Fractional Relaxation and Fractional Oscillation Models Involving Erdélyi-Kober Integrals

Abstract

We consider fractional relaxation and fractional oscillation equations involving Erdélyi-Kober integrals. In terms of Riemann-Liouville integrals, the equations we analyze can be understood as equations with timevarying coefficients. Replacing Riemann-Liouville integrals with Erdélyi-Kober-type integrals in certain fractional oscillation models, we obtain some more general integro-differential equations. The corresponding Cauchytype problems can be solved numerically, and, in some cases analytically, in terms of Saigo-Kilbas Mittag-Leffler functions. The numerical results are obtained by a treatment similar to that developed by K. Diethelm and N.J. Ford to solve the Bagley-Torvik equation. Novel results about the numerical approach to the fractional damped oscillator equation with time-varying coefficients are also presented.MSC 2010 : Primary 34C26; 26A33; 65L05; Secondary 326A33; 26A48; 34A08; 33E12

The use of partition polynomial series in Laplace inversion of composite functions with applications in fractional calculus

“…Anomalous relaxation is frequently encountered in studying various physical systems including dielectrics, polymers, and viscoelastic materials . It is also a common phenomenon in biological systems including human brain and DNA .…”

l-problem of moments for one-dimensional integro-differential equations with Erdélyi-Kober operators

“…где 0 ν > . Такой тип уравнений описывает процессы с последействием, к нему же могут быть сведены различные уравнения динамики осциллятора дробного порядка [9,10,12].…”

“…Т е о р е м а 3. Пусть дана проблема моментов (7), (14), где функция g и момент c заданы формулами (12) и (13). Тогда проблема моментов (7), (14) может быть поставлена и является разрешимой при выполнении следующих условий:…”

Periodic solutions of neutral fractional differential equations

“…We refer the reader to the works and the references listed therein for recent information on this subject. For some applications of fractional differential equations, see .…”