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**Rational biparameter homotopy perturbation method and Laplace-Padé coupled version.**
*(English)*
Zbl 1268.65094

Summary: 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 Riccati 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.

### MSC:

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |

### Keywords:

numerical examples; rational biparameter homotopy perturbation method; truncated power series; nonlinear differential equation; Van der Pol oscillator
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\textit{H. Vazquez-Leal} et al., J. Appl. Math. 2012, Article ID 923975, 21 p. (2012; Zbl 1268.65094)

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### References:

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