Luis Hernandez-Martinez

Modified HPMs Inspired by Homotopy Continuation Methods

Nonlinear differential equations have applications in the modelling area for a broad variety of phenomena and physical processes; having applications for all areas in science and engineering. At the present time, the homotopy perturbation method (HPM) is amply used to solve in an approximate or exact manner such nonlinear differential equations. This method has found wide acceptance for its versatility and ease of use. The origin of the HPM is found in the coupling of homotopy methods with perturbation methods. Homotopy methods are a well established research area with applications, in particular, an applied branch of such methods are the homotopy continuation methods, which are employed on the numerical solution of nonlinear algebraic equation systems. Therefore, this paper presents two modified versions of standard HPM method inspired in homotopy continuation methods. Both modified HPMs deal with nonlinearities distribution of the nonlinear differential equation. Besides, we will use a calcium-induced calcium released mechanism model as study case to test the proposed techniques. Finally, results will be discussed and possible research lines will be proposed using this work as a starting point.

Homotopy method with a formal stop criterion applied to circuit simulation

The continuous scaling for fabrication technologies of electronic circuits demands the design of new and improved simulation techniques for integrated circuits. Therefore, this work shows a new double bounded polynomial homotopy based on a polynomial formulation with four solution lines separated by a fixed distance. The new homotopy scheme presents a bounding between the two internal solution lines and the symmetry axis, which allows to establish a stop criterion for the simulation in DC. Besides, the initial and final points on this new double bounded homotopy can be set arbitrarily. Finally, mathematical properties for the new homotopy are introduced and exemplified using a benchmark circuit.

Powering Multiparameter Homotopy-Based Simulation with a Fast Path-Following Technique

The continuous scaling for fabrication technologies of electronic circuits demands the design of new and improved simulation techniques for integrated circuits. Therefore, this work shows how the hypersphere technique can be adapted and applied to trace a multiparameter homotopy. Besides, we present a path-following technique based on circles (evolved from hypersphere), which is faster, and simpler to be implemented than hypersphere technique. Last, a comparative analysis between both techniques applied to simulation of circuits with bipolar transistors will be shown.

Removal of Noise Oscillation Term Appearing in the Nonlinear Equation Solution

This paper suggests a novel modified Laplace method for removal of noise oscillation term appearing in the nonlinear equation solutions. The modified method overcomes the noise oscillation during the iteration procedure by suitable choice of an initial solution. Several examples are tested, and the obtained results suggest that this newly developed technique could lead to a promising tool and powerful improvement for many applications in differential and integral equations.

Variational iteration algorithm-II for solving linear and non-linear ODEs

In recent years, much attention has been paid to the application of the Variational iteration method (He, 1999) to various problems due to its simplicity. The variational iteration method was proposed first by the Chinese researcher He (2006) and was further developed by him (He, 2006, 2008, 2012; He and Wu, 2007). Recently, Wazwaz (2009) applied a universal variational iteration algorithm for linear and non-linear ODEs which caught an immediate attention in the mathematics community. The highly entertaining work, related to fractional calculus, shows that the variational iteration method (He, 2011) is a highly useful mathematical tool for discovery of real-life problems. This method has wider application because it reduces the size of computation and is a very powerful mathematical tool for various kinds of linear and nonlinear problems arising in differential equations/ fractional differential equations in a recent publication by He et al. (2010). “The variational iteration method should be followed”, for the problem to be completely eliminated, and a new iteration algorithm was suggested, and the algorithm was termed as the variational iteration algorithm-II (He, 2007; Faraz et al., 2010; Wu and Li, 2011). It is an alternative approach to linear and nonlinear differential equations using the variational iteration

Modified Differential Transform Method for Solving the Model of Pollution for a System of Lakes

This work presents the application of the differential transform method (DTM) to the model of pollution for a system of three lakes interconnected by channels. Three input models (periodic, exponentially decaying, and linear) are solved to show that DTM can provide analytical solutions of pollution model in convergent series form. In addition, we present the posttreatment of the power series solutions with the Laplace-Pade resummation method as a useful strategy to extend the domain of convergence of the approximate solutions. The Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant (RKF45) numerical solution of the lakes system problem is used as a reference to compare with the analytical approximations showing the high accuracy of the results. The main advantage of the proposed technique is that it is based on a few straightforward steps and does not generate secular terms or depend of a perturbation parameter.

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.

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.

Applying an iterative-decomposed piecewise-linear model to find multiple operating points

This paper presents a new methodology for obtaining all the operating points in piecewise-linear (PWL) electrical networks. The networks can include independent and controlled voltage or current sources, resistors or conductances, and PWL elements. The nonlinear behavior of the PWL elements is graphically described by one-dimensional PWL curves. The mathematical representation of every PWL curve is given by an iterative and decomposed PWL model. The model is denominated as iterative because the representation of the segments belonging to the PWL curve depends on the value of one parameter included in the formulation. It is also denoted as decomposed because the two axis variables (x and y), in the PWL curve, are included separately in a system of linear equations. In order to optimize the process of searching all the operating points, the methodology is aided by a graphical procedure for identifying such constitutive element segments involved into the existence of any DC solution. By a numerical example here presented, it is possible to confirm the efficiency of the methodology.

Path optimization for terrestrial robots using Homotopy Path Planning Method

The path planning method is an important element in the architecture of all mobile robots with certain degree of autonomy. An appropriate path planning method provides the robot the ability to move in a previously known environment while avoiding collisions with obstacles along the way. Additionally, this method should generate optimal trajectories from one point to another. In this work the optimal path is defined as the shortest one. Path planning is not an easy task, some algorithms and methods have been developed for solving this problem, however, its success is not guaranteed. On the other hand, Homotopy Path Planning Method (HPPM) is a new tool used to find collision-free paths, this takes the properties of the Homotopy Continuation Methods (HCM) to find a successful path. However, the path found by this method is not the optimal. In this work, a path planning method for a terrestrial mobile robot based on HPPM and Spherical Algorithm (SA) is presented. Furthermore, a new strategy able to obtain the optimal path is proposed. Finally, simulation results for several environment maps with hundreds of circular obstacles are shown.