Recently, a new fractional derivative operator has been introduced so that it presents the combination of the Riemann–Liouville integral and Caputo derivative. This paper aims to enhance the reproducing kernel Hilbert space method (RKHSM, for short) for solving certain fractional differential equations involving this new derivative. This is the first time that the application of the RKHSM is employed for solving some differential equations with the new operator. We illustrate the convergence analysis of the applicability and reliability of the suggested approaches. The results confirm that the RKHSM finds the true solution. Additionally, these numerical results indicate the effectiveness of the proposed method.

The purpose of this work is to analyze the nonlinear mechanical behavior of sandwich structures with a magnetorheological elastomer (MRE) core subjected to a permanent magnetic field. A detailed study is first carried out to characterize the mechanical behavior of these structures. The tests were carried out in three-point bending on beams of these complex materials for several distances between supports. An experimental study, of the mechanical behavior response realized using a Zwick 2.5 kN machine, allows to measure displacements as a function of force. The results deduced from the numerical simulation by the Abaqus software are compared with those obtained from the theoretical analysis. This study allows to show that these structures exhibit a non-linear behavior even at small deformations due to the rheological parameters which are more sensitive by the application of a magnetic field.

In this paper, an accurate technique is used to find an approximate solution to the fractional-order Duffing-Van der Pol (DVP, for short) oscillators equation which is reproducing kernel Hilbert space (RKHS, for short) method. The numerical results show that the n-term approximation is a rapidly convergent series representation and they present also the high accuracy and effectiveness of this method. The efficiency of the proposed method has been proved by the theoretical predictions and confirmed by the numerical experiments.