R. Castaneda-Sheissa

High Accurate Simple Approximation of Normal Distribution Integral

The integral of the standard normal distribution function is an integral without solution and represents the probability that an aleatory variable normally distributed has values between zero and 𝑥. The normal distribution integral is used in several areas of science. Thus, this work provides an approximate solution to the Gaussian distribution integral by using the homotopy perturbation method (HPM). After solving the Gaussian integral by HPM, the result served as base to solve other integrals like error function and the cumulative distribution function. The error function is compared against other reported approximations showing advantages like less relative error or less mathematical complexity. Besides, some integrals related to the normal (Gaussian) distribution integral were solved showing a relative error quite small. Also, the utility for the proposed approximations is verified applying them to a couple of heat flow examples. Last, a brief discussion is presented about the way an electronic circuit could be created to implement the approximate error function.

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.

Numerical continuation scheme for tracing the double bounded homotopy for analysing nonlinear circuits

A numerical continuation for tracing the double bounded homotopy (DBH) for obtaining DC solutions of nonlinear circuits is proposed. The double bounded homotopy is used to find multiple DC solutions with the advantage of having a stop criterion which is based on the property of having a double bounded trajectory. The key aspects of the implementation of the numerical continuation are presented in this paper. Besides, in order to trace and apply the stop criterion some blocks of the numerical continuation are modified and explained.

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.

Biparameter Homotopy-based Direct Current Simulation of Multistable Circuits

The microelectronics area constantly demands better and improved circuit simulation tools. This is the reason that this article is to present a biparameter homotopy with automated stop criterion, which is applied to direct current simulation of multistable circuits. This homotopy possesses the following characteristics: symmetry axis, double bounding solution line, arbitrary initial and final points, and lessen the nonlinearities that exist in the circuit. Besides, this method will be exemplified and discussed by using a benchmark multistable circuit.

Exploring collision-free path planning by using homotopy continuation methods

Autonomous and semi-autonomous robots play significant roles in space and terrestrial exploration, even more in unfavorable and dangerous environments. Although recent advances allow robots to evolve in many such environments, one of the most important problems remains the establishment of collision-free trajectories in static or partially (temporal) static environments. This paper presents a different approach to address this problem, proposing a methodology based on homotopy continuation methods (HCM) capable of generating collision-free trajectories in two and three dimensions. The basic idea behind the proposal relies on the construction of a nonlinear equation representing the map of the environment, making it possible to apply HCM methods to obtain collision-free paths. A series of simulations are presented to show the effectiveness of the method avoiding circular, semi-rectangular, spherical shaped and semi-parallelepipeds obstacles.

Transient and DC approximate expressions for diode circuits

In general terms, it is not possible to establish symbolic explicit analytic expressions of the operating point and transient analysis for circuits containing diodes modelled using an exponential function. Therefore, this work propose replacing the diode for an equivalent circuit obtained by using a power series and a Taylor series consecutively. Finally, we present a symbolic solution for some circuits that include diodes; resulting for the best case: for DC analysis a relative error of 1E-11 and for transient analysis a relative error ≤ 5E-4.

Approximations for Large Deflection of a Cantilever Beam under a Terminal Follower Force and Nonlinear Pendulum

In theoretical mechanics field, solution methods for nonlinear differential equations are very important because many problems are modelled using such equations. In particular, large deflection of a cantilever beam under a terminal follower force and nonlinear pendulum problem can be described by the same nonlinear differential equation. Therefore, in this work, we propose some approximate solutions for both problems using nonlinearities distribution homotopy perturbation method, homotopy perturbation method, and combinations with Laplace-Pade posttreatment. We will show the high accuracy of the proposed cantilever solutions, which are in good agreement with other reported solutions. Finally, for the pendulum case, the proposed approximation was useful to predict, accurately, the period for an angle up to yielding a relative error of 0.01222747.

New Aspects of Double Bounded Polynomial Homotopy

This work presents some new aspects about the mathematical properties of the double bounded polynomial homotopy like: isolated symmetry branches equation and a study about initial and final points of the path and curvature radius. The application of the homotopy is illustrated by solving a mathematical problem and a nonlinear circuit.