On a class of fractional differential equations for mathematical models of dynamic systems with memory
The differential equation with Riemann–Liouville fractional derivatives is considered. The equations of this class are proposed as a model fractional oscillating equations for the description, analysis and investigation of oscillatory processes in dynamic systems with memory. Such a kind of equations obtainment 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 can be reduced equivalently to the Volterra type integral equations with the differentiable kernels. This circumstance allow to use the method of successive approximation to resolve these integral equations. It’s indicated, that such a kind of differential equations can be interesting as mathematical models of nonlinear dynamic systems behavior.