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 novel family of transcendental Leal-functions with applications to science and engineering.

During the phenomena modelling process in the different areas of science and engineering is common to face nonlinear equations without exact solutions; thus, the need of employing numerical methods to obtain such solutions. Therefore, in order to provide new possibilities for the isolation of variables, we propose a novel family of transcendental functions with new algebraic properties including their integration and differentiation rules. Likewise, in order facilitate the numerical evaluation for every new family set of functions, a highly accurate series of approximations is proposed by employing analytical expressions in terms of known transcendental functions and polynomials combinations.

The novel Leal-polynomials for the multi-expansive approximation of nonlinear differential equations.

This work presents the novel Leal-polynomials (LP) for the approximation of nonlinear differential equations of different kind. The main characteristic of LPs is that they satisfy multiple expansion points and its derivatives as a mechanism to replicate behaviour of the nonlinear problem, giving more accuracy within the region of interest. Therefore, the main contribution of this work is that LP satisfies the successive derivatives in some specific points, resulting more accurate polynomials than Taylor expansion does for the same degree of their respective polynomials.

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.

Transforming the canonical piecewise-linear model into a smooth-piecewise representation.

A smoothed representation (based on natural exponential and logarithmic functions) for the canonical piecewise-linear model, is presented. The result is a completely differentiable formulation that exhibits interesting properties, like preserving the parameters of the original piecewise-linear model in such a way that they can be directly inherited to the smooth model in order to determine their parameters, the capability of controlling not only the smoothness grade, but also the approximation accuracy at specific breakpoint locations, a lower or equal overshooting for high order derivatives in comparison with other approaches, and the additional advantage of being expressed in a reduced mathematical form with only two types of inverse functions (logarithmic and exponential). By numerical simulation examples, this proposal is verified and well-illustrated.

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.

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.

Nonlinearities distribution Laplace transform-homotopy perturbation method.

This article proposes non-linearities distribution Laplace transform-homotopy perturbation method (NDLT-HPM) to find approximate solutions for linear and nonlinear differential equations with finite boundary conditions. We will see that the method is particularly relevant in case of equations with nonhomogeneous non-polynomial terms. Comparing figures between approximate and exact solutions we show the effectiveness of the proposed method.

Direct application of Padé approximant for solving nonlinear differential equations.

Unlabelled: This work presents a direct procedure to apply Padé method to find approximate solutions for nonlinear differential equations. Moreover, we present some cases study showing the strength of the method to generate highly accurate rational approximate solutions compared to other semi-analytical methods. The type of tested nonlinear equations are: a highly nonlinear boundary value problem, a differential-algebraic oscillator problem, and an asymptotic problem.

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.

A handy approximate solution for a squeezing flow between two infinite plates by using of Laplace transform-homotopy perturbation method.

This article proposes Laplace Transform Homotopy Perturbation Method (LT-HPM) to find an approximate solution for the problem of an axisymmetric Newtonian fluid squeezed between two large parallel plates. After comparing figures between approximate and exact solutions, we will see that the proposed solutions besides of handy, are highly accurate and therefore LT-HPM is extremely efficient.

Modified Taylor series method for solving nonlinear differential equations with mixed boundary conditions defined on finite intervals.

Abstract: In this article, we propose the application of a modified Taylor series method (MTSM) for the approximation of nonlinear problems described on finite intervals. The issue of Taylor series method with mixed boundary conditions is circumvented using shooting constants and extra derivatives of the problem. In order to show the benefits of this proposal, three different kinds of problems are solved: three-point boundary valued problem (BVP) of third-order with a hyperbolic sine nonlinearity, two-point BVP for a second-order nonlinear differential equation with an exponential nonlinearity, and a two-point BVP for a third-order nonlinear differential equation with a radical nonlinearity.

A handy approximation for a mediated bioelectrocatalysis process, related to Michaelis-Menten equation.

In this article, Perturbation Method (PM) is employed to obtain a handy approximate solution to the steady state nonlinear reaction diffusion equation containing a nonlinear term related to Michaelis-Menten of the enzymatic reaction. Comparing graphics between the approximate and exact solutions, it will be shown that the PM method is quite efficient.

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.