Fractional order Van der Pol model of electroviscoelasticity of liquid/liquid interfaces: analytical and numerical solution

Abstract

A new, recently developed, theory of electroviscoelasticity describes the behavior of electrified liquid-liquid interfaces in fine dispersed systems, and is based on a new constitutive model of liquids. Three possible mathematical formalisms have been considered related to this physical formalism, i.e to the theory of electroviscoelasticity: the first is tension tensor model, the second is Van del Pol derivative model and the third model is Van der pol derivative model of non-integer (fractional) order. The fractional integro-differential operators –(fractional calculus) is a generalization of integration and derivation to non-integer order (fractional) operators. Fractional derivatives/integrals provide an excellent instrument for the description of memory and heretidary properties of various materials and processes and, also obtaining more degrees of freedom in the model. In this paper, central parts of presentation will be solution using Caputo definition of fractional operators to obtain a (non)homogeneuos solution of a (nonlinear) equation of Van der Pol of non-integer order where it willbwe discussed a cases of the (alpha,beta) order, variable order approach alpha=alpha(t) and generalized differential transforms method. Finally, it is proposed adequate algorithms using a predictor-corrector approach and approximations of fractional derivatives/integrals.