Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives

Abstract

Two nonlinear anelastic models with fractional derivatives, describing the properties of a series of materials as polymers, and polycrystalline materials are presented in this paper. These models are studied analytically, using a variational iteration method. The paper clarifies the different ways in which the fractional differentiation operator can be defined. A Volterra series method of model parameters identification from the experimental data is also presented.

Rational approximation solution of the fractional Sharma–Tasso–Olever equation

“…In 1998, the variational iteration method [30] was first proposed to solve fractional differential equations with great success. Following the above idea, Draganescu, Momani and Odibat [31,32] applied the variational iteration method to more complex fractional differential equations, showing the effectiveness and accuracy of the used method. In 2002, the Adomian method [33] was suggested to solve fractional differential equations.…”

Application of He's variational iteration method for solving the Cauchy reaction–diffusion problem

“…In 1998 [17] applied VIM to a fractional differential equation arising in seepage flow. Following the idea of the above reference, Draganescu [11] applied VIM to nonlinear oscillator with fractional damping and then [10] to nonlinear viscoelastic models with fractional derivatives. Odibat and Momani [27] applied the method to nonlinear differential equations of fractional order with great success, see [6,26].…”

A note on optimal homotopy asymptotic method for the solutions of fractional order heat- and wave-like partial differential equations

“…These phenomena in science and engineering problems can be described very effectively by models using mathematical tools from fractional calculus [1][2][3][4]. Draganescu [5], Momani and Odibat [6][7][8] applied VIM to fractional differential equations. The Adomian decomposition method (ADM) was used to solve fractional differential equations [9] in 2002.…”