In this paper, the fractional equations of the mass-spring-damper system with Caputo and Caputo–Fabrizio derivatives are presented. The physical units of the system are preserved by introducing an auxiliary parameter σ. The input of the resulting equations is a constant and periodic source; for the Caputo case, we obtain the analytical solution, and the resulting equations are given in terms of the Mittag–Leffler function; for the Caputo–Fabrizio approach, the numerical solutions are obtained by the numerical Laplace transform algorithm. Our results show that the mechanical components exhibit viscoelastic behaviors producing temporal fractality at different scales and demonstrate the existence of Entropy 2015, 17 6290 material heterogeneities in the mechanical components. The Markovian nature of the model is recovered when the order of the fractional derivatives is equal to one.

We present new analytical approximated solutions for the space-time fractional nonlinear partial differential coupled mKdV equation. A homotopy analysis method is considered to obtain an infinite series solution. The effectiveness of this method is demonstrated by finding exact solutions of the fractional equation proposed, for the special case when the limit of the integral order of the time derivative is considered. The comparison shows a precise agreement between these solutions.

The main goal of this work is to present a new modified version of the Atangana-Baleanu fractional derivative with Mittag-Leffler non-singular kernel and strong memory. This proposal presents important advantages when specific initial conditions are impossed. The new modified version of the Atangana-Baleanu fractional derivative with Mittag-Leffler non-singular kernel has been constructed considering the fulfillment of the initial conditions with special interest because they are decisive in the obtaintion of analytical and numerical solutions of the fractional differential equations. The advantage of this new fractional derivative in the fulfilling of initial conditions plays a central role for the implementation of different perturbative analytical methods, such as the homotopy perturbation method and the modified homotopy perturbation method. These methods will be applied to solve nonlinear fractional differential equations. This novel modified derivative can be applied in the future in different mathematical modeling areas where satisfy the initial conditions is of great relevance to get more accurate description of real-world problems.

In this paper, with the aid of the Maple software, the new extended direct algebraic method is used as a powerful method to constructs some new solutions to the well-known nonlinear models, namely, the simplified modified Camassa-Holm equation. The space-time fractional derivatives are defined in the sense of the new conformable fractional derivative. It is shown that the method is straightforward and effective mathematical tool for solving nonlinear evolution equations in mathematical physics and engineering.