On a class of fractional differential equations for mathematical models of dynamic system with memory

Some differential equation with Riemann–Liouville fractional derivatives is considered. The class of these equations are proposed as a model fractional oscillating equation for the description, analysis and investigation of oscillatory processes in dynamic systems with memory. The obtainment such a kind of equations is based on the hypothesis supposed the existence of the non-ideal viscoelastic connection in the one-dimensional dynamic system, which is associated with the fractional analogy of Zener rheologic model of the viscoelastic body. It's shown, that the initial values problems with Cauchy type conditions is reduced equivalently to the Volterra type integral equations with the differentiable kernels. This circumstance allow to use the method of successive approximation to resolve that integral equations. It's indicated, that such a kind of differential equations may be interesting as mathematical models of nonlinear dynamic systems behavior.

differential and integral equations with fractional Riemann–Liouville operators, fractional oscillators, fractional oscillating equations, rheological model of viscoelastic body with memory, Mittag-Leffler type special functions, Volterra type integral equations with special functions in kernel