Abstract
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.
Highlights
Solving nonlinear differential equations is an important issue in sciences because many physical phenomena are modelled using such classes of equations
In order to improve accuracy, we propose the Laplace-Paderational biparameter homotopy perturbation method LPRBHPM, when the solution is expressed as the quotient of two truncated power series
The HPM method is based on the use of a power series, which transforms the original nonlinear differential equation into a series of linear differential equations
Summary
Solving nonlinear differential equations is an important issue in sciences because many physical phenomena are modelled using such classes of equations. One of the most powerful methods to approximately solve nonlinear differential equations is the homotopy perturbation method HPM 1–45. The HPM method is based on the use of a power series, which transforms the original nonlinear differential equation into a series of linear differential equations. Like HPM, the use of this quotient transforms the nonlinear differential equation into a series of linear differential equations. We propose an after-treatment to the approximate solutions with the Laplace-Pade LP transform in order to improve the accuracy of the solutions. This coupled method will be denominated as the LPRBHPM.
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