D. Pereyra-Diaz

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.

Analytical Solutions for Systems of Singular Partial Differential-Algebraic Equations

This paper proposes power series method (PSM) in order to find solutions for singular partial differential-algebraic equations (SPDAEs). We will solve three examples to show that PSM method can be used to search for analytical solutions of SPDAEs. What is more, we will see that, in some cases, Pade posttreatment, besides enlarging the domain of convergence, may be employed in order to get the exact solution from the truncated series solutions of PSM.

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.

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.

A generalization of the Bernoulli's method applied to brachistochrone-like problems

In this paper we study a generalization of the Johann Bernoullis solution of the brachistocrone problem. We will see that his method can be quickly extended in such a way that it can be used to solve other problems in a similar way using just elementary calculus methods. In addition, we will show that it is not necessary to know Eulers formalism for the calculus of variations, making it a handy and useful method for engineering applications. The provided examples will illustrate that this technique is equivalent to Eulers equation of the calculus of variations; for the particular case where one of the variables do not appear explicitly.

Laplace transform homotopy perturbation method for the approximation of variational problems

Abstract 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.

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.