Numerical solution of full fractional Duffing equations with Cubic-Quintic-Heptic nonlinearities

P. Pirmohabbati; A. H. Refahi Sheikhani; H. Saberi Najafi; A. Abdolahzadeh Ziabari; P. Pirmohabbati; A. H. Refahi Sheikhani; H. Saberi Najafi; A. Abdolahzadeh Ziabari

doi:10.3934/math.2020110

AIMS Mathematics

2020, Volume 5, Issue 2: 1621-1641. doi: 10.3934/math.2020110

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Research article

Numerical solution of full fractional Duffing equations with Cubic-Quintic-Heptic nonlinearities

1.
Department of Applied Mathematics, Faculty of Mathematical Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran
2.
Department of physics, Faculty of Science, Lahijan Branch, Islamic Azad University, Lahijan, Iran

Received: 21 September 2019 Accepted: 19 December 2019 Published: 06 February 2020
MSC : 26A33, 97N40, 34A08, 34C15

In this article, based on the operational matrix of fractional order integration, we introduce a method for the numerical solution of fractional strongly nonlinear Duffing oscillators with cubic-quintic-heptic nonlinear restoring force and then use it in some cases. For this purpose, concerning the Caputo sense, we implement the block-pulse wavelets matrix of fractional order integration. To reach this aim, we analyse the errors. The approach has been examined by some numerical examples and changes in coefficients as well as in the derivative of the equation too. It is shown that this method works well for all the parameters and order of the fractional derivative. Results indicate the precision and computational performance of the suggested algorithm.
- Duffing differential equation,
- block-pulse functions,
- fractional order,
- cubic-quintic-heptic,
- nonlinear oscillator
Citation: P. Pirmohabbati, A. H. Refahi Sheikhani, H. Saberi Najafi, A. Abdolahzadeh Ziabari. Numerical solution of full fractional Duffing equations with Cubic-Quintic-Heptic nonlinearities[J]. AIMS Mathematics, 2020, 5(2): 1621-1641. doi: 10.3934/math.2020110

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Abstract

In this article, based on the operational matrix of fractional order integration, we introduce a method for the numerical solution of fractional strongly nonlinear Duffing oscillators with cubic-quintic-heptic nonlinear restoring force and then use it in some cases. For this purpose, concerning the Caputo sense, we implement the block-pulse wavelets matrix of fractional order integration. To reach this aim, we analyse the errors. The approach has been examined by some numerical examples and changes in coefficients as well as in the derivative of the equation too. It is shown that this method works well for all the parameters and order of the fractional derivative. Results indicate the precision and computational performance of the suggested algorithm.

References

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