Journal ArticleDOI
Series solution of nonlinear eigenvalue problems by means of the homotopy analysis method
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TLDR
In this article, a general analytic approach for nonlinear eigenvalue problems is described and two physical problems are used as examples to show the validity of this approach for eigen value problems with either periodic or non-periodic eigenfunctions.Abstract:
A general analytic approach for nonlinear eigenvalue problems is described. Two physical problems are used as examples to show the validity of this approach for eigenvalue problems with either periodic or non-periodic eigenfunctions. Unlike perturbation techniques, this approach is independent of any small physical parameters. Besides, different from all other analytic techniques, it provides a simple way to ensure the convergence of series of eigenvalues and eigenfunctions so that one can always get accurate enough approximations. Finally, unlike all other analytic techniques, this approach provides great freedom to choose an auxiliary linear operator so as to approximate the eigenfunction more effectively by means of better base functions. This approach provides us a new way to investigate eigenvalue problems with strong nonlinearity.read more
Citations
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Journal ArticleDOI
A new application of the homotopy analysis method: Solving the Sturm–Liouville problems
Saeid Abbasbandy,Ahmad Shirzadi +1 more
TL;DR: In this article, the homotopy analysis method (HAM) is applied to numerically approximate the eigenvalues of the second and fourth-order Sturm-Liouville problems.
Journal ArticleDOI
Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems
Saeid Abbasbandy,Ahmad Shirzadi +1 more
TL;DR: It can be seen in this paper that the auxiliary parameter $\hbar,$ which controls the convergence of the HAM approximate series solutions, has another important application, predicting and calculating multiple solutions.
Journal ArticleDOI
An efficient method for quadratic Riccati differential equation
TL;DR: In this paper, a new form of homotopy perturbation method (NHPM) has been adopted for solving the quadratic Riccati differential equation, where the solution is considered as a Taylor series expansion converges rapidly to the exact solution of the nonlinear equation.
Journal ArticleDOI
A note on the homotopy analysis method
TL;DR: A theorem is proved here which generalizes some lemmas and theorems provided in Liao (2009) and Molabahrami and Khani (2007) and which significant applicability of the theorem obtained here in some practical situations is demonstrated.
Journal ArticleDOI
A study on the convergence of homotopy analysis method
TL;DR: An alternative framework of the homotopy analysis method is presented which can be used simply and effectively to handle nonlinear problems and the sufficient condition for convergence is addressed, showing the power of the method.
References
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Book
Perturbation Methods
Ali H. Nayfeh,Vimal Singh +1 more
TL;DR: This website becomes a very available place to look for countless perturbation methods sources and sources about the books from countries in the world are provided.
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Introduction to perturbation techniques
TL;DR: In this paper, the authors introduce the notion of forced Oscillations of the Duffing Equation and the Mathieu Equation for weakly nonlinear systems with quadratic and cubic nonlinearities.
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Beyond Perturbation: Introduction to the Homotopy Analysis Method
Shijun Liao,SA Sherif +1 more
TL;DR: In this paper, a simple bifurcation of a nonlinear problem multiple solutions of a Nonlinear Problem Nonlinear Eigenvalue Problem Thomas-Fermi Atom Model Volterra's Population Model Free Oscillation Systems with Odd Nonlinearity Free oscillations with Quadratic nonlinearity Limit Cycle in a Multidimensional System Blasius' viscous flow Boundary-layer Flow Boundarylayer Flow with Exponential Property Boundary Layer Flow with Algebraic Property Von Karman Swirling Flow Nonlinear Progressive Waves in Deep Water BIBLIOGR