A symmetric version of the Euler equations by using Generalized Bernoulli Method.

The aim of this article is to show a way to extend the usefulness of the Generalized Bernoulli Method (GBM) with the purpose to apply it for the case of variational problems with functionals that depend explicitly of all the variables. Moreover, after expressing the Euler equations in terms of this extension of GBM, we will see that the resulting equations acquire a symmetric form, which is not shared by the known Euler equations. We will see that this symmetry is useful because it allows us to recall these equations with ease. The presentation of three examples shows that by applying GBM, the Euler equations are obtained just as well as it does the known Euler formalism but with much less effort, which makes GBM ideal for practical applications. In fact, given a variational problem, GBM establishes the corresponding Euler equations by means of a systematic procedure, which is easy to recall, based in both elementary calculus and algebra without having to memorize the known formulas. Finally, in order to extend the practical applications of the proposed method, this work will employ GBM with the purpose to apply it for the case of solving isoperimetric problems.

Stress-Charge Nonlinear Physical Description and Tensor Symmetries for Piezoelectric Materials.

Nonlinear piezoelectric materials are raised as a great replacement for devices that require low power consumption, high sensitivity, and accurate transduction, fitting with the demanding requirements of new technologies such as the Fifth-Generation of telecommunications (5G), the Internet of Things (IoT), and modern radio frequency (RF) applications. In this work, the state equations that correctly predict the nonlinear piezoelectric phenomena observed experimentally are presented. Furthermore, we developed a fast methodology to implement the state equations in the main FEM simulation software, allowing an easy design and characterization of this type of device, as the symmetry structures for high-order tensors are shown and explained. The operation regime of each high-order tensor is discussed and connected with the main nonlinear phenomena reported in the literature. Finally, to demonstrate our theoretical deductions, we used the experimental measurements, which presented the nonlinear effects, which were reproduced through simulations, obtaining maximum percent errors for the effective elasticity constants, relative effective permittivity, and resonance frequencies of 0.79%, 2.9%, and 0.3%, respectively, giving a proof of the potential of the nonlinear state equations presented for the unifying of all nonlinear phenomena observed in the piezoelectric devices.

A practical proposal to obtain solutions of certain variational problems avoiding Euler formalism.

The aim of this article is to show the way to get both, exact and analytical approximate solutions for certain variational problems with moving boundaries but without resorting to Euler formalism at all, for which we propose two methods: the Moving Boundary Conditions Without Employing Transversality Conditions (MWTC) and the Moving Boundary Condition Employing Transversality Conditions (METC). It is worthwhile to mention that the first of them avoids the concept of transversality condition, which is basic for this kind of problems, from the point of view of the known Euler formalism. While it is true that the second method will utilize the above mentioned conditions, it will do through a systematic elementary procedure, easy to apply and recall; in addition, it will be seen that the Generalized Bernoulli Method (GBM) will turn out to be a fundamental tool in order to achieve these objectives.

Speed-up hyperspheres homotopic path tracking algorithm for PWL circuits simulations.

In the present work, we introduce an improved version of the hyperspheres path tracking method adapted for piecewise linear (PWL) circuits. This enhanced version takes advantage of the PWL characteristics from the homotopic curve, achieving faster path tracking and improving the performance of the homotopy continuation method (HCM). Faster computing time allows the study of complex circuits with higher complexity; the proposed method also decrease, significantly, the probability of having a diverging problem when using the Newton-Raphson method because it is applied just twice per linear region on the homotopic path. Equilibrium equations of the studied circuits are obtained applying the modified nodal analysis; this method allows to propose an algorithm for nonlinear circuit analysis. Besides, a starting point criteria is proposed to obtain better performance of the HCM and a technique for avoiding the reversion phenomenon is also proposed. To prove the efficiency of the path tracking method, several cases study with bipolar (BJT) and CMOS transistors are provided. Simulation results show that the proposed approach can be up to twelve times faster than the original path tracking method and also helps to avoid several reversion cases that appears when original hyperspheres path tracking scheme was employed.

Laplace transform homotopy perturbation method for the approximation of variational problems.

This article proposes the application of Laplace Transform-Homotopy Perturbation Method and some of its modifications in order to find analytical approximate solutions for the linear and nonlinear differential equations which arise from some variational problems. As case study we will solve four ordinary differential equations, and we will show that the proposed solutions have good accuracy, even we will obtain an exact solution. In the sequel, we will see that the square residual error for the approximate solutions, belongs to the interval [0.001918936920, 0.06334882582], which confirms the accuracy of the proposed methods, taking into account the complexity and difficulty of variational problems.

Modified hyperspheres algorithm to trace homotopy curves of nonlinear circuits composed by piecewise linear modelled devices.

We present a homotopy continuation method (HCM) for finding multiple operating points of nonlinear circuits composed of devices modelled by using piecewise linear (PWL) representations. We propose an adaptation of the modified spheres path tracking algorithm to trace the homotopy trajectories of PWL circuits. In order to assess the benefits of this proposal, four nonlinear circuits composed of piecewise linear modelled devices are analysed to determine their multiple operating points. The results show that HCM can find multiple solutions within a single homotopy trajectory. Furthermore, we take advantage of the fact that homotopy trajectories are PWL curves meant to replace the multidimensional interpolation and fine tuning stages of the path tracking algorithm with a simple and highly accurate procedure based on the parametric straight line equation.

Approximate Solutions for Flow with a Stretching Boundary due to Partial Slip.

The homotopy perturbation method (HPM) is coupled with versions of Laplace-Padé and Padé methods to provide an approximate solution to the nonlinear differential equation that describes the behaviour of a flow with a stretching flat boundary due to partial slip. Comparing results between approximate and numerical solutions, we concluded that our results are capable of providing an accurate solution and are extremely efficient.