Abstract
In this paper, the homotopy analysis method (HAM) has been employed to obtain the approximate analytical solution of the nonlinear Harry-Dym (HD) equation, which is one of the most important soliton equations. Utilizing the HAM, thereby employing the initial approximation, variations of the 7th-order approximation of the Harry-Dym equation is obtained. It is found that effect of the nonzero auxiliary parameter on convergence rate of the series solution is undeniable. It is also shown that, to some extent, order of the fractional derivative plays a fundamental role in the prediction of convergence. The final results reported by the HAM have been compared with the exact solution as well as those obtained through the other methods.
Highlights
The nonlinear Harry-Dym equation is [1], ut = u3uxxx, (1)with the initial approximation, u ( x,0=) a − b 2 2 x (2)where a and b are constants
Some papers are devoted to compare the homotopy analysis method (HAM) with the other analytical methods e.g., optimal homotopy asymptotic method (OHAM), differential transformation method (DTM), homotopy perturbation method (HPM), general series expansion method, harmonic balance method (HBM) which can be found in Refs. [31] [32] [33] [34] [35]
It has been shown that Equation (1), which can be transformed to the Korteweg-de Vries (KdV) equation, is an integrable nonlinear partial differential equations (NPDEs)
Summary
Department of Mechanical Engineering, College of Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran How to cite this paper: Khoshrouye Ghiasi, E. and Saleh, R. (2017) A Mathematical Approach Based on the Homotopy Analysis Method: Application to Solve the Nonlinear Harry-Dym (HD) Equation. Applied Mathematics, 8, 1546-1562. https://doi.org/10.4236/am.2017.811113 Received: October 3, 2017 Accepted: November 10, 2017 Published: November 13, 2017
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
