9 research outputs found
HPM Approximations for Trajectories: From a Golf Ball Path to Mercury’s Orbit
In this work, we propose the approximated analytical solutions for two highly nonlinear problems using the homotopy perturbation method (HPM). We obtained approximations for a golf ball trajectory model and a Mercury orbit’s model. In addition, to enlarge the domain of convergence of the first case study, we apply the Laplace-Padé resummation method to the HPM series solution. For both case studies, we were able to obtain approximations in good agreement with numerical methods, depicting the basic nature of the trajectories of the phenomena
Classical Perturbation Method for the Solution of a Model of Diffusion and Reaction
In this paper, we employ perturbation method (PM) to solve nonlinear problems. As case study PM is employed to obtain approximate solutions for the nonlinear differential equation that models the diffusion and reaction in porous catalysts. We find that the square residual error (S.R.E) of our solutions is in the range and this requires only the third order approximation of PM, which shows the effectiveness of the method
Fixed-Term Homotopy
A new tool for the solution of nonlinear differential equations is presented. The Fixed-Term Homotopy (FTH) delivers a high precision representation of the nonlinear differential equation using only a few linear algebraic terms. In addition to this tool, a procedure based on Laplace-Padé to deal with the truncate power series resulting from the FTH method is also proposed. In order to assess the benefits of this proposal, two nonlinear problems are solved and compared against other semianalytic methods. The obtained results show that FTH is a power tool capable of generating highly accurate solutions compared with other methods of literature
Analytical Solution of a Nonlinear Index-Three DAEs System Modelling a Slider-Crank Mechanism
The slider-crank mechanism (SCM) is one of the most
important mechanisms in modern technology. It appears in most combustion
engines including those of automobiles, trucks, and other small engines. The
SCM model considered here is an index-three nonlinear system of
differential-algebraic equations (DAEs), and therefore difficult to integrate
numerically. In this work, we present the application of the differential
transform method (DTM) to obtain an approximate analytical solution of the
SCM model in convergent series form. In addition, we propose a
posttreatment of the power series solution with the Padé resummation
method to extend the domain of convergence of the approximate series
solution. The main advantage of the proposed technique is that it does not
require an index reduction and does not generate secular terms or depend on
a perturbation parameter
Rational Biparameter Homotopy Perturbation Method and Laplace-Padé Coupled Version
The fact that most of the physical phenomena are modelled by nonlinear differential equations
underlines the importance of having reliable methods for solving them. This work presents the rational
biparameter homotopy perturbation method (RBHPM) as a novel tool with the potential to find approximate
solutions for nonlinear differential equations. The method generates the solutions in the form of a quotient
of two power series of different homotopy parameters. Besides, in order to improve accuracy, we propose
the Laplace-Padé rational biparameter homotopy perturbation method (LPRBHPM), when the solution is
expressed as the quotient of two truncated power series. The usage of the method is illustrated with two
case studies. On one side, a Ricatti nonlinear differential equation is solved and a comparison with the
homotopy perturbation method (HPM) is presented. On the other side, a nonforced Van der Pol Oscillator
is analysed and we compare results obtained with RBHPM, LPRBHPM, and HPM in order to conclude
that the LPRBHPM and RBHPM methods generate the most accurate approximated solutions