Application of fractional calculus to the electroviscoelastic behaviour of liquid/liquid interfaces

Abstract

Using fractional-order alpha (0,1) , capacitive and inductive elements one can obtain linear and/or nonlinear fractional integral-differential equation of the van der Pol type, i.e. more generalized mathematical formalism for description of the electroviscoelastic behavior of liquid/liquid interfaces e.g. in finely dispersed systems, emulsions or double emulsions. First, in this paper, it is obtained nonlinear van der Pol model applying (alpha,beta ) order and variable order approach, alpha =alpha (t) . Also,it is proposed applying generalized differential transform method using Caputo definition of fractional operators to obtain a analytical homogeneous solution of a linear equation of van der Pol model of non-integer order. Finally, it is proposed numerical algorithm using approximation of Caputo derivative for calculating the corresponding homogeneous solution of linear fractional differential equation of the van der Pol type.