiv
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