Anharmonic vibrations of a nano-sized oscillator with fractional damping

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].…”

The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method

“…The linear and nonlinear cases are discussed in their work and several test examples are given to show the efficiency of this procedure. Draganescu [53] applied VIM to nonlinear oscillator with fractional damping and then [54] to nonlinear viscoelastic models with fractional derivatives. Odibat and Momani [55] applied the method to nonlinear differential equations of fractional order with great success.…”

Numerical comparison for the solutions of anharmonic vibration of fractionally damped nano-sized oscillator

“…It is an established fact (Draganescu and Capalnasan, 2003;Draganescu, 2006;Dra˘ga˘nescu et al, 2010;Ke and Espinosa, 2004) that mechanical motion of the elements of micro-and nano-devices is examined inconnection with nonlinear forces of quantum nature similar to the Casimir force. Moreover, the anelastic properties of materials are nonlinear in nature (see Draganescu, 2006;Dra˘ga˘nescu et al, 2010;Ke and Espinosa, 2004;He, 2008a, and the references therein). The Casimir effect consists in the electrical polarization of two perfectly conducting bodies, the Casimir force taking significant values when the separation between these bodies is reduced to less than 100 nm.…”

“…The Casimir effect consists in the electrical polarization of two perfectly conducting bodies, the Casimir force taking significant values when the separation between these bodies is reduced to less than 100 nm. On the other hand (Cleland, 2003;Draganescu and Capalnasan, 2003;Draganescu, 2006;Dra˘ga˘-nescu et al, 2010;Ke and Espinosa, 2004;Ghorbani and Nadjfi, 2007;He, 2008a), it was found that in materials like plastics and nano-wires, the most adequate kind of damping is the fractional damping. Recently, Dra˘ga˘nescu et al (2010) used Adomian's decomposition method for solving the governing problem.…”

“…The mechanical properties of the micro-and nano-devices can be described in terms of classical or quantum mechanics (Cleland, 2003;Draganescu and Capalnasan, 2003;Draganescu, 2006;Dra˘ga˘nescu et al, 2010;Ke and Espinosa, 2004;Ghorbani and Nadjfi, 2007;He, 2008a). It is an established fact (Draganescu and Capalnasan, 2003;Draganescu, 2006;Dra˘ga˘nescu et al, 2010;Ke and Espinosa, 2004) that mechanical motion of the elements of micro-and nano-devices is examined inconnection with nonlinear forces of quantum nature similar to the Casimir force.…”