Abstract

Physical Phenomena’s located around us are primarily nonlinear in nature and their solutions are of highest significance for scientists and engineers. In order to have a better representation of these physical models, fractional calculus is used. Fractional order oscillation equations are included among these nonlinear phenomena’s. To tackle with the nonlinearity arising, in these phenomena’s we recommend a new method. In the proposed method, Picard’s iteration is used to convert the nonlinear fractional order oscillation equation into a fractional order recurrence relation and then Legendre wavelets method is applied on the converted problem. In order to check the efficiency and accuracy of the suggested modification, we have considered three problems namely: fractional order force-free Duffing–van der Pol oscillator, forced Duffing–van der Pol oscillator and higher order fractional Duffing equations. The obtained results are compared with the results obtained via other techniques.

Highlights

  • IntroductionImportance of fractional calculus [1,2,3] has increased a lot especially over the past few decades

  • Importance of fractional calculus [1,2,3] has increased a lot especially over the past few decades.Physical phenomena, describing fractional oscillation equations [4,5,6], are mainly nonlinear in nature.In general, exact solutions of these governing fractional oscillation equations are not available.different techniques for finding approximate analytical solutions of such problems were developed

  • A systematic technique, is employed and executed successfully to solve the emerging problems modeled from nonlinear fractional oscillation phenomena

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Summary

Introduction

Importance of fractional calculus [1,2,3] has increased a lot especially over the past few decades. It is to be highlighted that Abd-Elhameed and Youssri [22] introduced new spectral solutions of multi-term fractional order initial value problems with error analysis in the recent past. Abd-Elhameed et al [23] extended new spectral second kind Chebyshev wavelets algorithm for solving linear and nonlinear second order differential equations involving singular and. It is worth mentioning that Youssri et al and Doha et al [24,25] developed an excellent scheme which is called Ultraspherical wavelets method and applied the same on Lane–. Wavelet-Picard Method (LWPM) to solve the nonlinear fractional oscillation equations. Solutions obtained by LWPM are compared with Variational Iterational Method (VIM) using exact Lagrange multiplied and Ultraspherical Wavelets Collocation Method (UWCM) [24].

Legendre Wavelets
Applications
Conclusions
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