29 research outputs found
Economical Runge-Kutta methods
For the numerical solution of the Cauchy problem, this paper presents
an elegant idea of saving one function call for a special class of Runge-Kutt methods
by using information from the previous step. The stability analysis and practical error
estimation by embedded formulas of order (3,2), (5,4) are studied. Numerical examples
are, also, considered, which proved that these methods are in competition with the best
methods now exist
A NEW FRACTIONAL MODEL OF SINGLE DEGREE OF FREEDOM SYSTEM, BY USING GENERALIZED DIFFERENTIAL TRANSFORM METHOD
Generalized differential transform method (GDTM) is a powerful method to solve the fractional differential equations. In this paper, a new fractional model for systems with single degree of freedom (SDOF) is presented, by using the GDTM. The advantage of this method compared with some other numerical methods has been
shown. The analysis of new approximations, damping and acceleration of systems are also described. Finally, by reducing damping and analysis of the errors, in one of the fractional cases, we have shown that in addition to having a suitable solution for the
displacement close to the exact one, the system enjoys acceleration once crossing the equilibrium point
A High Accurate Approximation for a Galactic Newtonian Nonlinear Model Validated by Employing Observational Data
This article proposes Perturbation Method (PM) to solve nonlinear problems. As case study PM is employed to provide a detailed study of a nonlinear galactic model. Our approach is rather elementary and seeks to explain as much detail as possible the material of this work.In particular our solution gives rise qualitatively, to the known flat rotation curves. In fact, we compare the numerical solution and the obtained approximation by employing observational data proving the validity and high accuracy of the model under study
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-Padé 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
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
Power Series Extender Method for the Solution of Nonlinear Differential Equations
We propose a power series extender method to obtain approximate solutions
of nonlinear differential equations. In order to assess the benefits of this proposal, three nonlinear
problems of different kind are solved and compared against the power series solution obtained using
an approximative method. The problems are homogeneous Lane-Emden equation of α index, governing
equation of a burning iron particle, and an explicit differential-algebraic equation related to battery
model simulations. The results show that PSEM generates highly accurate handy approximations
requiring only a few steps. The main advantage of PSEM is to extend the domain of convergence of
the power series solutions of approximative methods as Taylor series method, homotopy perturbation
method, homotopy analysis method, variational iteration method, differential transform method, and Adomian
decomposition method, among many others. From the application of PSEM, it results in handy easy
computable expressions that extend the domain of convergence of high order power series solutions