Exploring the Novel Continuum-Cancellation Leal-Method for the Approximate Solution of Nonlinear Differential Equations

Hector Vazquez-Leal

doi:10.1155/2020/4967219

Abstract

This work presents the novel continuum-cancellation Leal-method (CCLM) for the approximation of nonlinear differential equations. CCLM obtains accurate approximate analytical solutions resorting to a process that involves the continuum cancellation (CC) of the residual error of multiple selected points; such CC process occurs during the successive derivatives of the differential equation resulting in an accuracy increase of the inner region of the CC-points and, thus, extends the domain of convergence and accuracy. Users of CCLM can propose their own trial functions to construct the approximation as long as they are continuous in the CC-points, that is, it can be polynomials, exponentials, and rational polynomials, among others. In addition, we show how the process to obtain the approximations is straightforward and simple to achieve and capable to generate compact, and easy, computable expressions. A convergence control is proposed with the aim to establish a solid scheme to obtain optimal CCLM approximations. Furthermore, we present the application of CCLM in several examples: Thomas–Fermi singular equation for the neutral atom, magnetohydrodynamic flow of blood in a porous channel singular boundary-valued problem, and a system of initial condition differential equations to model the dynamics of cocaine consumption in Spain. We present a computational convergence study for the proposed approximations resulting in a tendency of the RMS error to zero as the approximation order increases for all case studies. In addition, a computation time analysis (using Fortran) for the proposed approximations presents average times from 3.5 nanoseconds to 7 nanoseconds for all the case studies. Thence, CCML approximations can be used for intensive computing simulations.

Highlights

All kinds of nonlinear differential equations arise when modelling different problems without knowing an exact solution
We can mention some recent methods: (i) e power series extender method [5,6,7] is used to construct analytical approximate solutions using a scheme of matching among the Taylor series of a trial function (TF) and the Taylor series of the Discrete Dynamics in Nature and Society exact solution. is method extends the power of traditional Taylor series providing approximations with a large convergence domain, resorting to a process of resummation of the Taylor series into a compact expression (TF)
We propose the straightforward novel continuumcancellation Leal-method (CCLM) for the approximation of nonlinear differential equations

Summary

Introduction

All kinds of nonlinear differential equations arise when modelling different problems without knowing an exact solution. It is important to develop methods that provide, at least, analytical approximate solutions for nonlinear ODEs of the problem to solve. Another way to study the behaviour of nonlinear ODEs is through the application of numerical methods. (i) e power series extender method [5,6,7] is used to construct analytical approximate solutions using a scheme of matching among the Taylor series of a trial function (TF) and the Taylor series of the Discrete Dynamics in Nature and Society exact solution (obtained from the differential equation). We propose the straightforward novel continuumcancellation Leal-method (CCLM) for the approximation of nonlinear differential equations.

The Basic Procedure of CCLM

Study Cases

Numerical Simulation and Discussion

Conclusions