An Overview: Ordinary and Partial Differential Equations of Fractional Order on Examples in the Dynamics of Rheological Systems

An overview of ordinary and partial differential equations of fractional order, which arose as a new result of research and mathematical description of the constitutive relations of new models of rheological materials of fractional type, as well as the rheological longitudinal dynamics of these materials, is given. Systems of differential equations of fractional order are also presented, which describe the dynamics of rheological models of discrete dynamic systems such as rheological oscillators or creepers. Here we present ordinary and partial differential equations of fractional order, which mathematically describe the constitutive relations of new models of rheological Kelvin-Voight materials of fractional type, i.e. rheological Maxwell materials of fractional type, and the rheological longitudinal dynamics of these materials. Systems of fractional order differential equations, dynamics of rheological models of discrete dynamic systems in which standard light coupling sets of the Kelvin-Voight/Kelvin-Voight-Faraday model of fractional type or the Maxwell/Maxwell-Faraday model of fractional type with piezoelectric effects are incorporated are also presented. The methodology of approximate analytical solution of some of the presented ordinary and partial differential equations of fractional order will also be indicated.