Time domain analysis of the fractional order weighted distributed parameter Maxwell model

In this paper, we derive in time domain the fundamental solution and relevant properties of the fractional order weighted distributed parameter Maxwell model (FOWDPMM). The weight function is replaced by the truncated Fourier series, which is leading to three basic fractional order distributed parameter elements. The inverse Laplace transforms of the distributed parameter operators are derived by cutting the complex plane and computing the complex path integral along the Hankel path. The asymptotic property and boundary problem are discussed by using the inverse Laplace transform, the energy of the weight function and the band width of the Fourier series. The relaxation modulus of FOWDPMM is solved as well, which is closely related to some other viscoelastic phenomena as creep, precondition and hysteresis. A number of novel characteristics of FOWDPMM such as power-law decay and intermediate phenomenon are concluded as well. Several illustrated examples are provided to validate the concepts.